Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Size: px
Start display at page:

Download "Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam"

Transcription

1 Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam

2 Contents 1. Introduction Hypothesis Testing Hypothesis Tests Concerning the Mean Hypothesis Tests Concerning Variance Other Issues: Nonparametric Inference Summary Next Steps This document should be read in conjunction with the corresponding reading in the 2014 Level I CFA Program curriculum. Some of the graphs, charts, tables, examples, and figures are copyright 2013, CFA Institute. Reproduced and republished with permission from CFA Institute. All rights reserved. Required disclaimer: CFA Institute does not endorse, promote, or warrant the accuracy or quality of the products or services offered by Irfanullah Financial Training. CFA Institute, CFA, and Chartered Financial Analyst are trademarks owned by CFA Institute. Copyright Irfanullah Financial Training. All rights reserved. Page 2

3 1. Introduction Analysts come across many statements and ideas about how financial markets work. In this reading, we will discuss how analysts can decide whether these statements are true or false using hypothesis testing. If we can reduce an idea to a definite statement about the value of a quantity, such as the population mean, the idea becomes a statistically testable statement of hypothesis. This hypothesis can then be tested through hypothesis testing tools. Hypothesis testing is the process of making judgments about a larger group (a population) on the basis of a smaller group actually observed (a sample). The results of such a test then help us evaluate whether our hypothesis or assertion is true or false. 2. Hypothesis Testing A hypothesis is defined as a statement about one or more populations. In order to test a hypothesis, we follow the following steps: (i) Stating the hypothesis (ii) Identifying the appropriate test statistic and its probability distribution (iii) Specifying the significance level (iv) Stating the decision rule (v) Collecting the data and calculating the test statistic (vi) Making the statistical decision (vii) Making the economic or investment decision We will go through the steps of hypothesis testing using an example to illustrate the process. Suppose you are a researcher and believe that the average return on all Asian stocks was greater than 2%. In this case, you are making a statement about the population mean (µ) of all Asian stocks. The first step is stating the hypothesis. We always state two hypotheses the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis is the hypothesis to be tested. This Copyright Irfanullah Financial Training. All rights reserved. Page 3

4 is usually the hypothesis that the researcher wants to reject. If this hypothesis is false, we are led to the alternative hypothesis. The alternative hypothesis is the hypothesis accepted when the null hypothesis is rejected. This is usually the hypothesis we are trying to evaluate. An easy way to differentiate among the two is to remember that the null hypothesis always contains some form of the equal sign. Continuing with our example, you believe that the mean (µ) of Asian stocks is greater than 2%. Therefore our null and alternative hypotheses are: H0: µ 2 (Null) Ha: µ > 2 (Alternate) The value 2% is known as µ0. The next step in hypothesis testing is identifying the appropriate test statistic and its probability distribution. A test statistic is a quantity, calculated based on a sample, whose value is the basis for deciding whether or not to reject the null hypothesis. The formula for a test statistic is: Test statistic = Sample statistic Value of the parameter under H 0 Standard error of the sample statistic Continuing with our example, let us further suppose that the sample mean of 36 observations of Asian stocks is 4 and the standard deviation of the population is 4. In this case, our sample statistic will be 4, the value of the parameter under H0 will be 2 and the standard error of the sample statistic will be Recall that the standard error is calculated using the following formula: σ x = σ n In this example, the standard error is calculated as: σ x = 4 36 = 4 6 = 0.67 Therefore, our test statistic is calculated as: Copyright Irfanullah Financial Training. All rights reserved. Page 4

5 Test statistic= = 3 We now need to determine what probability distribution this test statistic follows. There are generally four distributions for test statistics: The t-distribution The z-distribution The chi-square (χ 2 ) distribution The F-distribution In our example, we can conduct a z-test based on the central limit theorem because our sample has many observations. Hence, we can assume that it follows the standard normal distribution. The third step is specifying the significance level. When the test statistic has been calculated, we can either reject the null hypothesis or fail to reject the null hypothesis. This decision is based on comparing the calculated value of the test statistic to a specified possible value(s). The specified possible value(s) that we compare to the test statistic are based on the level of significance. Continuing with our example of Asian stocks, suppose we want to test our hypothesis at the 5% significance level, so the value that corresponds to the significance level is Graphically, this is shown below: This is a one-tailed test because we are trying to assess whether the population mean is greater than 2% or not. Hence, we are only interested in the right tail of the distribution. If we were trying to assess whether the population mean is less than 2% we would have been interested in the left tail and the value corresponding to the significance level would have been Copyright Irfanullah Financial Training. All rights reserved. Page 5

6 The fourth step is stating the decision rule. In order to test the null hypothesis, we compare the test statistic with the critical value calculated in the step before that relates to the desired level of significance. If we find that the calculated value of the test statistic is greater than the critical value, we reject the null hypothesis. This means that the result is statistically significant. If the test statistic is less than the critical value, we fail to reject the null hypothesis. This means that the result is not statistically significant. The critical value is also known as the rejection point for the test statistic. Graphically, this point separates the acceptance and rejection regions for a set of values of the test statistic. This is shown below: The region to the left of the test statistic is the acceptance region. This represents the set of values for which we do not reject (accept) the null hypothesis. The region to the right of the test statistic is known as the rejection region. The fifth step is collecting the data and calculating the test statistic. The quality of data used in conducting the test is important. We should be careful to check for measurement errors in the data as well as any instances of sample selection bias and time-period bias. Sample selection bias refers to the bias introduced by systematically excluding some members of the population according to a particular attribute. Time-period bias refers to the possibility that when we use a time-series sample, our statistical conclusion may be sensitive to the starting and ending dates of the sample. Copyright Irfanullah Financial Training. All rights reserved. Page 6

