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


 Barry Lawrence
 1 years ago
 Views:
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 tdistribution The zdistribution The chisquare (χ 2 ) distribution The Fdistribution In our example, we can conduct a ztest 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 onetailed 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 timeperiod bias. Sample selection bias refers to the bias introduced by systematically excluding some members of the population according to a particular attribute. Timeperiod bias refers to the possibility that when we use a timeseries 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 decisionmaking 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 twotailed hypothesis tests are conducted. In a twotailed 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 twotailed test, two critical values exist one positive and one negative. For a twosided test at the 5% level of significance, we calculate the zvalues 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 pvalue. The pvalue (also known as probability value) is an alternative approach to hypothesis testing. The pvalue is the smallest level of significance at which the null hypothesis can be rejected. The pvalue is analogous to the teststatistic. High teststatistic means low pvalue and low teststatistic denotes high pvalue. The smaller the pvalue, the stronger is the evidence against the null hypothesis and in favor of the alternative hypothesis. We can use the pvalues in the hypothesis testing framework presented earlier as an alternative to using rejection points. If the pvalue 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 ttest and a ztest. The ttest is a hypothesis test using a statistic (tstatistic) that follows a tdistribution. 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 tstatistic. 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 tstatistic. In order to calculate the tstatistic we use the following formula: t n 1 = X μ 0 s/ n where t n 1 = tstatistic with n1 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 tdistribution table. Since this is a twotailed 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 ttest 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 ztest instead of a ttest for tests concerning a mean. The justifications for using a ztest are: In large samples, the sample mean should follow the normal distribution at least approximately, fulfilling the normality assumption of the ztest. In large samples, the difference between the rejection points for the ttest and ztest 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 ttest, 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 ttests 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 ttest 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 tdistribution 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 tvalue for 0.10/2 = 0.05 using df = 48. The tvalue 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 ttest 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 ttest. 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 teststatistic 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 ttest. 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 twotailed test we have a probability of 2.5% (0.025) in each tail. This critical value is determined from a ttable using a onetailed 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 chisquare 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 chisquare distribution is bounded below by 0. The graph below shows the shape of a chisquare 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 chisquare 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 chisquare 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 chisquare 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 chisquare 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 Ftest. An Ftest is the ratio of sample variances. For an Ftest 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 Fdistribution, like the chisquare distribution, is a family of asymmetrical distributions bounded from below by 0. Each Fdistribution 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 Ftests, 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 Ftest 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 Ftables. 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 Fstatistic: 18/13 = 1.4 df = 119 for the numerator and denominator α = 0.01 which means in each tail. From the Ftable: critical value = 1.6 Since the Fstat is less than the critical value, do not reject the null hypothesis. 5. Other Issues: Nonparametric Inference The hypothesistesting 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 twosided test and Formulations 2 and 3 are onesided 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 twosided 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 pvalue is the smallest level of significance at which the null hypothesis can be rejected. The smaller the pvalue, the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis. The pvalue approach to hypothesis testing does not involve setting a significance level; rather it involves computing a pvalue 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 tstatistic (zstatistic). In the unknown variance case, given large samples (generally, samples of 30 or more observations), the zstatistic may be used in place of the tstatistic because of the force of the central limit theorem. The tdistribution is a symmetrical distribution defined by a single parameter: degrees of freedom. Compared to the standard normal distribution, the tdistribution 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 ttest 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 ttest 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 tstatistic. 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 chisquare (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 Ftest (the ratio of the sample variances). The Fstatistic 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 Ftest, 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
MCQ TESTING OF HYPOTHESIS
MCQ TESTING OF HYPOTHESIS MCQ 13.1 A statement about a population developed for the purpose of testing is called: (a) Hypothesis (b) Hypothesis testing (c) Level of significance (d) Teststatistic MCQ
More informationLecture Topic 6: Chapter 9 Hypothesis Testing
Lecture Topic 6: Chapter 9 Hypothesis Testing 9.1 Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether a statement about the value of a population parameter should
More informationTwoSample TTest from Means and SD s
Chapter 07 TwoSample TTest from Means and SD s Introduction This procedure computes the twosample ttest and several other twosample tests directly from the mean, standard deviation, and sample size.
