STAT 400 Lecture AL1 Examples for 7.2, 8.2 Spring 2015 Dalpiaz Two independent samples: X 1 , X 2 , … , X n 1 Y 1 , Y 2 , … , Y n 2 from population 1 from population 2 mean µ 1 , std. dev. σ 1 mean µ 2 , std. dev. σ 2 If n 1 and n 2 are large, or population 1 and population 2 are approximately normal, then ( )YX− is (approximately) normal with mean µ 1 – µ 2 and standard deviation 222121 nn σσ+. A confidence interval for µ 1 – µ 2 is ( )2221212YX nnz σσ⋅ +α±−. Test statistic for testing H 0 : µ 1 – µ 2 = δ 0 is Z = ()2221210 σσδYXnn+−−. If σ 1 and σ 2 are unknown, use ()2221212YX nsnst +α±−⋅ “Conservative” approach: the number of degrees of freedom = the smaller of n 1 – 1 and n 2 – 1. Welch’s T: the number of degrees of freedom = −+−+22222212112222121 1111nsnnsnnsns If n 1 and n 2 are large, t α/2 can be approximated by z α/2.1. Dr. Statman claims that his new revolutionary study method “Study While You Sleep” is more effective than the traditional study methods. In an experiment, 250 students enrolled in the same section of STAT 100 at UIUC were divided into two groups. One hundred students volunteered to study using SWYS method, and the other 150 students did whatever students usually do. At the end of the semester, the averages of the total number of points (out of 500) were compared for the two groups. Note: This is NOT a good experiment design! SWYS Traditional (sample) average total points 450 410 (sample) standard deviation 20 45 a) Construct a 95% confidence interval for the difference in the average total points for SWYS and traditional study methods. b) Perform the appropriate test at a 1% level of significance. c) Test H 0 : µ S – µ T ≤ 30 vs. H 1 : µ S – µ T > 30 at α = 0.05.2. Two work designs are being considered for possible adoption in an assembly plant. A time study is conducted with 10 workers using design A and 12 workers using design B. The sample means and sample standard deviations of their assembly times (in minutes) are Design A Design B Sample Mean 78.3 85.6 Sample Standard deviation 4.8 6.5 Construct a 90% confidence interval for the difference in the mean assembly times between design A and Design B. Use Welch’s T. If we can assume that population 1 and population 2 standard deviations are equal (i.e., σ 1 = σ 2 = σ ), then we can use ( )21pooled211YX nnst +α±− ⋅⋅ where ( ) ( )211212222112pooled−+−+−=⋅⋅ nnsnsns. Then the number of degrees of freedom = n 1 + n 2 – 2.3. A national equal employment opportunities committee is conducting an investigation to determine if women employees are as well paid as their male counterparts in comparable jobs. Random samples of 14 males and 11 females in junior academic positions are selected, and the following calculations are obtained from their salary data. Male Female Sample Mean $48,530 $47,620 Sample Standard deviation 780 750 Assume that the populations are normally distributed with equal variances. a) Construct a 95% confidence interval for the difference between the mean salaries of males and females in junior academic positions. b) What is the p-value of the test H 0 : µ Male = µ Female vs. H 1 : µ Male ≠ µ Female?The t Distribution r t 0.40 t 0.25 t 0.10 t 0.05 t 0.025 t 0.01 t 0.005 7 0.263 0.711 1.415 1.895 2.365 2.998 3.499 19 0.257 0.688 1.328 1.729 2.093 2.539 2.861 23 0.256 0.685 1.319 1.714 2.069 2.500 2.807 ∞ 0.253 0.674 1.282 1.645 1.960 2.326 2.576 Matched Pair Comparison: Pair Difference 1 X 1 Y 1 D 1 = X 1 – Y 1 2 X 2 Y 2 D 2 = X 2 – Y 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ n X n Y n D n = X n – Y n Assume that the differences D i = X i – Y i are a random sample from normal distribution with mean δ and standard deviation σ D . A confidence interval for δ is nst D2D⋅α±. The number of degrees of freedom = n – 1. Test statistic for testing H 0 : δ = δ 0 is T = ns D0δD −.4. A new revolutionary diet-and-exercise plan is introduced. Eight participants were weighed in the beginning of the program, and then again a week later. The results were as follows: Participant 1 2 3 4 5 6 7 8 Weight Before 213 222 232 201 230 188 218 182 Weight After 207 220 224 198 219 183 220 175 Pounds Lost 6 2 8 3 11 5 – 2 7 a) Construct a 90% confidence interval for the average number of pounds lost during one week on that plan. b) Is there enough evidence to conclude that the average weight loss is less than 7 pounds per week? (Use α = 0.05.) What is the p-value of this
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