### Transcription of Hypothesis Testing with z Tests - University of Michigan

1 **Hypothesis** **Testing** . **with** Z **Tests** . Arlo Clark-Foos Review: Standardization Allows us to easily see how one score (or sample). compares **with** all other scores (or a population). CDC Example: p Jessica Jessica is 15 years old and in. tall For 15 year old girls, = , = z=. ( X ) ( ). = = CDC Example: p Jessica 1. Percentile: How many 15 year old girls are shorter than Jessica? 50% + = CDC Example: p Jessica 2. What percentage of 15 year old girls are taller than Jessica? 50% - OR 100% - = CDC Example: p Jessica 3. What percentage of 15 year old girls are as far from the mean as Jessica (tall or short)? % + = CDC Example: p Manuel Manuel is 15 years old and in. tall For 15 year old boys, = 67, = z=. ( X ) ( 67). = = Consult z table for CDC Example: p Manuel 1. Percentile Negative z, below mean: 50% - = CDC Example: p Manuel 2.

2 Percent Above Manuel 100% - = %. CDC Example: p Manuel 3. Percent as extreme as Manuel + = Percentages g to z Scores SAT Example: = 500, = 100. You find out you are at 63rd percentile Consult z table for 13% z = .33. 33. X = .33(100) + 500 = 533. z Table and Distribution of Means Remember that if we use distribution of means, we are using a sample and need to use standard error. How do UMD students measure up on the GRE? = 554, = 99 M = 568, N = 90. 99. M = = 554 M = = = N 90. UMD & GRE Example p z=. (M M ) (568 554). = = M Consult z table for z = %. 50% + 40 99% = 90 99%. Assumptions p of **Hypothesis** yp **Testing** g 1. 1. The DV is measured on an interval scale 2. Participants are randomly selected 3. 3. The distribution of the population is approximately normal Robust: These hyp. hyp **Tests** are those that produce fairly accurate results even when the data suggest that the ppopulation p might g not meet some of the assumptions.

3 P Parametric **Tests** Nonparametric **Tests** Assumptions p of **Hypothesis** yp **Testing** g **Testing** g Hypotheses yp (6. ( Steps). p). 1. 1. Identify the population, comparison distribution, inferential test, and assumptions 2. 2. State the null and research hypotheses 3. Determine characteristics of the comparison distribution Whether this is the whole population or a control group, we need to find the mean and some measure of spread (variability). **Testing** g Hypotheses yp (6. ( Steps). p). 4. 4. Determine critical values or cutoffs How extreme must our data be to reject the null? Critical Values: Test statistic values beyond which we will reject the null **Hypothesis** (cutoffs). p levels ( ): Probabilities used to determine the critical value 5. Calculate test statistic ( , z statistic). 6. Make a decision Statistically Significant: Instructs us to reject the null **Hypothesis** because the pattern in the data differs from what h we would ld expect bby chance h alone.

4 L The z Test: An Example p = , = , M = , N = 97. 1. Populations, distributions, and assumptions Populations: 1. All students at UMD who have taken the test (not just our sample). 2. All students nationwide who have taken the test Distribution: Sample distribution of means Test & Assumptions: z test 1. Data are interval 2. 2. We hope random selection (otherwise, (otherwise less generalizable). 3. Sample size > 30, therefore distribution is normal The z Test: An Example p 2. 2. State the null and research hypotheses H0: 1 2. H1: 1 > 2. OR. H0: 1 = 2. H1: 1 2. The z Test: An Example p 3. 3. Determine characteristics of comparison distribution. Population: p = ,, = Sample: M = , N = 97. M = = = N 97. The z Test: An Example p 4. 4. Determine critical value (cutoffs). In Behavioral Sciences, we use p = .05. p = .05 = 5% in each tail 50% - = Consult z table for z = The z Test: An Example p 5.)

5 5. Calculate test statistic z=. (M M ) ( ). = = M 6. M k a Decision Make D ii Increasing g Sample p Size By increasing sample size, one can increase the value of the test statistic, thus increasing probability of finding g a significant g effect Increasing g Sample p Size Example: Psychology GRE scores Population: = 554, = 99. Sample: M = 568, N = 90. 99. M = = = N 90. z=. (M M ) (568 554). = = M Increasing g Sample p Size Example: Psychology GRE scores Population: = 554, = 99. Sample: M = 568, N = 200. 99. M = = = N 200. z=. (M M ) (568 554). = = M