Example OneSample TTest Researchers are interested in whether

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Example: One-Sample T-Test • Researchers are interested in whether the pulse rate of long-distance

Example: One-Sample T-Test • Researchers are interested in whether the pulse rate of long-distance runners differs from that of other athletes • They randomly sample 8 long-distance runners, measure their resting pulse, and obtain the following data: 45, 42, 64, 58, 49, 48, 56 • The average resting pulse of athletes in the general population is 60 beats per minute • Test the null hypothesis at the 0. 05 level of significance

Example: One-Sample T-Test HO: Pulse of long-distance runners = 60 HA: Pulse of long-distance

Example: One-Sample T-Test HO: Pulse of long-distance runners = 60 HA: Pulse of long-distance runners differs from 60 • • • Mean = 416/8 = 52 SS = 374; s 2 = 53. 4; s = 7. 3; SE = 2. 6 df = 7 t = (52 -60)/2. 6 = 3 tcrit (from table) at alpha of 0. 05 = 2. 365 Reject null hypothesis. There is a difference

Paired T-Test • Used when two samples are not independent of each other •

Paired T-Test • Used when two samples are not independent of each other • Observations in one sample can be paired with observations in the other sample • For example: – Before and after observations on the same subjects – A comparison of two different measurements or treatments on the same subjects

Paired T-Test: Procedure • Calculate the difference (di = yi − xi) between the

Paired T-Test: Procedure • Calculate the difference (di = yi − xi) between the two observations on each pair, making sure you distinguish between positive and negative differences • Calculate the mean difference • Calculate the standard deviation of the differences (sd) and use this to calculate the standard error of the mean difference • SE = sd / n • Calculate t = d / SE • Degrees of freedom = n − 1 • Use Table B. 3 to obtain tcrit

Example • Four individuals with high levels of cholesterol went on a special diet,

Example • Four individuals with high levels of cholesterol went on a special diet, avoiding high cholesterol foods and taking special supplements. Using the. 05 level of significance, was there a significant decrease in cholesterol level? • Their total cholesterol levels before and after the diet were as follows: Before After 287 255 305 269 243 245 309 247

Example Before 287 305 245 309 After 255 269 243 247 Difference 32 36

Example Before 287 305 245 309 After 255 269 243 247 Difference 32 36 2 62 132 (Total) Mean difference = 132/4 = 33 Standard deviation = 1812/3 = 24. 6 Standard error = 12. 3 t = 33/12. 3 = 2. 683 df = 3 tcrit = 2. 353 (one-sided test at α = 0. 05) Reject null hypothesis d-d 1 3 31 29 (d-d)2 1 9 961 841 1812 (SS)

CI for Difference in Means • Recall example of verbal skills in 8 -year

CI for Difference in Means • Recall example of verbal skills in 8 -year boys and girls C. I. = X 1 – X 2 tcrit (SX 1 – X 2) • tcrit = 2. 878 for α =0. 01 • 99% CI = 37 -31 ± 2. 878(2) = 6 ± 5. 756 = (0. 244; 11. 756) • This interval does not include zero, therefore there is a difference between boys and girls

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The OneSample tTest Advanced Research Methods in PsychologyThe OneSample tTest Advanced Research Methods in Psychology