7 The sixth step is making the statistical decision. In our example, because the test statistic z = 3 is greater than the critical value of 1.645, we reject the null hypothesis in favor of the alternative hypothesis that the average return on all Asian stocks is greater than 2%. The seventh and final step is making the economic or investment decision. This takes into account not only the statistical decision (step 6) but also all pertinent economic issues. Based on these results, an investor might want to invest in Asian stocks. Therefore, a range of nonstatistical considerations, such as the investor s tolerance for risk, transaction costs and impact on existing portfolio, might also enter the decision-making process. Let us now consider the left tail. Suppose we believed that the average return on all Asian stocks was less than 2%. Our sample size is of 36 observations with a sample mean of -3. The standard deviation of the population is 4. In this case, our null and alternative hypotheses would have been: H0: µ 2 (Null) Ha: µ < 2 (Alternate) The standard error of the sample will be unchanged at 0.67: σ x = σ n = 4 36 = 0.67 The test statistic will be: Test statistic = Sample statistic Value of the parameter under H 0 Standard error of the sample statistic = = 7.5 The critical values corresponding to a 5% level of significance will be When we consider the left tail of the distribution, our decision rule is then as follows: Reject the null hypothesis if the test statistic is less than the critical value and vice versa. Since our calculated test statistic of is less than the critical value of -1.65, we reject the null hypothesis. Copyright Irfanullah Financial Training. All rights reserved. Page 7

8 Let us now discuss how two-tailed hypothesis tests are conducted. In a two-tailed test, we reject the null in favor of the alternative if the evidence indicates that the population parameter is either smaller or larger than the value of the parameter under H0. Suppose we believe that the average return on all Asian stocks was not 0%. We take a sample of 36 observations with a sample mean of 1 and a population standard deviation of 4. In this case our null and alternative hypotheses will be: H0: µ = 0 (Null) Ha: µ 0 (Alternate) The standard error of the sample will be unchanged at 0.67: σ x = σ n = 4 36 = 0.67 The test statistic will be: Test statistic = Sample statistic Value of the parameter under H 0 Standard error of the sample statistic = = 1.5 In a two-tailed test, two critical values exist one positive and one negative. For a two-sided test at the 5% level of significance, we calculate the z-values that correspond to 0.05/2 = level of significance. These are and Therefore, we reject the null hypothesis if we find that the test statistic is less than or greater than We fail to reject the null hypothesis if test statistic Graphically, this can be shown as: Copyright Irfanullah Financial Training. All rights reserved. Page 8

9 The above figure also illustrates the relationship between confidence intervals and hypothesis tests. The 5% level of significance in the hypothesis tests corresponds to a 95% confidence interval. When the hypothesized value of the population parameter (in this case µ0) under the null hypothesis is outside the corresponding confidence interval, the null hypothesis is rejected. When the hypothesized value of the population parameter is inside the corresponding confidence interval, the null hypothesis is not rejected. We could use confidence intervals to test hypotheses; practitioners, however, usually do not. Computing a test statistic is more efficient. Furthermore, only when we compute a test statistic can we obtain a p-value. The p-value (also known as probability value) is an alternative approach to hypothesis testing. The p-value is the smallest level of significance at which the null hypothesis can be rejected. The p-value is analogous to the test-statistic. High test-statistic means low p-value and low test-statistic denotes high p-value. The smaller the p-value, the stronger is the evidence against the null hypothesis and in favor of the alternative hypothesis. We can use the p-values in the hypothesis testing framework presented earlier as an alternative to using rejection points. If the p-value is less than our specified level of significance, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis. When we conduct hypothesis testing, there are four possible outcomes: (i) (ii) (iii) (iv) We reject a false null hypothesis. This is a correct decision. We reject a true null hypothesis. This is called a Type I error. We do not reject a false null hypothesis. This is called a Type II error. We do not reject a true null hypothesis. This is a correct decision. Decision H0 True H0 False Do not reject H0 Correct Decision Type II Error Reject H0 (accept Ha) Type I Error Correct Decision The probability of a Type I error in testing a hypothesis is denoted by the Greek letter alpha, α. This probability is also known as the level of significance of the test. For example, a level of Copyright Irfanullah Financial Training. All rights reserved. Page 9

10 significance of 0.05 for a test means that there is a 5% probability of rejecting a true null hypothesis. The probability of a Type II error is denoted by the Greek letter, β. All else equal, if we decrease the probability of a Type I error by specifying a smaller significance level, we increase the probability of making a Type II error because we will reject the null less frequently, including when it is false. The only way to reduce the probabilities of both types of errors simultaneously is to increase the sample size. The power of a test is the probability of correctly rejecting the null i.e. the probability of rejecting the null when it is false. When more than one test statistic is available to conduct a hypothesis test, we should prefer the most powerful. To summarize, α is the probability of a Type I error and β is the probability of a Type II error. The power of a test is equal to 1 P (Type II error). 3. Hypothesis Tests Concerning the Mean Hypothesis tests concerning the mean are very common and are usually concerned with the following: Tests concerning a single mean Tests concerning differences between means Tests concerning mean differences 3.1 Tests Concerning a Single Mean One of the decisions we need to make in hypothesis testing is deciding which test statistic and which corresponding probability distribution to use. We generally choose between a t-test and a z-test. The t-test is a hypothesis test using a statistic (t-statistic) that follows a t-distribution. As discussed in the previous reading, this is defined by a single parameter known as the degrees of freedom (df). Copyright Irfanullah Financial Training. All rights reserved. Page 10

