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Comparing the Means of Two Dependent Populations. The Paired T-test …. Do males earn higher average starting salaries than females?. (in $1,000s) Males Females 34 28 32 30 29 22 35 32 Sample Mean: $33 $28.
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Comparing the Means of Two Dependent Populations The Paired T-test ….
Do males earn higher average starting salaries than females? (in $1,000s) Males Females 34 28 32 30 29 22 35 32 Sample Mean: $33 $28 Real question is whether males and females in the same job earn different average salaries. Better then to compare the difference in salaries in “pairs” of males and females.
Same example, but now a Paired Study JobMales Females Difference=M-F Non-Profit 22 20.5 1.5 Education 29 28 1.0 Doctor 80 78 2.0 Scientist 35 32 3.0 (in $1,000s) Sample Average Difference = $1.9 How likely is it that a paired sample would have a difference as large as $1,900 if the true difference were 0? Problem reduces to a One-Sample T-test on differences!!!!
Hypotheses for Paired T-test Does the average difference of the population, D, differ from 0? Null hypothesis: H0: D = 1 - 2 = 0 Alternative hypotheses: HA: D = 1 - 2 0 HA: D = 1 - 2 > 0 HA: D = 1 - 2 < 0
The Paired-T Test Statistic • If: • there are n pairs • and the differences are normally distributed Then: The following test statistic, which follows a t-distribution with n-1 d.f., gives us our P-value:
The Paired-T Confidence Interval • If: • there are n pairs • and the differences are normally distributed Then: The following confidence interval, with t following t-distribution with n-1 d.f. estimates the actual population difference:
Paired T in Minitab • Enter your paired data in two columns. • Select Stat >> Basic Statistics >> Paired T • In the boxes labeled First (Second) Sample, select the variables that contains your first (second) samples. • Under Options, specify the confidence level, the null mean value (usually 0), and the alternative hypothesis. Select OK. • Select OK.
Ways Pairing Can Occur • When, in “before and after” studies, the same subjects are measured twice. • When subjects serve as their own control by receiving both of two different treatments. • When subjects in one group are “matched” with a similar subject in the second group.
What is the effect of exercise on pulse rates? H0: D = Before - After = 0 vs. HA: D < 0 Paired T for Before - After N Mean StDev SE Mean Before 64 69.41 10.44 1.30 After 64 80.84 11.40 1.43 Difference 64 -11.44 8.07 1.01 95% CI for mean difference: (-13.45, -9.42) T-Test of mean difference = 0 (vs < 0): T-Value = -11.34 P-Value = 0.000
What is the effect of alcohol on useful consciousness? H0: D = No alcohol - Alcohol = 0 vs. HA: D > 0 Paired T for NoAlcohol - Alcohol N Mean StDev SE Mean NoAlcohol 10 546.6 238.8 75.5 Alcohol 10 351.0 210.9 66.7 Difference 10 195.6 230.5 72.9 95% CI for mean difference: (30.7, 360.5) T-Test of mean difference= 0(vs > 0): T-Value = 2.68 P-Value = 0.013
For males, does average actual and ideal weights differ from 0? H0: D = Actual - Ideal = 0 vs. HA: D 0 Paired T for M_Actual - M_Ideal N Mean StDev SE Mean M_Actual 83 172.72 28.65 3.14 M_Ideal 83 169.60 22.09 2.43 Difference 83 3.12 17.01 1.87 95% CI for mean difference: (-0.59, 6.83) T-Test of mean difference = 0 (vs not = 0):T-Value = 1.67 P-Value = 0.098
For females, does average actual and ideal weights differ from 0? H0: D = Actual - Ideal = 0 vs. HA: D 0 Paired T for F_Actual - F_Ideal N Mean StDev SE Mean F_Actual 88 138.20 28.00 2.98 F_Ideal 88 125.64 14.01 1.49 Difference 88 12.57 18.68 1.99 95% CI for mean difference: (8.61, 16.53) T-Test of mean difference = 0 (vs not = 0): T-Value = 6.31 P-Value = 0.000
