Transcript day13
The t Tests Independent Samples The t Test for Independent Samples • Observations in each sample are independent (not from the same population) each other. • We want to compare differences between sample means. t ( X 1 X 2 ) ( 1 2 ) hyp sX X 2 1 Sampling Distribution of the Difference Between Means • Imagine gathering two samples from the same population. • And then subtracting the mean of one from the mean of the other… • If you create a sampling distribution of the difference between the means… – Given the null hypothesis, we expect the mean of the sampling distribution of differences, 1- 2, to be 0. – We must estimate the standard deviation of the sampling distribution of the difference between means (i.e., the standard error of the difference between means). Pooled Estimate of the Population Variance • Using the assumption of homogeneity of variance, both s1 and s2 are estimates of the same population variance. • If this is so, rather than make two separate estimates, each based on some small sample, it is preferable to combine the information from both samples and make a single pooled estimate of the population variance. 2 2 (n 1)s (n 1)s 2 1 2 2 sp 1 (n1 1) (n2 1) Pooled Estimate of the Population Variance • The pooled estimate of the population variance becomes the average of both sample variances, once adjusted for their degrees of freedom. – Multiplying each sample variance by its degrees of freedom ensures that the contribution of each sample variance is proportionate to its degrees of freedom. – You know you have made a mistake in calculating the pooled estimate of the variance if it does not come out between the two estimates. – You have also made a mistake if it does not come out closer to the estimate from the larger sample. • The degrees of freedom for the pooled estimate of the variance equals the sum of the two sample sizes minus two, or (n1-1) +(n2-1). Estimating Standard Error of the Difference Between Means 2 2 (n 1)s (n 1)s 2 1 2 2 sp 1 (n1 1) (n2 1) sX X 1 t 2 s 2p n1 s 2p n2 ( X 1 X 2 ) ( 1 2 ) hyp sX X 2 1 The t Test for Independent Samples: An Example • Stereotype Threat “Trying to develop the test itself.” “This test is a measure of your academic ability.” The t Test for Independent Samples: An Example • State the research question. – Does stereotype threat hinder the performance of those individuals to which it is applied? • State the statistical hypotheses. H o : 1 2 0 H 1 : 1 2 0 or H o : 1 2 H 1 : 1 2 The t Test for Independent Samples: An Example • Set the decision rule. .05 df (n1 1) (n2 1) (11 1) (12 1) 21 t crit 1.721 The t Test for Independent Samples: An Example • Calculate the test statistic. Control 4 9 12 8 9 13 12 13 13 7 6 Control Sq 16 81 144 64 81 169 144 169 169 49 36 106 1122 Threat 7 8 7 2 6 9 7 10 5 0 10 8 79 Threat Sq 49 64 49 4 36 81 49 100 25 0 100 64 621 t ( X 1 X 2 ) ( 1 2 ) hyp sX X 2 1 X1 79 6.58 12 X2 106 9.64 11 The t Test for Independent Samples: An Example • Calculate the test statistic. 2 12 ( 621 ) ( 79 ) s12 9.18 12(11) 11(1122) (106) s 10.05 11(10) sX X 1 2 2 2 (n1 1)s (n2 1)s s (n1 1) (n2 1) 2 p 2 1 2 2 sX X 1 2 (12 1)9.18 (11 1)10.05 s 9.59 (12 1) (11 1) 2 p 2 s 2p n1 s 2p n2 9.59 9.59 1.29 12 11 The t Test for Independent Samples: An Example • Calculate the test statistic. t ( X 1 X 2 ) ( 1 2 ) hyp sX X 2 1 X 1 6.58 sx X 1 2 X 2 9.64 9.59 9.59 1.29 12 11 6.58 9.64 t 2.37 1.29 The t Test for Independent Samples: An Example • Decide if your result is significant. – Reject H0, - 2.37< - 1.721 • Interpret your results. – Stereotype threat significantly reduced performance of those to whom it was applied. Assumptions 1) The observations within each sample must be independent. 2) The two populations from which the samples are selected must be normal. 3) The two populations from which the samples are selected must have equal variances. – This is also known as homogeneity of variance, and there are two methods for testing that we have equal variances: • • 4) a) informal method – simply compare sample variances b) Levene’s test – We’ll see this on the SPSS output Random Assignment To make causal claims 5) Random Sampling To make generalizations to the target population Handout Example Effect Size 1) Simply report the actual results of the study. – – (a) Most direct method. (b) Can be misleading. 2) Calculate Cohen’s d or Δ (preferred). (a) Magnitude of effect size is standardized by measuring the mean difference between two treatments in terms of the standard deviation. (b) d = (M1-M2)/sp2 (c) Evaluate using the following criteria: • • • i) .20 small effect ii) .50 medium effect iii) > .80 large effect Effect Size: Example • In the study evaluating stereotype threat, the null hypothesis was rejected, with M1=6.58, M2=9.64, and sp2 = 9.59. • Calculate Cohen’s d, and evaluate the magnitude of this measure (small, medium, or large). • Compare effect size to z table to determine where the mean of one group is relative to the other. Type 1 Error & Type 2 Error Scientist’s Decision Reject null hypothesis Null hypothesis is true Null hypothesis is false Type 1 Error Fail to reject null hypothesis Type 1 Error probability = Correct Decision Probability = 1- Correct decision probability = 1 - Type 2 Error probability = = Cases in which you reject null hypothesis when it is really true Type 2 Error = Cases in which you fail to reject null hypothesis when it is false Power and sample size estimation • Power is the probability of correctly rejecting a null hypothesis. – In social sciences we typically use .80. • What determines the power of a study – Effect size – Sample size – Variance –α – One vs. two tailed tests If you want to know • Sample Size – Need to know •α •β •Δ • Power – Need to know •α • N per condition •Δ Calculating sample size • Remember stereotype threat example – Δ = .99 • Say we want to perform a test with – power (1-β) = .80 – Two tailed alpha = .05 Solving for n: 1-β = .80 Solving for n: 1-β = .80 one tailed Solving for n: 1-β = .90 Solving for n: 1-β = .90 one tailed Calculating power • Say we did the same study with n of 5 in each condition (N =10) • We want to know how much power we have to find d or Δ = .99. • Again we are using a two tailed test with α = .05. Using Piface What if I did a one tailed test? Spss: Homework Hint • For the two sample t tests you will need to create two variables, cond (X) and score (Y)