Hypothesis Testing. Experimental Hypotheses. Revision of Important Concepts. New Definition: Sampling Distribution. Hypothesis Testing
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1 Revision of Important Concepts Hypothesis Testing Week 7 Statistics Dr. Sancho Moro Remember that we used z scores to compute probabilities related to the normal (or gausssian) distribution. Confidence intervals tell us the probable limits of finding an observation around the average. The lower & upper limit for 95% CI are respectively z= and z= This week we will learn to compute probabilities to test the probability that the hypothesis we make about data are true or false. 1 2 New Definition: Sampling Distribution Sampling Distribution of the mean: properties Most of the times we only measure one sample of reduced size (n) instead of all the elements (N) of a population. eg. sample the intelligence scores of 10 students in one university. The sampling distribution is the group of all possible samples and tells us what degree of sample-to-sample variability we can expect. A sampling distribution can be constructed by taking an infinite number of random samples of size n from a population. The sampling distribution is a theoretical concept in statistics but it could be be obtained empirically by repeating the measured sample of size n infinite times (or simulating it in a computer). 3 The mean of the sampling distribution is equal to the true mean (µ) of the population. The standard deviation of the sampling distribution is also known as the standard error of the mean (SEM) = x (SEM ) n The SEM determines the sample-to-sample variability. When the sample size n=1 each random sample contains one score of the original population so the SEM equals the variability of the population (). When the sample size is infinite, each sample contains all the scores in the population and the error is zero. 4 Experimental Hypotheses Experimental hypotheses describe the predicted outcome or relationship we may or may not find in an experiment. Hypothesis Testing Examples: 1- financial crisis in US may affect European markets (or not?) 2- spending more in internet advertising may increase sales (or not?) 3- taking vitamin supplements may protect you against colds (or not?) Inferential Statistics 5 Statistical analysis of the data will inform or guide you to choose the best decision or at leats to determine whether there is a relationship between two (or more) variables 6 1
2 Null Hypothesis (H 0 ) The null hypothesis describes the population parameters that the sample data represent if the predicted relationship does not exist. Alternative Hypothesis (H 1 ) The alternative hypothesis describes the population parameters that the sample data represent if the predicted relationship exists. Example: a financial crisis in US does not affect European markets Example: a financial crisis in US does affect European markets 7 8 Experimental design and steps during hypothesis testing Example: When there is a clear effect or relationship we reject the Null Hypothesis (=no effect). 1- Begin with a hypothesis about the data e.g. vitamin C protects against colds 2- Design the experiment to collect data e.g. 2 groups of people: vitc vs placebo (=no vitamin C) How many colds per person during a 5-year period? 3- Setup the null hypothesis (Ho= vitamin C has no effect. The two groups are equal) 3- Retain H 0 or Reject it and accept the alternative hypothesis H 1 depending on the probability of the statistical test Example: When there is not an effect (or is very weak) we accept the Null Hypothesis (=no effect). Types of Statistical TESTS Z-test : when sigma is known t-test : when sigma is unknown
3 The z-test The z-test is the procedure for comparing the z- score obtained for one sample mean with the sampling distribution of means. With a z-test we can determine whether our sample is different (H 1 ) or not (H 0 ) from the population z Computing z The z-score is computed using the same formula as before where obt µ = = N Z-test: Sampling Distribution Showing the Region of Rejection for α = 0.05 in a Two-tailed Test Example: Suppose that the average normal intelligence (IQ) score is µ =100 (µ is the true mean of the total population) with std dev =10. If we take one sample of 50 individuals and we obtain a mean IQ of x 1 = 70 then z 1 =-3. The probability of obtaining such sample is less than 5%. We can then reject the null hypothesis and say that our sample is significantly smaller (different) from the normal population. Possible Results of Rejecting or Retaining H 0 exercise: reproduce this table in a blank page and explain the possible errors in decision making. 16 Rejecting H 0 When the obtained z obt is below or above a critical value z crit,we rejecth 0 and accept H a. The value z crit depends on alpha. For alpha=0.05 z crit = ± 1.96 When we reject H 0 and accept H a we say the difference of the data from the population are significant at the chosen level of alpha (usually 0.05 or 0.01). The sample is likely to differ from the population. Failing to Reject H 0 When the z obt does not fall beyond the critical value, the statistic does not lie within the region of rejection, so we do not reject H 0 When we fail to reject H 0 we say the results are nonsignificant. Nonsignificant indicates that the sample does not differ from the population