11 In hypothesis testing, analysts require the population standard deviation. In practice, however, analysts estimate the population standard deviation by calculating a sample standard deviation because the population variance is unknown. Hence, in hypothesis tests concerning the population mean of a normally distributed population with unknown variance, the theoretically correct test statistic is the t-statistic. Consider a simple example. Fund Alpha has been in existence for 20 months and has achieved a mean monthly return of 2.00% with a sample standard deviation of 5%. The expected monthly return for a fund of this nature is 1.60%. Assuming monthly returns are normally distributed, are the actual results consistent with an underlying or population mean monthly return of 1.60%? The null and alternative hypotheses for this example will be: H0: µ = 1.60 (Null) versus Ha: µ 1.60 (Alternate) Since we only have the sample standard deviation, the appropriate test statistic in this case will be the t-statistic. In order to calculate the t-statistic we use the following formula: t n 1 = X μ 0 s/ n where t n 1 = t-statistic with n-1 degrees of freedom X = the sample mean µ0 = the hypothesized value of the population mean s = the sample standard deviation n = sample size Using this formula, we see that the value of the test statistic is The critical values at a 0.05 level of significance can be calculated from the t-distribution table. Since this is a two-tailed test, we should be careful to look at a 0.05/2 = level of significance with df = 20 1 = 19. This gives us two values of +2.1 and Since our test statistic of 0.35 lies between -2.1 and +2.1, i.e. the acceptance region, we do not reject the null hypothesis. Copyright Irfanullah Financial Training. All rights reserved. Page 11

12 Given at least approximate normality, the t-test is always called for when we deal with small samples and do not know the population variance. For large samples and unknown population variance, analysts sometimes use a z-test instead of a t-test for tests concerning a mean. The justifications for using a z-test are: In large samples, the sample mean should follow the normal distribution at least approximately, fulfilling the normality assumption of the z-test. In large samples, the difference between the rejection points for the t-test and z-test become quite small. If the population sampled is normally distributed with known variance, then the test statistic for a hypothesis test concerning a single population mean is: z = X μ 0 σ/ n If the population sampled has unknown variance and the sample is large, in place of the t-test, an alternative test statistic is: z = X μ 0 s/ n In both the equations: σ = the known population standard deviation s = the sample standard deviation µ0 = the hypothesized value of the population mean 3.2 Tests Concerning Differences between Means We often want to know whether a mean value differs between two groups. In this section, we discuss how to study the difference between the means of two independent and normally distributed populations. We can use two kinds of t-tests for a test concerning differences between means. In one case the population variances, although unknown, can be assumed to be equal. In the second case the population variances are assumed to be unknown and unequal. Copyright Irfanullah Financial Training. All rights reserved. Page 12

13 Given two populations, we often want to test whether the population means of the first and second populations are equal or whether one is larger than the other. To test this, we usually formulate the following hypotheses: (i) H0: µ1 - µ2 = 0 versus Ha: µ1 - µ2 0 This is used when we believe the population means are not equal. (ii) H0 : µ1 - µ2 0 versus Ha: µ1 - µ2 > 0 This is used when we believe the mean of the first population is greater than the mean of the second population. (iii) H0 : µ1 - µ2 0 versus Ha: µ1 - µ2 < 0 This is used when we believe the mean of the first population is less than the mean of the second population. where µ1 = population mean of the first population µ2 = population mean of the second population Unknown but Equal Population Variance: When we can assume that the two populations are normally distributed and that the unknown population variances are equal, the t-test based on independent random samples is given by: t = (X 1 X 2) (μ 1 μ 2 ) ( s p 2 + s 2 p ) 1/2 n 1 n 2 The number of degrees of freedom is n1 + n2 2. The term s p 2 is known as the pooled estimator of the common variance. A pooled estimate is an estimate drawn from the combination of two different samples. It is calculated by the following formula: s p 2 = (n 1 1)s ((n 2 1)s 2 2 n 1 + n 2 2 Unknown and Unequal Population Variance: When we can assume that the two populations are normally distributed and that the unknown population variances are unequal, an approximate t- test based on independent random samples is given by: Copyright Irfanullah Financial Training. All rights reserved. Page 13

14 t = (X 1 X 2) (μ 1 μ 2 ) ( s s 2 2 ) 1/2 n 1 n 2 In this formula, we use the tables of the t-distribution using the modified degrees of freedom. The modified degrees of freedom are calculated using the following formula: df = ( s s 2 2 ) 2 n 1 n 2 (s 2 1 /n 1 ) 2 + (s 2 2 /n 2 ) 2 n 1 n 2 Worked Example 1 You believe the mean return on NYSE stocks was different from the mean on NSE stocks last month. To test your hypothesis you collect the following data: Sample Size (n) Sample Mean (X ) Sample Standard Deviation (s) NSE 20 2% 4 NYSE 40 3% 5 Determine whether to reject the null hypothesis at the 0.10 level of significance. Solution: The first step is to formulate the null and alternative hypotheses. Since we want to test whether the two means were equal or different, we define the hypotheses as: H0: µ1 - µ2 = 0 Ha: µ1 - µ2 0 Since the population standard deviation is unknown and we cannot assume that it is equal, we use the following formula to calculate the test statistic: t = (X 1 X 2) (μ 1 μ 2 ) (2 3) (0) ( s s 2 = = ) 1/2 ( n 1 n )1/2 Next, we calculate the modified degrees of freedom: Copyright Irfanullah Financial Training. All rights reserved. Page 14

15 df = ( s s 2 2 ) 2 n 1 n 2 = (s 2 1 /n 1 ) 2 + (s 2 2 /n 2 ) 2 n 1 n 2 ( )2 (4 2 /20) 2 + (52 /40) = 48 For a 0.10 level of significance, we find the t-value for 0.10/2 = 0.05 using df = 48. The t-value is therefore ta/2= and Since our test statistic of lies in the acceptance region, we fail to reject the null hypothesis. 3.3 Tests concerning Mean Differences In the previous section, in order to perform hypothesis tests on differences between means of two populations, we assumed that the samples were independent. What if the samples were not independent? For example, suppose you want to conduct tests on the mean monthly return on Toyota stock and mean monthly return on Honda stock. These two samples are believed to be dependent. In such situations, we conduct a t-test that is based on data arranged in paired observations. The test is also sometimes known as a paired comparisons test. Paired observations are observations that are dependent because they have something in common. A paired comparisons test is a statistical test for differences in dependent items. We will now discuss the process for conducting such a t-test. Suppose that we gather data regarding the mean monthly returns on stocks of Toyota and Honda for the last 20 months. Month Mean return of Toyota Mean monthly return of Difference in mean stock Honda stock monthly returns (di) 1 0.5% 0.4% 0.1% 2 0.7% 1.0% -0.3% 3 0.3% 0.7% -0.4% % 0.6% 0.3% Average 0.750% 0.600% 0.075% Here is a simplified process for conducting the hypothesis test: Step 1: Define the null and alternate hypotheses We believe the mean difference is not 0. Hence the null and alternate hypotheses are: Copyright Irfanullah Financial Training. All rights reserved. Page 15

