Sociology 5811: Lecture 11: T-Tests for Difference in Means Copyright © 2005 by Evan Schofer Do not copy or distribute without permission

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Hypothesis Testing Definition: Two-tailed test: A hypothesis test in which the  -area of interest falls in both tails of a Z or T distribution. Example: H0: m = 4; H1: m ≠ 4 Definition: One-tailed test: A hypothesis test in which the  -area of interest falls in just one tail of a Z or T distribution. Example: H0:  > or = 4; H1:  < 4 Example: H0:  4 This is called a “directional” hypothesis test.

Hypothesis Tests About Means A one-tailed test: H1:  < 4 Entire  -area is on left, as opposed to half (  /2) on each side. SO: the critical t-value changes. 4

Hypothesis Tests About Means T-value changes because the alpha area (e.g., 5%) is all concentrated in one size of distribution, rather than split half and half. One tail vs. Two-tail:  

Looking Up T-Tables How much does the 95% t-value change when you switch from a 2-tailed to 1-tailed test? Two-tailed test (20 df): t=2.086 One-tailed test (20df) t=1.725

Review: Hypothesis Tests T-value changes because the alpha area (e.g., 5%) is all concentrated in one size of distribution, rather than split half and half. One tail vs. Two-tail:   Concentrating the alpha area in one tail reduces the critical T-value needed to reject H0

Tests for Differences in Means A more useful application: Two groups Issue: Whenever you compare two groups, you’ll observe different means Question: Is the difference due to the particular sample, but populations have the same mean? Or can we infer that the populations are different? Example: Test scores for 20 boys, 20 girls in the 2nd grade Y-bar boys = 72.75, s = 8.80 Y-bar girls = 78.20, s = 9.55

Example: Boy’s Test Scores

Example: Girl’s Test Scores

Differences in Means Inferential statistics can help us determine if group population means are really different The hypotheses we must consider are: An alternate (equivalent) formulation:

Differences in Means Issue: How likely is it to draw means with a difference of -5.45, if the difference in population means is really 0? If common, we can’t conclude anything If rare, we can conclude that the population means differ.

Strategy for Mean Difference We never know true population means So, we never know true value of difference in means So, we don’t know if groups really differ If we can figure out the sampling distribution of the difference in means… We can guess the range in which it typically falls If it is improbable for the sampling distribution to overlap with zero, then the population means probably differ An extension of the Central Limit Theorem provides information necessary to do calculations!

Strategy for Mean Difference Logic of tests about differences in means: The C.L.T. defines how sample means (Y-bars) cluster around the true mean: The center and width of the sampling distribution This tells us the range of values where Y-bars fall For any two means, the difference will also fall in a certain range: Group 1 means range from about 6.0 to 8.0 Group 2 means range from about 1.0 to 2.0 Estimates of the difference in means will range from about 4.0 to 7.0!

A Corollary of the C.L.T. Visually: If each group has a sampling distribution of the mean, the difference does too: Sampling distribution of differences in means

A Corollary of the C.L.T. Example: If population means are 7 and 10, observed difference in means will cluster around 3 If group 1 sample mean is 7.4, group 2 is 9.8… Difference is 2.4

A Corollary of the C.L.T. Example: If two groups have similar means, the difference will be near zero When group means are similar, difference are usually near zero. But, even if group means are identical, difference in sample means won’t be exactly zero in most cases.

Sampling Distribution for Difference in Means The mean (Y-bar) is a variable that changes depending on the particular sample we took Similarly, the differences in means for two groups varies, depending on which two samples we chose The distribution of all possible estimates of the difference in means is a sampling distribution! The “sampling distribution of differences in means” It reflects the full range of possible estimates of the difference in means.

A Corollary of the C.L.T For any two random samples (of size N 1, N 2 ), with means  1,  2 and S.D.  1,  2 : The sampling distribution for the difference of two means is normal, with mean and S.D:

A Corollary of the C.L.T We can calculate the standard error of differences in means It is the standard deviation of the sampling distribution of differences in means: This formula tells us the dispersion of our estimates of the difference in means.