More informationStatistics for Management IISTAT 362Final Review
Statistics for Management IISTAT 362Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to
More informationTHE 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 informationMAT X Hypothesis Testing  Part I
MAT 2379 3X Hypothesis Testing  Part I Definition : A hypothesis is a conjecture concerning a value of a population parameter (or the shape of the population). The hypothesis will be tested by evaluating
More information6.1 The Elements of a Test of Hypothesis
University of California, Davis Department of Statistics Summer Session II Statistics 13 August 22, 2012 Date of latest update: August 20 Lecture 6: Tests of Hypothesis Suppose you wanted to determine
More informationChapter 11: Two Variable Regression Analysis
Department of Mathematics Izmir University of Economics Week 1415 20142015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions
More informationLAB 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 informationWooldridge, Introductory Econometrics, 4th ed. Multiple regression analysis:
Wooldridge, Introductory Econometrics, 4th ed. Chapter 4: Inference Multiple regression analysis: We have discussed the conditions under which OLS estimators are unbiased, and derived the variances of
More informationTesting: is my coin fair?
Testing: is my coin fair? Formally: we want to make some inference about P(head) Try it: toss coin several times (say 7 times) Assume that it is fair ( P(head)= ), and see if this assumption is compatible
More informationHypothesis Testing hypothesis testing approach formulation of the test statistic
Hypothesis Testing For the next few lectures, we re going to look at various test statistics that are formulated to allow us to test hypotheses in a variety of contexts: In all cases, the hypothesis testing
More information9Tests of Hypotheses. for a Single Sample CHAPTER OUTLINE
9Tests of Hypotheses for a Single Sample CHAPTER OUTLINE 91 HYPOTHESIS TESTING 91.1 Statistical Hypotheses 91.2 Tests of Statistical Hypotheses 91.3 OneSided and TwoSided Hypotheses 91.4 General
More informationMath 62 Statistics Sample Exam Questions
Math 62 Statistics Sample Exam Questions 1. (10) Explain the difference between the distribution of a population and the sampling distribution of a statistic, such as the mean, of a sample randomly selected
More informationStatistical foundations of machine learning
Machine learning p. 1/45 Statistical foundations of machine learning INFOF422 Gianluca Bontempi Département d Informatique Boulevard de Triomphe  CP 212 http://www.ulb.ac.be/di Machine learning p. 2/45
More informationI. Basics of Hypothesis Testing
Introduction to Hypothesis Testing This deals with an issue highly similar to what we did in the previous chapter. In that chapter we used sample information to make inferences about the range of possibilities
More informationOutline of Topics. Statistical Methods I. Types of Data. Descriptive Statistics
Statistical Methods I Tamekia L. Jones, Ph.D. (tjones@cog.ufl.edu) Research Assistant Professor Children s Oncology Group Statistics & Data Center Department of Biostatistics Colleges of Medicine and Public
More informationHypothesis Testing. Concept of Hypothesis Testing
Quantitative Methods 2013 Hypothesis Testing with One Sample 1 Concept of Hypothesis Testing Testing Hypotheses is another way to deal with the problem of making a statement about an unknown population
More informationSociology 6Z03 Topic 15: Statistical Inference for Means
Sociology 6Z03 Topic 15: Statistical Inference for Means John Fox McMaster University Fall 2016 John Fox (McMaster University) Soc 6Z03: Statistical Inference for Means Fall 2016 1 / 41 Outline: Statistical