16 H0: µd = µd0 versus Ha: µd µd0 µd stands for the population mean difference and µd0 stands for the hypothesized value for the population mean difference. Step 2: Calculate the test-statistic Determine the sample mean difference using: n d = 1 n d i i=0 For the data given, the sample mean difference is Calculate the sample standard deviation. The process for calculating the sample standard deviation has been discussed in an earlier reading. The simplest method is to plug the numbers (0.1, -0.3, ) into a financial calculator. The entire data set has not been provided. We ll take it as a given that the sample standard deviation is 0.150%. Use this to calculate the standard error of the mean difference as follows: s d = s d n For our data this is / 20 = We now have the required data to calculate the test statistic using a t-test. This is calculated using the following formula using n - 1 degrees of freedom: t = d μ d0 s d For our data the test statistic is ( ) / = 2.23 Step 3: Determine the critical value based on the level of significance We will use a 5% level of significance. Since this is a two-tailed test we have a probability of 2.5% (0.025) in each tail. This critical value is determined from a t-table using a one-tailed probability of and df = 20 1 = 19. This value is Step 4: Compare the test statistic with the critical value and make a decision In our case the test statistic (2.23) is greater than the critical value (2.093). Hence we can reject the null hypothesis. Our conclusion: the data seems to indicate that the mean difference is not 0. Copyright Irfanullah Financial Training. All rights reserved. Page 16

17 The hypothesis test presented above is based on the belief that the population mean difference is not equal to 0. If we believe that the population mean difference is greater than 0, the null and alternate hypotheses will be written as: H0: µd µd0 versus Ha: µd > µd0 If we believe that the population mean difference is less than 0, the null and alternate hypotheses will be written as: H0: µd µd0 versus Ha: µd < µd0 4. Hypothesis Tests Concerning Variance Variance is a widely used quantitative measure of risk in investments and so analysts should be familiar with hypothesis tests concerning variance. We discuss two kinds of tests: - Tests concerning the value of a single population variance - Tests concerning the differences between two population variances 4.1 Tests Concerning a Single Variance In tests concerning the variance of a single normally distributed population, we use the chisquare test statistic, denoted by χ 2. The chi-square distribution is asymmetrical and like the t- distribution, is a family of distributions. This means that a different distribution exists for each possible value of degrees of freedom, n - 1. Since the variance is a squared term, the minimum value can only be 0. Hence, the chi-square distribution is bounded below by 0. The graph below shows the shape of a chi-square distribution: The term σ 2 represents the true population variance and σ0 2 represents the hypothesized variance. There are three hypotheses that can be formulated: Copyright Irfanullah Financial Training. All rights reserved. Page 17

18 (i) (ii) (iii) H0 : σ 2 = σ0 2 versus Ha : σ 2 σ0 2 This is used when we believe the population variance is not equal to 0. H0 : σ 2 σ0 2 versus Ha : σ 2 < σ0 2 This is used when we believe the population variance is less than 0 or any other specified value. H0 : σ 2 σ0 2 versus Ha : σ 2 > σ0 2 This is used when we believe the population variance is greater than 0 or any other specified value. After drawing a random sample from a normally distributed population, we calculate the test statistic using the following formula using n - 1 degrees of freedom: χ 2 = (n 1)(s2 ) σ 0 2 where n = sample size s = sample variance We then determine the critical values using the level of significance and degrees of freedom. The table below is a snapshot of the chi-square distribution table which is used to calculate the critical value. Copyright Irfanullah Financial Training. All rights reserved. Page 18

19 Suppose our degrees of freedom are 19 and we are testing at the 0.05 level of significance. In this case, the critical value will be This critical value is then compared with the test statistic calculated earlier and a decision is reached on whether or not the null hypothesis can be rejected. It is important to note that the chi-square test is sensitive to violations of its assumptions. If the sample is not actually random or if it does not come from a normally distributed population, inferences based on a chi-square test are likely to be faulty. Worked Example 2 Consider Fund Alpha which we discussed in an earlier example. This fund has been in existence for 20 months. During this period the standard deviation of monthly returns has been 5%. You want to test a claim by the fund manager that the standard deviation of monthly returns is less than 6%. Solution: Copyright Irfanullah Financial Training. All rights reserved. Page 19

20 The null and alternate hypotheses are formulated as mentioned below. Please note that the standard deviation is 6%. Since we are dealing with population variance, we will square this number to arrive at a variance of 36%: H0: σ 2 36 versus Ha: σ 2 < 36 We then calculate the value of the chi-square test statistic: 2 = (n - 1) s 2 / σ0 2 = 19 x 25/36 = Next we determine the rejection point based on df = 19 and significance = Using the chisquare table, we find that this number is Since the test statistic (13.19) is higher than the rejection point (10.117) we cannot reject H0. In other words, the sample standard deviation is not small enough to validate the fund manager s claim that population standard deviation is less than 6%. 4.2 Tests Concerning the Equality (Inequality) of Two Variances In order to test the equality or inequality of two variances, we use an F-test. An F-test is the ratio of sample variances. For an F-test to be valid, it is important that the samples be independent and that the populations from which the samples are taken are normally distributed. The F-distribution, like the chi-square distribution, is a family of asymmetrical distributions bounded from below by 0. Each F-distribution is defined by two values of degrees of freedom, called the numerator and denominator degrees of freedom. As shown in the figure below, the F- distribution is skewed to the right and is truncated at zero on the left hand side. As shown in the graph, the rejection region is always in the right side tail of the distribution. When working with F-tests, there are three hypotheses that can be formulated: (i) H0 : σ1 2 = σ2 2 versus Ha : σ1 2 σ2 2 This is used when we believe the two population variances are not equal. Copyright Irfanullah Financial Training. All rights reserved. Page 20

