Independent samples t-tests

Introducing t tests Z vs t Degrees of freedom Types of t test P values One sample Dependent Independent P values

The t Test for a Single Sample The single sample t test is used to compare a single sample to a population with a known mean but an unknown variance. The formula for the t statistic is similar in structure to the Z It is named the “Student’s t” because its main principles were developed by William S. Gosset, who published articles anonymously using the name “Student”. Gossett was a mathematician in Ireland who was employed by Guinness to solve the problem of how to make beer less variable, and especially to find the cause of bad batches. Creating experimental batches of beer was very expensive, so Gosset was forced to conduct experiments using only a few batches of different strains of barley. Adding to the problem was that he had no idea of the variability of a given strain of barley (the population’s variance). Gosset discovered the t distribution to solve the problem. Guinness did not allow its scientists to publish papers (fearing they would reveal brewery secrets), so Gossett anonymously published his results under the name “Student”.

The t Test for Independent Samples Observations in each sample are independent (not related to) each other. We want to compare differences between sample means, not a mean of differences.

Sampling Distribution of the Difference Between Means Imagine two sampling distributions of the mean... And then subtracting one from 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.

Example: Independent Samples t test Does staying up all night affect your creativity? Group 1 stays up all night, Group 2 gets a full night’s sleep Next morning: everyone thinks of uses for a bucket full of hungry cats Ho: no difference in # of ideas between groups Ha: sleep deprivation will lead to more, or fewer, creative ideas Group 1: n = 35, mean # ideas = 24.0, standard deviation = 12.2 Group 2: n = 29, mean # ideas = 16.5, standard deviation = 11.8 -2.00 2.00 df = n1 + n2 - 2 = 62

Example continued 2.00 < 2.5 tcrit < t obs -2.00 2.00 Reject Null & conclude sleep deprivation increases creativity 2.00

Pooled Variance estimate, S2p Assumes pop. variances are equal Mean of each variance, proportional to df S2p = (SS1 + SS2)/df df = n1 + n2 -2 Plug in for both s12 and s22 for estimated standard error Cohen’s d:(x-bar1 - x-bar2)/standard deviation (square root of pooled variance): √s2p

Confidence Interval Formulas z test CI: Single Sample t CI: Dependent t CI: Independent t CI:

Homework ch. 14 Does consuming caffeine increase perceived attractiveness of others? A group of people are given a cup of coffee and asked to rate random pictures from 1-10 (mean rating = 7.5, s = 4, n = 36); another group is given decaf coffee and follows the same procedure (mean = 6.0, s = 3, n = 36). 1. State the null hypothesis 2. Calculate t and find t* 3. Find your decision and interpretation/conclusion 4. (optional) find a 99% confidence interval for the difference in perceived attractiveness of others between subjects who do or don’t consume caffeine.