Acknowledge error Smaller samples, less spread

Transcription

1 Hypothesis Testing with t Tests Al Arlo Clark-Foos kf

2 Using Samples to Estimate Population Parameters Acknowledge error Smaller samples, less spread s = Σ ( X M N 1 ) 2

3 The t Statistic Indicates the distance of a sample mean from a population mean in terms of the standard error s s M = N

4 Hypothesis Tests: Single-Sample Sample t Tests Guinness is the best beer available, it does not need advertising as its quality will sell it, and those who do not drink it are to be sympathized with rather than advertised to to.

5 Hypothesis Tests: Single-Sample Sample t Tests Hypothesis test in which we compare data from one sample to a population for which we know the mean but not the standard deviation. Degrees of Freedom: The number of scores that are free to vary when estimating a population parameter from a sample df = N 1 (for a Single-Sample t Test)

6 One Tailed vs. Two Tailed Tests

7 Six Steps for Hypothesis Testing 1. Identify 2. State the hypotheses 3. Characteristics of the comparison distribution 4. Critical values 5. Calculate 6. Decide

8 Single-Sample Sample t Test: Example Participation in therapy sessions Contract to attend 10 sessions μ = 4.6 Sample: 6, 6, 12, 7, 8

9 Single-Sample Sample t Test: Example 1. Identify Pop 1: All clients who sign contract Pop 2: All clients who do not sign contract Distribution of means Test: Population mean is known but not standard deviation single-sample t test Assumptions

10 Single-Sample Sample t Test: Example 2. State the null and research hypotheses H 0 : Clients who sign the contract will attend the same number of sessions as those who do not sign the contract. μ 1 = μ 2 H 1 : Clients who sign the contact will attend a different number of sessions than those who do not sign the contract. μ 1 μ 2

11 Single-Sample Sample t Test: Example 3. Determine characteristics of comparison distribution μ M = μ = 4.6, 2 Σ ( X M ) s M = s = = N 1 s s = = M N 5 = 1.114

12 Single-Sample Sample t Test: Example 4. Determine critical values or cutoffs df = N-1 = 5-1 = 4, p =.05 t = ± 2.776

13 Single-Sample Sample t Test: Example 5. Calculate the test statistic t ( M μ s M = s M ) = ( ) = Mk Make a decision ii > 2.776, we reject the null Clients who sign a contract will attend more sessions than those who do not sign a contract.

14 Reporting Results in APA Format 1. Write the symbol for the test statistic (e.g., z or t) 2. Write the degrees of freedom in parentheses 3. Write an equal sign and then the value of the test statistic (2 decimal places) 4. Write a comma and then whether the p value associated with the test statistic was less than or greater than the cutoff p value of.05 t(4) = 2.87, p <.05

15 Hypothesis tests for two samples Conditions must be equivalent (controlled) for a fair test Beer Tasting: Between-groups vs. Withingroups

16 Paired-Samples t Test Used to compare 2 means for a within- groups design, a situation in which every participant p is in both samples (paired/dependent) Difference Scores: X 1 Y 1, X 2 Y 2,

17 Paired-Samples t Test 1. Identify 2. State null & research hyp. 3. Determine characteristics of comparison distribution H 0 : μ 1 = μ 2

18 Paired-Samples t Test 4. Determine cutoffs 5. Calculate test statistic t = 6. Mk Make a decision ii ( M μm ) s M

19 Independent Samples t Test Used to compare 2 means for a between- groups design, a situation in which each participant p is assigned to only one condition.

20 Independent Samples t Test What percentage of cartoons do men and woman consider funny? Women: Men:

21 Independent Samples t Test 1. Identify Population 1: Women exposed to humorous cartoons Population 2: Men exposed to humorous cartoons Distribution: Differences between means Not mean differences Assumptions are the same We meet one here: DV is interval/scale

22 Independent Samples t Test 2. State null and research hypotheses H 0 : μ 1 = μ 2 Women will categorize the same number of cartoons as funny as will men. H 1 : μ 1 μ 2 Women will categorize e a different number of cartoons funny as as will men.

23 Independent Samples t Test 3. Determine characteristics of comparison distribution H 0 : μ 1 = μ 2 Measure of spread: a) Calculate variance for each sample b) Pool variances, accounting for sample size c) Convert from squared standard deviation to squared standard error d) Add the two variances e) Take square root to get estimated standard error for distribution of differences between means.

24 Independent Samples t Test a) Calculate variance for each sample s 2 2 Σ( X M) X = = = N s 2 2 Σ ( Y M ) Y = = = N

25 Independent Samples t Test Degrees of Freedom df X = N 1 = 4 1 = 3 df Y = N 1 = 5 1 = 4 df Total = df X + df Y = = 7

26 Pooled Variance b) Pool variances, accounting for sample size Weighted average of the two estimates of variance one from each sample that are calculated when conducting an independent d samples t test. s = df s + df s 2 X 2 Y 2 Pooled X Y dftotal dftotal = s Pooled s = = Pooled

27 Independent Samples t Test c) Convert from squared standard deviation to squared standard error 2 s Pooled = s Pooled 2 spooled sm = = = s X M = = = Y N 4 N d) Add the two variances s = s + s = = Difference M M X Y e) Take square root to get estimated standard error for distribution of differences between means. s Difference = s = = Difference

28 Independent Samples t Test 4. Determine critical values df Total = 7 p =.05 t = ± Calculate a test statistic t ( M M ) ( μ μ ) ( M M ) = = s s X Y X Y X Y Difference Difference t ( ) = =

29 Independent Samples t Test 6. Make a decision Fail to reject null hypothesis Fail to reject null hypothesis Men and women find cartoons equally humorous