A Corollary of the C.L.T Hypothesis tests using Z-distribution depend on: N being large N of both groups > 60, ideally > 100 And, we must estimate population standard deviations based on samples standard deviations:

Z-Values for Mean Differences Finally, we can calculate a Z-value using the Z- score formula: This will be compared to a critical Z-value

Z-Values for Mean Differences Visually: Large Z Small Z Question: In which case can we reject H0? Answer: If observed Z is large, it is improbable that difference in means of populations is zero.

Z-Values for Mean Differences Back to the example: Test score differences for boys and girls Y-bar boys = 72.75, s = 8.80 Y-bar girls = 78.20, s = 9.55 Pretend our total N (of both groups) is “large” Choose  =.05, two-tailed test: critical Z = 1.96

Z-Values for Mean Differences Strategy: Calculate Z-value using formula:

Z-Values for Mean Differences Strategy: Calculate Z-value using formula: Observed Z = 1.87, critical Z = 1.96 Question: Can we reject H0? Answer: NO! We are less than 95% confident Also, our N is too small to do a Z-test.

Mean Differences for Small Samples Sample Size: rule of thumb Total N (of both groups) > 100 can safely be treated as “large” in most cases Total N (of both groups) < 100 is possibly problematic Total N (of both groups) < 60 is considered “small” in most cases If N is small, the sampling distribution of mean difference cannot be assumed to be normal Again, we turn to the T-distribution.

Mean Differences for Small Samples To use T-tests for small samples, the following criteria must be met: 1. Both samples are randomly drawn from normally distributed populations 2. Both samples have roughly the same variance (and thus same standard deviation) To the extent that these assumptions are violated, the T-test will become less accurate Check histogram to verify! But, in practice, T-tests are fairly robust.

Mean Differences for Small Samples For small samples, the estimator of the Standard Error is derived from the variance of both groups (i.e. it is “pooled”) Formulas:

Probabilities for Mean Difference A T-value may be calculated: Where (N1 + N2 – 2) refers to the number of degrees of freedom –Recall, t is a “family” of distributions –Look up t-dist for “N1 + N2 -2” degrees of freedom.

T-test for Mean Difference Back to the example: 20 boys & 20 girls Boys: Y-bar = 72.75, s = 8.80 Girls: Y-bar = 78.20, s = 9.55 Let’s do a hypothesis test to see if the means differ: Use  -level of.05 H0: Means are the same (  boys =  girls ) H1: Means differ (  boys ≠  girls ).

T-test for Mean Difference Calculate t-value:

T-Test for Mean Difference We need to calculate the Standard Error of the difference in means:

T-Test for Mean Difference We also need to calculate the Standard Error of the difference in means:

T-test for Mean Difference Plugging in Values:

T-test for Mean Difference

T-Test for Mean Difference Question: What is the critical value for  =.05, two-tailed T-test, 38 degrees of freedom (df)? Answer: Critical Value = approx Observed T-value = 1.88 Can we reject the null hypothesis (H0)? Answer: No! Not quite! We reject when t > critical value

T-Test for Mean Difference The two-tailed test hypotheses were: Question: What hypotheses would we use for the one-tailed test?

T-Test for Mean Difference Question: What is the critical value for  =.05, one-tailed T-test, 38 degrees of freedom (df)? Answer: Around (40 df) One-tailed test: T =1.88 > We can reject the null hypothesis!!! Moral of the story: If you have strong directional suspicions ahead of time, use a one-tailed test. It increases your chances of rejecting H0. But, it wouldn’t have made a difference at  =.01

T-Test for Mean Difference Question: What if you wanted to compare 3 or more groups, instead of just two? Example: Test scores for students in different educational tracks: honors, regular, remedial Can you use T-tests for 3+ groups? Answer: Sort of… You can do a T-test for every combination of groups e.g., honors & reg, honors & remedial, reg & remedial But, the possibility of a Type I error proliferates… 5% for each test With 5 groups, chance of error reaches 50% Solution: ANOVA.