More informationStudy 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 information93.4 Likelihood ratio test. NeymanPearson lemma
93.4 Likelihood ratio test NeymanPearson lemma 91 Hypothesis Testing 91.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental
More informationInferential Statistics
Inferential Statistics Sampling and the normal distribution Zscores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
More informationRegression Analysis. Pekka Tolonen
Regression Analysis Pekka Tolonen Outline of Topics Simple linear regression: the form and estimation Hypothesis testing and statistical significance Empirical application: the capital asset pricing model
More informationNull Hypothesis H 0. The null hypothesis (denoted by H 0
Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property
More informationSimple Linear Regression Chapter 11
Simple Linear Regression Chapter 11 Rationale Frequently decisionmaking situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related
More informationHypothesis Testing. Hypothesis Testing CS 700
Hypothesis Testing CS 700 1 Hypothesis Testing! Purpose: make inferences about a population parameter by analyzing differences between observed sample statistics and the results one expects to obtain if
More informationTest of Hypotheses. Since the NeymanPearson approach involves two statistical hypotheses, one has to decide which one
Test of Hypotheses Hypothesis, Test Statistic, and Rejection Region Imagine that you play a repeated Bernoulli game: you win $1 if head and lose $1 if tail. After 10 plays, you lost $2 in net (4 heads
More informationHypothesis Testing with One Sample. Introduction to Hypothesis Testing 7.1. Hypothesis Tests. Chapter 7
Chapter 7 Hypothesis Testing with One Sample 71 Introduction to Hypothesis Testing Hypothesis Tests A hypothesis test is a process that uses sample statistics to test a claim about the value of a population
More informationChapter 8. Professor Tim Busken. April 20, Chapter 8. Tim Busken. 8.2 Basics of. Hypothesis Testing. Works Cited
Chapter 8 Professor April 20, 2014 In Chapter 8, we continue our study of inferential statistics. Concept: Inferential Statistics The two major activities of inferential statistics are 1 to use sample
More informationModule 5 Hypotheses Tests: Comparing Two Groups
Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this
More informationUnit 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 informationHypothesis Testing. Dr. Bob Gee Dean Scott Bonney Professor William G. Journigan American Meridian University
Hypothesis Testing Dr. Bob Gee Dean Scott Bonney Professor William G. Journigan American Meridian University 1 AMU / BonTech, LLC, JourniTech Corporation Copyright 2015 Learning Objectives Upon successful
More informationGeneral Procedure for Hypothesis Test. Five types of statistical analysis. 1. Formulate H 1 and H 0. General Procedure for Hypothesis Test
Five types of statistical analysis General Procedure for Hypothesis Test Descriptive Inferential Differences Associative Predictive What are the characteristics of the respondents? What are the characteristics
More informationChapter 7. Section Introduction to Hypothesis Testing
Section 7.1  Introduction to Hypothesis Testing Chapter 7 Objectives: State a null hypothesis and an alternative hypothesis Identify type I and type II errors and interpret the level of significance Determine
More informationStatistical Concepts and Market Return
Statistical Concepts and Market Return 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 2 2. Some Fundamental Concepts... 2 3. Summarizing Data Using Frequency Distributions...