21 (ii) (iii) H0 : σ1 2 σ2 2 versus Ha : σ1 2 > σ2 2 This is used when we believe the variance of the first population is greater than the variance of the second population. H0 : σ1 2 σ2 2 versus Ha : σ1 2 < σ2 2 This is used when we believe the variance of the first population is less than the variance of the second population. The term σ1 2 represents the population variance of the first population and σ2 2 represents the population variance of the second population. The formula for the test statistic of the F-test is calculated by the following formula: F = s 1 2 where 2 s 1 = the sample variance of the first population with n observations 2 s 2 = the sample variance of the second population with n observations df1 = n1 1 numerator degrees of freedom df2 = n2 1 denominator degrees of freedom s 2 2 A convention is to put the larger sample variance in the numerator and the smaller sample variance in the denominator. When we follow this convention, the value of the test statistic is always greater than or equal to 1. The test statistic is then compared with the critical values found using the two degrees of freedom and the F-tables. Finally a decision is made whether to reject or not reject the null hypothesis. Worked Example 3 You are investigating whether the population variance of the Indian equity market changed after the deregulation of You collect 120 months of data before and after deregulation. Variance of returns before deregulation was 13. Variance of returns after deregulation was 18. Solution: Copyright Irfanullah Financial Training. All rights reserved. Page 21

22 Null and alternate hypothesis: H0: σ1 2 = σ2 2 versus HA: σ1 2 σ2 2 F-statistic: 18/13 = 1.4 df = 119 for the numerator and denominator α = 0.01 which means in each tail. From the F-table: critical value = 1.6 Since the F-stat is less than the critical value, do not reject the null hypothesis. 5. Other Issues: Nonparametric Inference The hypothesis-testing procedures we have discussed so far have two characteristics in common: They are concerned with parameters, such as the mean and variance Their validity depends on a set of assumptions Any procedure which has either of the two characteristics is known as a parametric test. Nonparametric tests are not concerned with a parameter and/or make few assumptions about the population from which the sample comes. We use nonparametric procedures in three situations: Data does not meet distributional assumptions Data given in ranks (Example: relative size of company and use of derivatives) Hypothesis does not concern a parameter (Example: is a sample random or not) The Spearman rank correlation coefficient test is a popular nonparametric test. The coefficient is calculated based on the ranks of two variables within their respective samples. Summary (Note: This summary has been taken from the curriculum.) In this reading, we have presented the concepts and methods of statistical inference and hypothesis testing. A hypothesis is a statement about one or more populations. The steps in testing a hypothesis are as follows: 1. Stating the hypotheses. Copyright Irfanullah Financial Training. All rights reserved. Page 22

23 2. Identifying the appropriate test statistic and its probability distribution. 3. Specifying the significance level. 4. Stating the decision rule. 5. Collecting the data and calculating the test statistic. 6. Making the statistical decision. 7. Making the economic or investment decision. We state two hypotheses: The null hypothesis is the hypothesis to be tested; the alternative hypothesis is the hypothesis accepted when the null hypothesis is rejected. There are three ways to formulate hypotheses: 1. H0: θ = θ0 versus Ha: θ θ0 2. H0: θ θ0 versus Ha: θ > θ0 3. H0: θ θ0 versus Ha: θ < θ0 where θ0 is a hypothesized value of the population parameter and θ is the true value of the population parameter. In the above, Formulation 1 is a two-sided test and Formulations 2 and 3 are one-sided tests. When we have a suspected or hoped for condition for which we want to find supportive evidence, we frequently set up that condition as the alternative hypothesis and use a onesided test. To emphasize a neutral attitude, however, the researcher may select a not equal to alternative hypothesis and conduct a two-sided test. A test statistic is a quantity, calculated on the basis of a sample, whose value is the basis for deciding whether to reject or not reject the null hypothesis. To decide whether to reject, or not to reject, the null hypothesis, we compare the computed value of the test statistic to a critical value (rejection point) for the same test statistic. In reaching a statistical decision, we can make two possible errors: We may reject a true null hypothesis (a Type I error), or we may fail to reject a false null hypothesis (a Type II error). The level of significance of a test is the probability of a Type I error that we accept in conducting a hypothesis test. The probability of a Type I error is denoted by the Greek letter alpha, α. The standard approach to hypothesis testing involves specifying a level of significance (probability of Type I error) only. The power of a test is the probability of correctly rejecting the null (rejecting the null when it is false). Copyright Irfanullah Financial Training. All rights reserved. Page 23

