An inferential procedure to use sample data to understand a population Procedures

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1 Hypothesis Test An inferential procedure to use sample data to understand a population Procedures Hypotheses, the alpha value, the critical region (z-scores), statistics, conclusion Two types of errors Effect size and power Two-tailed vs. one-tailed Two tailed tests are more conservative.

2 The Problems of Using z We don t know the population standard deviation.

3 t-tests

4 Hypothesis Test with t-statistics Procedures are similar to those using z- statistics, except Finding the critical region from the t-distribution table Using estimated standard errors to compute the scores

5 t Statistics Use estimated standard errors to replace standard errors Estimated standard errors are from sample statistics, rather than population parameter 2 SS SS Sample variance S (unbiased variance) n 1 df Estimated standard error 2 S S M n

6 t Statistics Use estimated standard errors to replace standard errors Estimated standard errors are from sample statistics, rather than population parameter. Degree of freedom Critical region

7

8 Example 9.1 Infants, even newborns, prefer to look at attractive faces compared to less attractive faces (Slater, et al., 1998). Subjects: infants from 1 to 6 days IV: Face in photo DV: time to look at a photo (in second) Method: showing two photographs of women's faces (one significantly more attractive than the other) 20 seconds in both M =13 (attractive face), SS = 72, n = 9

9 Steps Hypothesis H 0 : m attractive = 10 H 1 : m attractive 10 Critical region df = n - 1 = 8

10 Steps Calculation M=13, n=9, SS = 72 t =? Conclusion t s s M 2 M s m M 2 s n SS n 1 SS df

11 Effect size Cohen s d estimated d mean difference sample standard deviation M m s

12 Effect size Percentage of Variance Explained r 2 variability accounted for 2 total variability t 2 t df

13 Confidence Intervals for Estimating μ Alternative technique for describing effect size Estimates μ from the sample mean (M) Based on the reasonable assumption that M should be near μ Based on the estimated standard error of the mean (s M )

14 Confidence Intervals for Estimating μ (continue) Every sample mean has a corresponding t: t Rearrange the equations solving for μ: M m s M m M ts M

15 Distribution with df = 8 m M tsm * We are 80% confident that the average time to look at the pretty face is 13 seconds with an error seconds.

16 Report t-test Results The subjects averaged M = 13 seconds on the more attractive face with SD = 3.0. Statistical analysis indicated that the time spent on the attractive face was significantly more than would be expected by chance, t(8) = 3.00, p <.05, r 2 = 52.94%

17 How about this? Statistical analysis indicated that the time spent on the attractive face was significantly more than would be expected by chance, t(8) = 3.00, p <.05, r 2 = 52.94%. The subjects averaged M = 13 seconds on the plain side of the apparatus with SD = 3.00.

18 General Rule Report the descriptive statistics first. Mean, standard deviation, Present inferential statistics. z, t, F,

19 How about One Tailed?

20 This t-test Is Better, But It still requires the knowledge of the population mean. Often, we don t know the population mean. In practice, not just inferring population parameters based on samples, but also checking the mean difference between two populations based on two samples?

21 Two Different t-tests Checking whether scores from two groups are different? Between-subjects design Checking whether scores from different treatments are different Within-subjects design

22 t-test for Two Independent Samples

23 Between-Subjects Design

24 The t-test for Independent Measures Hypotheses The null hypothesis: no difference between two population H 0 : m 1 = m 2 The Alternative hypothesis: there is a mean difference H 1 : m 1 m 2

25 Set the Criteria The alpha value The critical region How to determine the df? We have two samples Two dfs The overall df is the sum of two dfs df = df 1 + df 2

26 Compute Statistics The t value sample mean hypothesized population mean t = estimated standard error t M m s M For independent measures sample mean diff. hypothesized population mean diff. t = estimated standard error sample mean diff. M 1 -M 2 = = estimated standard error S (M1-M2)

27 HOW TO COMPUTE S(M1-M2)?

28 Estimated Standard Error Measure of standard or average distance between sample statistic (M 1 -M 2 ) and the population parameter s ( M M ) s 1 2 n n s Unbiased only if n 1 = n 2

29 Pooled Variance Pooled variance (s p 2 provides an unbiased basis for calculating the standard error) s 2 p SS df 1 1 SS df 2 2 df df1 df2 ( n1 1) ( n2 1)

30 Make a Decision Rejecting the null hypothesis M 1 - M 2 0 The mean difference between sample represents the the mean difference between populations Not rejecting the null hypothesis No evidence to show the sample means are different

31 Ex Impact of TV time on high school performance IV: Sesame street DV: high school grade

32 Steps df = df1+ df2 = (n 1) + (n2 1) = = 18 a 0.01

33 Calculation (M 1 -M 2 ) (m 1 -m 2 ) t = s p s ( M 1 -M 2 ) s (M1-M2)

34 Effect Size Cohen s d Use s p r 2 r t t df

35 Assumptions for independentmeasures t-test Independent observation within each sample Two population are normal. Two population have equal variances. Homogeneity of variance Hartley s F-max test: F 2 s (largest) max 2 s (smallest)

36 t-test for Two Related Samples

37 Within-Subjects Design Subjects are compared with themselves Different conditions No groups to compare Data in different treatments are actually related To study the impact of treatments, we can still use t-test Slightly different from comparing independent samples

38 The Interest in Related Samples The mean difference between two population The difference between scores in different samples In related samples, we can put scores into pairs based on subjects This is impossible in independent samples The target: the score difference D = X 2 -X 1

39

40 Example 11.1 Neutral word vs. swearing Both conditions for each subject N S vs. S N

41 Procedures Hypotheses H 0 : m D = 0 H 1 : m D 0 The alpha value and critical region 5% as usual t statistic M t M D Estimated standard error D s m D sm D s 2 SS df s n 2

42 Ex M D = SS = s 2 = s M D

43 Uses and Assumptions of Repeated Measures t-test Fewer subjects Study change over time Reduce individual differences Order effects Independent observations within each treatment. Normal distribution.

44 T-test in statistics software SPSS R Excel

45 Homework of This Week Chapter 9: 20 Chapter 10: 22 Chapter 11: 22, 24

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