More informationA Logic of Prediction and Evaluation
5  Hypothesis Testing in the Linear Model Page 1 A Logic of Prediction and Evaluation 5:12 PM One goal of science: determine whether current ways of thinking about the world are adequate for predicting
More informationWording of Final Conclusion. Slide 1
Wording of Final Conclusion Slide 1 8.3: Assumptions for Testing Slide 2 Claims About Population Means 1) The sample is a simple random sample. 2) The value of the population standard deviation σ is known
More informationTwoSample TTests Assuming Equal Variance (Enter Means)
Chapter 4 TwoSample TTests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when the variances of
More informationRecall this chart that showed how most of our course would be organized:
Chapter 4 OneWay 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 informationDescriptive Statistics
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
More informationNCSS Statistical Software
Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, twosample ttests, the ztest, the
More informationMINITAB ASSISTANT WHITE PAPER
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. OneWay
More informationIntroduction to Hypothesis Testing. Copyright 2014 Pearson Education, Inc. 91
Introduction to Hypothesis Testing 91 Learning Outcomes Outcome 1. Formulate null and alternative hypotheses for applications involving a single population mean or proportion. Outcome 2. Know what Type
More informationStatistiek I. ttests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. John Nerbonne 1/35
Statistiek I ttests John Nerbonne CLCG, Rijksuniversiteit Groningen http://wwwletrugnl/nerbonne/teach/statistieki/ John Nerbonne 1/35 ttests To test an average or pair of averages when σ is known, we
More informationStatistical Inference and ttests
1 Statistical Inference and ttests Objectives Evaluate the difference between a sample mean and a target value using a onesample ttest. Evaluate the difference between a sample mean and a target value
More informationPASS Sample Size Software. Linear Regression
Chapter 855 Introduction Linear regression is a commonly used procedure in statistical analysis. One of the main objectives in linear regression analysis is to test hypotheses about the slope (sometimes
More informationLesson 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 informationAP Statistics 2001 Solutions and Scoring Guidelines
AP Statistics 2001 Solutions and Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use
More informationEconometrics The Multiple Regression Model: Inference
Econometrics The Multiple Regression Model: João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, March 2011 1 / 24 in
More informationSTT 430/630/ES 760 Lecture Notes: Chapter 6: Hypothesis Testing 1. February 23, 2009 Chapter 6: Introduction to Hypothesis Testing
STT 430/630/ES 760 Lecture Notes: Chapter 6: Hypothesis Testing 1 February 23, 2009 Chapter 6: Introduction to Hypothesis Testing One of the primary uses of statistics is to use data to infer something
More informationThe calculations lead to the following values: d 2 = 46, n = 8, s d 2 = 4, s d = 2, SEof d = s d n s d n
EXAMPLE 1: Paired ttest and tinterval DBP Readings by Two Devices The diastolic blood pressures (DBP) of 8 patients were determined using two techniques: the standard method used by medical personnel
More informationSUBMODELS (NESTED MODELS) AND ANALYSIS OF VARIANCE OF REGRESSION MODELS
1 SUBMODELS (NESTED MODELS) AND ANALYSIS OF VARIANCE OF REGRESSION MODELS We will assume we have data (x 1, y 1 ), (x 2, y 2 ),, (x n, y n ) and make the usual assumptions of independence and normality.
More informationStatistics 104: Section 7
Statistics 104: Section 7 Section Overview Reminders Comments on Midterm Common Mistakes on Problem Set 6 Statistical Week in Review Comments on Midterm Overall, the midterms were good with one notable
More informationTwoSample TTests Allowing Unequal Variance (Enter Difference)
Chapter 45 TwoSample TTests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when no assumption
More informationUnit 29 ChiSquare GoodnessofFit Test
Unit 29 ChiSquare GoodnessofFit Test Objectives: To perform the chisquare hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni
More informationMath 10 MPS Homework 6 Answers to additional problems
Math 1 MPS Homework 6 Answers to additional problems 1. What are the two types of hypotheses used in a hypothesis test? How are they related? Ho: Null Hypotheses A statement about a population parameter
More information41.2. Tests Concerning asinglesample. Introduction. Prerequisites. Learning Outcomes