24 A decision rule consists of determining the rejection points (critical values) with which to compare the test statistic to decide whether to reject or not to reject the null hypothesis. When we reject the null hypothesis, the result is said to be statistically significant. The (1 α) confidence interval represents the range of values of the test statistic for which the null hypothesis will not be rejected at an α significance level. The statistical decision consists of rejecting or not rejecting the null hypothesis. The economic decision takes into consideration all economic issues pertinent to the decision. The p-value is the smallest level of significance at which the null hypothesis can be rejected. The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis. The p-value approach to hypothesis testing does not involve setting a significance level; rather it involves computing a p-value for the test statistic and allowing the consumer of the research to interpret its significance. For hypothesis tests concerning the population mean of a normally distributed population with unknown (known) variance, the theoretically correct test statistic is the t-statistic (zstatistic). In the unknown variance case, given large samples (generally, samples of 30 or more observations), the z-statistic may be used in place of the t-statistic because of the force of the central limit theorem. The t-distribution is a symmetrical distribution defined by a single parameter: degrees of freedom. Compared to the standard normal distribution, the t-distribution has fatter tails. When we want to test whether the observed difference between two means is statistically significant, we must first decide whether the samples are independent or dependent (related). If the samples are independent, we conduct tests concerning differences between means. If the samples are dependent, we conduct tests of mean differences (paired comparisons tests). When we conduct a test of the difference between two population means from normally distributed populations with unknown variances, if we can assume the variances are equal, we use a t-test based on pooling the observations of the two samples to estimate the common (but unknown) variance. This test is based on an assumption of independent samples. When we conduct a test of the difference between two population means from normally distributed populations with unknown variances, if we cannot assume that the variances are equal, we use an approximate t-test using modified degrees of freedom given by a formula. This test is based on an assumption of independent samples. Copyright Irfanullah Financial Training. All rights reserved. Page 24

25 In tests concerning two means based on two samples that are not independent, we often can arrange the data in paired observations and conduct a test of mean differences (a paired comparisons test). When the samples are from normally distributed populations with unknown variances, the appropriate test statistic is a t-statistic. The denominator of the t- statistic, the standard error of the mean differences, takes account of correlation between the samples. In tests concerning the variance of a single, normally distributed population, the test statistic is chi-square (χ 2 ) with n 1 degrees of freedom, where n is sample size. For tests concerning differences between the variances of two normally distributed populations based on two random, independent samples, the appropriate test statistic is based on an F-test (the ratio of the sample variances). The F-statistic is defined by the numerator and denominator degrees of freedom. The numerator degrees of freedom (number of observations in the sample minus 1) is the divisor used in calculating the sample variance in the numerator. The denominator degrees of freedom (number of observations in the sample minus 1) is the divisor used in calculating the sample variance in the denominator. In forming an F-test, a convention is to use the larger of the two ratios, s1 2 / s2 2 or s2 2 / s1 2, as the actual test statistic. A parametric test is a hypothesis test concerning a parameter or a hypothesis test based on specific distributional assumptions. In contrast, a nonparametric test either is not concerned with a parameter or makes minimal assumptions about the population from which the sample comes. A nonparametric test is primarily used in three situations: when data do not meet distributional assumptions, when data are given in ranks, or when the hypothesis we are addressing does not concern a parameter. The Spearman rank correlation coefficient is calculated on the ranks of two variables within their respective samples. Next Steps Work through the examples presented in the curriculum. Copyright Irfanullah Financial Training. All rights reserved. Page 25

26 Solve the practice problems in the curriculum. Solve the IFT Practice Questions associated with this reading. Review the learning outcomes presented in the curriculum. Make sure that you can perform the implied actions. Copyright Irfanullah Financial Training. All rights reserved. Page 26

THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7. THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

More information

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

More information

Study Guide for the Final Exam

Study Guide for the Final Exam Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

More information

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

More information

Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Two-Sample T-Tests Assuming Equal Variance (Enter Means) Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

More information

Recall this chart that showed how most of our course would be organized:

Recall this chart that showed how most of our course would be organized: Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

More information

NCSS Statistical Software

NCSS Statistical Software Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

More information

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

More information

Two-Sample T-Tests Allowing Unequal Variance (Enter Difference)

Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption

More information

How To Check For Differences In The One Way Anova

How To Check For Differences In The One Way Anova MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

More information

3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples 3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

More information

Lecture Notes Module 1

Lecture Notes Module 1 Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

More information

Comparing Means in Two Populations

Comparing Means in Two Populations Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we

More information

t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon

t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com

More information

NCSS Statistical Software

NCSS Statistical Software Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

More information

individualdifferences

individualdifferences 1 Simple ANalysis Of Variance (ANOVA) Oftentimes we have more than two groups that we want to compare. The purpose of ANOVA is to allow us to compare group means from several independent samples. In general,

More information

Time Value of Money. 2014 Level I Quantitative Methods. IFT Notes for the CFA exam

Time Value of Money. 2014 Level I Quantitative Methods. IFT Notes for the CFA exam Time Value of Money 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction...2 2. Interest Rates: Interpretation...2 3. The Future Value of a Single Cash Flow...4 4. The

More information

Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)

Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1) Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the

More information

Section 13, Part 1 ANOVA. Analysis Of Variance

Section 13, Part 1 ANOVA. Analysis Of Variance Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability

More information

Chapter 7 Notes - Inference for Single Samples. You know already for a large sample, you can invoke the CLT so:

Chapter 7 Notes - Inference for Single Samples. You know already for a large sample, you can invoke the CLT so: Chapter 7 Notes - Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a

More information

Hypothesis testing - Steps

Hypothesis testing - Steps Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

More information

MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS

MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance

More information

Tutorial 5: Hypothesis Testing

Tutorial 5: Hypothesis Testing Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrc-lmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................

More information

Using Excel for inferential statistics

Using Excel for inferential statistics FACT SHEET Using Excel for inferential statistics Introduction When you collect data, you expect a certain amount of variation, just caused by chance. A wide variety of statistical tests can be applied

More information

HYPOTHESIS TESTING: POWER OF THE TEST

HYPOTHESIS TESTING: POWER OF THE TEST HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

More information

Quantitative Methods for Finance

Quantitative Methods for Finance Quantitative Methods for Finance Module 1: The Time Value of Money 1 Learning how to interpret interest rates as required rates of return, discount rates, or opportunity costs. 2 Learning how to explain

More information

5.1 Identifying the Target Parameter

5.1 Identifying the Target Parameter University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying

More information

Simple Regression Theory II 2010 Samuel L. Baker

Simple Regression Theory II 2010 Samuel L. Baker SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

More information

Permutation Tests for Comparing Two Populations

Permutation Tests for Comparing Two Populations Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of