4 Interpreting Nonsignificant Results When we fail to reject H 0, we do not prove that H 0 is true. A difference may still exist but our measurements did not reveal it (eg. sample size too small, not a sensitive test etc.) Is impossible to prove H 0 is true. But we can report that we have no convincing evidence that a difference or relationship exist 19 Assumptions of the z-test 1. We have randomly selected one sample 2. The variable is at least approximately normally distributed in the population and involves an interval or ratio scale 3. We know the mean of the population of raw scores under some other condition of the independent variable 4. We know the true standard deviation of the population ( ) described by the null hypothesis 20 One vs Two-Tailed tests A two-tailed test is used to test whether a sample differs from the mean and we do not predict the direction in which scores will change. We know a posteriori (=after the experiment) that our sample is higher or lower than the mean. A one-tailed test is used to test whether the sample is different from the mean but we expect a priori (before the experiment) the direction of change (e.g. training improves performance, it nevers degrades performance). Setting up for a Two-Tailed Test 1.Choose alpha. Common values are 0.05 and Locate the region of rejection. For a twotailed test, this will involve defining an area in both tails of the sampling distribution. 3.Determine the critical value. Using the chosen alpha, find the z crit value that gives the appropriate region of rejection. Most of the times is safer to use two-tailed tests because making a priori predictions can result in errors Z-test: Sampling Distribution Showing the Region of Rejection for α = 0.05 in a Two-tailed Test Two-Tailed Hypotheses In a two-tailed test, the null hypothesis states that the population mean equals a given value. For example, H 0 : µ = 100. In a two-tailed test, the alternative hypothesis states that the population mean does not equal the same given value as in the null hypothesis. For example, H a : µ
5 Case1: One-Tailed increase The One-Tailed Test In a one-tailed test, if it is hypothesized that the independent variable causes an increase in scores, then the null hypothesis is that the population mean is less than or equal to a given value and the alternative hypothesis is that the population mean is greater than the same value. For example: H 0 : µ 50 H a : µ > A Sampling Distribution Showing the Region of Rejection for a One-tailed Test of Whether Scores Increase Case 2: One-Tailed decrease In a one-tailed test, if it is hypothesized that the independent variable causes a decrease in scores, then the null hypothesis is that the population mean is greater than or equal to a given value and the alternative hypothesis is that the population mean is less than the same value. For example: H 0 : µ 50 H a : µ < A Sampling Distribution Showing the Region of Rejection for a One-tailed Test of Whether Scores Decrease Choosing One-Tailed Versus Two-Tailed Tests Use a one-tailed test only when confident of the direction in which the dependent variable scores will change. When in doubt, use a two-tailed test
6 Summary of the z-test 1.Determine the experimental hypotheses and create the statistical hypothesis 2. Compute and compute z obt 3. Set up the sampling distribution 4. Compare zobt to z crit (for a two-tailed you can find zcrit with normsinv(alpha/2) Errors in Statistical Decision Making Type I Errors (p= α, alpha) A Type I error is defined as rejecting H 0 when H 0 is true A Type I error we conclude that there is a difference but in fact there is not. The theoretical probability of a Type I error equals α Type II Errors (p=β, beta) A Type II error is defined as retaining H 0 when H 0 is false (and H a is true) In a Type II error, the sample mean is so close to the µ described by H 0 that we conclude that the predicted relationship does not exist when it really does. It can be avoided by increasing sample size. The probability of a Type II error is β Power Summary The goal of research is to reject H 0 when H 0 is false The probability of rejecting H 0 when it is false is called power. Power increases with sample size and equals 1- β It s not enough to show that there is a significant difference, we need to take samples large enough to have sufficient power (usually 80-90%) so that the probability of a type II error β is only 10-20%
7 Example Use the following data set and conduct a two-tailed z-test to determine if is significantly different from a population with µ = 11 and standard deviation of = = N Example 1. H 0 : µ = 11; H a : µ Choose α = Reject H 0 if z obt > or if z obt < = z µ = = obt = Example Since z obt lies within the rejection region, we reject H 0 and accept H a. Therefore, we conclude that the mean of our sample is significantly different µ 11. Exercise Systolic blood pressure for males 35 to 44 years of age has mean 128 and standard deviation. The medical directors of a large company looks at the medical records of 72 executives in this age group and finds that the mean systolic blood pressure in this sample is Is this evidence that the company s executives have a different male blood pressure from the general population?
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