Tests Concerning asinglesample 41.2 Introduction This Section introduces you to the basic ideas of hypothesis testing in a nonmathematical way by using a problem solving approach to highlight the concepts
More informationCHAPTER 15: Tests of Significance: The Basics
CHAPTER 15: Tests of Significance: The Basics The Basic Practice of Statistics 6 th Edition Moore / Notz / Fligner Lecture PowerPoint Slides Chapter 15 Concepts 2 The Reasoning of Tests of Significance
More informationLecture 1: t tests and CLT
Lecture 1: t tests and CLT http://www.stats.ox.ac.uk/ winkel/phs.html Dr Matthias Winkel 1 Outline I. z test for unknown population mean  review II. Limitations of the z test III. t test for unknown population
More informationChapter 08. Introduction
Chapter 08 Introduction Hypothesis testing may best be summarized as a decision making process in which one attempts to arrive at a particular conclusion based upon "statistical" evidence. A typical hypothesis
More informationt Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon
ttests 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 informationUCLA STAT 13 Statistical Methods  Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates
UCLA STAT 13 Statistical Methods  Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally
More informationUnit 24 Hypothesis Tests about Means
Unit 24 Hypothesis Tests about Means Objectives: To recognize the difference between a paired t test and a twosample t test To perform a paired t test To perform a twosample t test A measure of the amount
More information13 TwoSample T Tests
www.ck12.org CHAPTER 13 TwoSample T Tests Chapter Outline 13.1 TESTING A HYPOTHESIS FOR DEPENDENT AND INDEPENDENT SAMPLES 270 www.ck12.org Chapter 13. TwoSample T Tests 13.1 Testing a Hypothesis for
More informationSelected Nonparametric and Parametric Statistical Tests for TwoSample Cases 1
Selected Nonparametric and Parametric Statistical Tests for TwoSample Cases The Tstatistic is used to test differences in the means of two groups. The grouping variable is categorical and data for the
More informationHypothesis Testing. Bluman Chapter 8
CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 81 Steps in Traditional Method 82 z Test for a Mean 83 t Test for a Mean 84 z Test for a Proportion 85 2 Test for
More informationSampling and Hypothesis Testing
Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus
More informationPower & Effect Size power Effect Size
Power & Effect Size Until recently, researchers were primarily concerned with controlling Type I errors (i.e. finding a difference when one does not truly exist). Although it is important to make sure
More information1 Confidence intervals
Math 143 Inference for Means 1 Statistical inference is inferring information about the distribution of a population from information about a sample. We re generally talking about one of two things: 1.
More information3. Nonparametric methods
3. Nonparametric methods If the probability distributions of the statistical variables are unknown or are not as required (e.g. normality assumption violated), then we may still apply nonparametric tests
More informationHypothesis Testing  II
3σ 2σ +σ +2σ +3σ Hypothesis Testing  II Lecture 9 0909.400.01 / 0909.400.02 Dr. P. s Clinic Consultant Module in Probability & Statistics in Engineering Today in P&S 3σ 2σ +σ +2σ +3σ Review: Hypothesis
More informationChapter 8 Introduction to Hypothesis Testing
Chapter 8 Student Lecture Notes 81 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate
More informationAP Statistics 1998 Scoring Guidelines
AP Statistics 1998 Scoring Guidelines These materials are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced Placement
More informationNCSS Statistical Software
Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, twosample ttests, the ztest, the
More informationHow to Conduct a Hypothesis Test
How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some
More informationAP Statistics 2007 Scoring Guidelines
AP Statistics 2007 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to college
More information4. Introduction to Statistics
Statistics for Engineers 41 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation
More informationNull and Alternative Hypotheses. Lecture # 3. Steps in Conducting a Hypothesis Test (Cont d) Steps in Conducting a Hypothesis Test
Lecture # 3 Significance Testing Is there a significant difference between a measured and a standard amount (that can not be accounted for by random error alone)? aka Hypothesis testing H 0 (null hypothesis)
More informationChiSquare Distribution. is distributed according to the chisquare distribution. This is usually written
ChiSquare Distribution If X i are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable is distributed according to the chisquare distribution. This