More information

Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures

Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone:

More information

Non-Inferiority Tests for One Mean

Non-Inferiority Tests for One Mean Chapter 45 Non-Inferiority ests for One Mean Introduction his module computes power and sample size for non-inferiority tests in one-sample designs in which the outcome is distributed as a normal random

More information

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics. Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

More information

Fairfield Public Schools

Fairfield Public Schools Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

More information

Section 7.1. Introduction to Hypothesis Testing. Schrodinger s cat quantum mechanics thought experiment (1935)

Section 7.1. Introduction to Hypothesis Testing. Schrodinger s cat quantum mechanics thought experiment (1935) Section 7.1 Introduction to Hypothesis Testing Schrodinger s cat quantum mechanics thought experiment (1935) Statistical Hypotheses A statistical hypothesis is a claim about a population. Null hypothesis

More information

Stat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015

Stat 411/511 THE RANDOMIZATION TEST. Charlotte Wickham. stat511.cwick.co.nz. Oct 16 2015 Stat 411/511 THE RANDOMIZATION TEST Oct 16 2015 Charlotte Wickham stat511.cwick.co.nz Today Review randomization model Conduct randomization test What about CIs? Using a t-distribution as an approximation

More information

Rank-Based Non-Parametric Tests

Rank-Based Non-Parametric Tests Rank-Based Non-Parametric Tests Reminder: Student Instructional Rating Surveys You have until May 8 th to fill out the student instructional rating surveys at https://sakai.rutgers.edu/portal/site/sirs

More information

Unit 26 Estimation with Confidence Intervals

Unit 26 Estimation with Confidence Intervals Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

More information

Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm

Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm Mgt 540 Research Methods Data Analysis 1 Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm http://web.utk.edu/~dap/random/order/start.htm

More information

Statistics Review PSY379

Statistics Review PSY379 Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses

More information

Sample Size and Power in Clinical Trials

Sample Size and Power in Clinical Trials Sample Size and Power in Clinical Trials Version 1.0 May 011 1. Power of a Test. Factors affecting Power 3. Required Sample Size RELATED ISSUES 1. Effect Size. Test Statistics 3. Variation 4. Significance

More information

Introduction to Hypothesis Testing OPRE 6301

Introduction to Hypothesis Testing OPRE 6301 Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about

More information

Simple Linear Regression Inference

Simple Linear Regression Inference Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

More information

Chapter 4: Statistical Hypothesis Testing

Chapter 4: Statistical Hypothesis Testing Chapter 4: Statistical Hypothesis Testing Christophe Hurlin November 20, 2015 Christophe Hurlin () Advanced Econometrics - Master ESA November 20, 2015 1 / 225 Section 1 Introduction Christophe Hurlin

More information

CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE

CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE 1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,

More information

SCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES

SCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES SCHOOL OF HEALTH AND HUMAN SCIENCES Using SPSS Topics addressed today: 1. Differences between groups 2. Graphing Use the s4data.sav file for the first part of this session. DON T FORGET TO RECODE YOUR

More information

Introduction to Hypothesis Testing

Introduction to Hypothesis Testing I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true

More information

Chapter 2. Hypothesis testing in one population

Chapter 2. Hypothesis testing in one population Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance

More information

1.5 Oneway Analysis of Variance

1.5 Oneway Analysis of Variance Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

More information

Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics

Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This

More information

Principles of Hypothesis Testing for Public Health

Principles of Hypothesis Testing for Public Health Principles of Hypothesis Testing for Public Health Laura Lee Johnson, Ph.D. Statistician National Center for Complementary and Alternative Medicine johnslau@mail.nih.gov Fall 2011 Answers to Questions

More information

NCSS Statistical Software. One-Sample T-Test

NCSS Statistical Software. One-Sample T-Test Chapter 205 Introduction This procedure provides several reports for making inference about a population mean based on a single sample. These reports include confidence intervals of the mean or median,

More information

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as... HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

More information

KSTAT MINI-MANUAL. Decision Sciences 434 Kellogg Graduate School of Management

KSTAT MINI-MANUAL. Decision Sciences 434 Kellogg Graduate School of Management KSTAT MINI-MANUAL Decision Sciences 434 Kellogg Graduate School of Management Kstat is a set of macros added to Excel and it will enable you to do the statistics required for this course very easily. To

More information

Part 2: Analysis of Relationship Between Two Variables

Part 2: Analysis of Relationship Between Two Variables Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

More information

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.

More information

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as... HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

More information

How To Test For Significance On A Data Set

How To Test For Significance On A Data Set Non-Parametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A non-parametric equivalent of the 1 SAMPLE T-TEST. ASSUMPTIONS: Data is non-normally distributed, even after log transforming.

More information

Statistical tests for SPSS

Statistical tests for SPSS Statistical tests for SPSS Paolo Coletti A.Y. 2010/11 Free University of Bolzano Bozen Premise This book is a very quick, rough and fast description of statistical tests and their usage. It is explicitly

More information

A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment

More information

CHAPTER 14 NONPARAMETRIC TESTS

CHAPTER 14 NONPARAMETRIC TESTS CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences

More information

Confidence Intervals for One Standard Deviation Using Standard Deviation

Confidence Intervals for One Standard Deviation Using Standard Deviation Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from

More information

In the past, the increase in the price of gasoline could be attributed to major national or global

In the past, the increase in the price of gasoline could be attributed to major national or global Chapter 7 Testing Hypotheses Chapter Learning Objectives Understanding the assumptions of statistical hypothesis testing Defining and applying the components in hypothesis testing: the research and null

More information

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of

More information

How To Read The Book \"Financial Planning\"

How To Read The Book \Financial Planning\ Time Value of Money Reading 5 IFT Notes for the 2015 Level 1 CFA exam Contents 1. Introduction... 2 2. Interest Rates: Interpretation... 2 3. The Future Value of a Single Cash Flow... 4 4. The Future Value

More information

22. HYPOTHESIS TESTING

22. HYPOTHESIS TESTING 22. HYPOTHESIS TESTING Often, we need to make decisions based on incomplete information. Do the data support some belief ( hypothesis ) about the value of a population parameter? Is OJ Simpson guilty?

