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 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 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 • 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 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, 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 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 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