More informationWeek 7 Lecture: Twoway Analysis of Variance (Chapter 12) Twoway ANOVA with Equal Replication (see Zar s section 12.1)
Week 7 Lecture: Twoway Analysis of Variance (Chapter ) We can extend the idea of a oneway ANOVA, which tests the effects of one factor on a response variable, to a twoway ANOVA which tests the effects
More informationWe have already discussed hypothesis testing in study unit 13. In this
14 study unit fourteen hypothesis tests applied to means: two related samples We have already discussed hypothesis testing in study unit 13. In this study unit we shall test a hypothesis empirically in
More informationStatistical Inference
Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this
More informationStatistics: revision
NST 1B Experimental Psychology Statistics practical 5 Statistics: revision Rudolf Cardinal & Mike Aitken 3 / 4 May 2005 Department of Experimental Psychology University of Cambridge Slides at pobox.com/~rudolf/psychology
More informationCHAPTERS 46: Hypothesis Tests Read sections 4.3, 4.5, 5.1.5, Confidence Interval vs. Hypothesis Test (4.3):
CHAPTERS 46: Hypothesis Tests Read sections 4.3, 4.5, 5.1.5, 6.1.3 Confidence Interval vs. Hypothesis Test (4.3): The purpose of a confidence interval is to estimate the value of a parameter. The purpose
More informationLecture 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 informationChapter Five. Hypothesis Testing: Concepts
Chapter Five The Purpose of Hypothesis Testing... 110 An Initial Look at Hypothesis Testing... 112 Formal Hypothesis Testing... 114 Introduction... 114 Null and Alternate Hypotheses... 114 Procedure for
More informationChapter 4: Constrained estimators and tests in the multiple linear regression model (Part II)
Chapter 4: Constrained estimators and tests in the multiple linear regression model (Part II) Florian Pelgrin HEC SeptemberDecember 2010 Florian Pelgrin (HEC) Constrained estimators SeptemberDecember
More informationChapter 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 information3.4 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 information7 Hypothesis testing  one sample tests
7 Hypothesis testing  one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X
More informationBox plots & ttests. Example
Box plots & ttests Box Plots Box plots are a graphical representation of your sample (easy to visualize descriptive statistics); they are also known as boxandwhisker diagrams. Any data that you can
More informationData Analysis and Uncertainty Part 3: Hypothesis Testing/Sampling
Data Analysis and Uncertainty Part 3: Hypothesis Testing/Sampling Instructor: Sargur N. University at Buffalo The State University of New York srihari@cedar.buffalo.edu Topics 1. Hypothesis Testing 1.
More informationDevelop hypothesis and then research to find out if it is true. Derived from theory or primary question/research questions
Chapter 12 Hypothesis Testing Learning Objectives Examine the process of hypothesis testing Evaluate research and null hypothesis Determine one or twotailed tests Understand obtained values, significance,
More informationHypothesis Testing for Two Variances
Hypothesis Testing for Two Variances The standard version of the twosample t test is used when the variances of the underlying populations are either known or assumed to be equal In other situations,
More informationFor eg: The yield of a new paddy variety will be 3500 kg per hectare scientific hypothesis. In Statistical language if may be stated as the random
Lecture.9 Test of significance Basic concepts null hypothesis alternative hypothesis level of significance Standard error and its importance steps in testing Test of Significance Objective To familiarize
More informationHypothesis tests: the ttests
Hypothesis tests: the ttests Introduction Invariably investigators wish to ask whether their data answer certain questions that are germane to the purpose of the investigation. It is often the case that
More informationComparing 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 informationChapter 1 Hypothesis Testing
Chapter 1 Hypothesis Testing Principles of Hypothesis Testing tests for one sample case 1 Statistical Hypotheses They are defined as assertion or conjecture about the parameter or parameters of a population,
More informationChapter 7 Part 2. Hypothesis testing Power
Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship
More informationOn Importance of Normality Assumption in Using a TTest: One Sample and Two Sample Cases
On Importance of Normality Assumption in Using a TTest: One Sample and Two Sample Cases Srilakshminarayana Gali, SDM Institute for Management Development, Mysore, India. Email: lakshminarayana@sdmimd.ac.in
More information