More information

II. DISTRIBUTIONS distribution normal distribution. standard scores

II. DISTRIBUTIONS distribution normal distribution. standard scores Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

More information

Outline. Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test

Outline. Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test The t-test Outline Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test - Dependent (related) groups t-test - Independent (unrelated) groups t-test Comparing means Correlation

More information

Time Value of Money. 2014 Level I Quantitative Methods. IFT Notes for the CFA exam

Time Value of Money. 2014 Level I Quantitative Methods. IFT Notes for the CFA exam Time Value of Money 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 2 2. Interest Rates: Interpretation... 2 3. The Future Value of a Single Cash Flow... 4 4. The

More information

t-test Statistics Overview of Statistical Tests Assumptions

t-test Statistics Overview of Statistical Tests Assumptions t-test Statistics Overview of Statistical Tests Assumption: Testing for Normality The Student s t-distribution Inference about one mean (one sample t-test) Inference about two means (two sample t-test)

More information

Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing. Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

More information

Tests for Two Proportions

Tests for Two Proportions Chapter 200 Tests for Two Proportions Introduction This module computes power and sample size for hypothesis tests of the difference, ratio, or odds ratio of two independent proportions. The test statistics

More information

Projects Involving Statistics (& SPSS)

Projects Involving Statistics (& SPSS) Projects Involving Statistics (& SPSS) Academic Skills Advice Starting a project which involves using statistics can feel confusing as there seems to be many different things you can do (charts, graphs,

More information

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

More information

Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:

Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name: Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours

More information

Statistical Functions in Excel

Statistical Functions in Excel Statistical Functions in Excel There are many statistical functions in Excel. Moreover, there are other functions that are not specified as statistical functions that are helpful in some statistical analyses.

More information

Financial Statement Analysis: An Introduction

Financial Statement Analysis: An Introduction Financial Statement Analysis: An Introduction 2014 Level I Financial Reporting and Analysis IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Scope of Financial Statement Analysis... 3 3. Major

More information

Pearson's Correlation Tests

Pearson's Correlation Tests Chapter 800 Pearson's Correlation Tests Introduction The correlation coefficient, ρ (rho), is a popular statistic for describing the strength of the relationship between two variables. The correlation

More information

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this

More information

An Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10- TWO-SAMPLE TESTS

An Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10- TWO-SAMPLE TESTS The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10- TWO-SAMPLE TESTS Practice

More information

1 Nonparametric Statistics

1 Nonparametric Statistics 1 Nonparametric Statistics When finding confidence intervals or conducting tests so far, we always described the population with a model, which includes a set of parameters. Then we could make decisions

More information

ELEMENTARY STATISTICS

ELEMENTARY STATISTICS ELEMENTARY STATISTICS Study Guide Dr. Shinemin Lin Table of Contents 1. Introduction to Statistics. Descriptive Statistics 3. Probabilities and Standard Normal Distribution 4. Estimates and Sample Sizes

More information

Data Analysis Tools. Tools for Summarizing Data

Data Analysis Tools. Tools for Summarizing Data Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool

More information

Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption

Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption Last time, we used the mean of one sample to test against the hypothesis that the true mean was a particular

More information

UNDERSTANDING THE DEPENDENT-SAMPLES t TEST

UNDERSTANDING THE DEPENDENT-SAMPLES t TEST UNDERSTANDING THE DEPENDENT-SAMPLES t TEST A dependent-samples t test (a.k.a. matched or paired-samples, matched-pairs, samples, or subjects, simple repeated-measures or within-groups, or correlated groups)

More information

2 Sample t-test (unequal sample sizes and unequal variances)

2 Sample t-test (unequal sample sizes and unequal variances) Variations of the t-test: Sample tail Sample t-test (unequal sample sizes and unequal variances) Like the last example, below we have ceramic sherd thickness measurements (in cm) of two samples representing

More information

Statistics 2014 Scoring Guidelines

Statistics 2014 Scoring Guidelines AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home

More information

Introduction. Statistics Toolbox

Introduction. Statistics Toolbox Introduction A hypothesis test is a procedure for determining if an assertion about a characteristic of a population is reasonable. For example, suppose that someone says that the average price of a gallon

More information

Comparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples

Comparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The

More information

An Introduction to Statistics using Microsoft Excel. Dan Remenyi George Onofrei Joe English

An Introduction to Statistics using Microsoft Excel. Dan Remenyi George Onofrei Joe English An Introduction to Statistics using Microsoft Excel BY Dan Remenyi George Onofrei Joe English Published by Academic Publishing Limited Copyright 2009 Academic Publishing Limited All rights reserved. No

More information

Confidence Intervals for the Difference Between Two Means

Confidence Intervals for the Difference Between Two Means Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

More information

Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing

Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters

More information

Standard Deviation Estimator

Standard Deviation Estimator CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of

More information

Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

More information

Chapter 9. Two-Sample Tests. Effect Sizes and Power Paired t Test Calculation

Chapter 9. Two-Sample Tests. Effect Sizes and Power Paired t Test Calculation Chapter 9 Two-Sample Tests Paired t Test (Correlated Groups t Test) Effect Sizes and Power Paired t Test Calculation Summary Independent t Test Chapter 9 Homework Power and Two-Sample Tests: Paired Versus

More information

SENSITIVITY ANALYSIS AND INFERENCE. Lecture 12

SENSITIVITY ANALYSIS AND INFERENCE. Lecture 12 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this

More information

COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES.

COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. 277 CHAPTER VI COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. This chapter contains a full discussion of customer loyalty comparisons between private and public insurance companies

More information