1. Measurements in the Lab. Juan makes a measurement in a chemistry laboratory and records the result in his lab report. Suppose that if Juan makes this measurement repeatedly, the standard deviation of his measurements will be σ = 10 milligrams. Juan repeats the measurement four times and records the mean x- of his four measurements.
(a) What is the standard deviation of Juan's mean result? (That is, if Juan kept on making four measurements and averaging them, what would be the standard deviation of all his x- values?)
(b) How many times must Juan repeat the measurement to reduce the standard deviation of x- to 2? Explain to someone who knows no statistics the advantage of reporting the average of several measurements rather than the result of a single measurement.
2. The Medical College Admission Test. Almost all medical schools in the United States require students to take the Medical College Admission Test (MCAT). To estimate the mean score μ. of those who took the MCAT on your campus, you will obtain the scores of an SRS of students. The scores follow a Normal distribution, and from published information you know that the standard deviation is 6.5. Suppose that (unknown to you) the mean score of those taking the MCAT on your campus is 25.0.
(a) If you choose one student at random, what is the probability that the student's score is between 20 and 30?
(b) You sample 25 students. What is the sampling distribution of their average score x-?
(c) What is the probability that the mean score of your sample is between 20 and 30?
3. What Is Power? The Trial Urban District Assessment (TUDA) mea¬sures educational progress within participating large urban districts. TUDA gives a reading test scored from 0 to 500. A score of 243 is a "basic" reading level for eighth-graders. Suppose scores on the TUDA reading test for eighth-graders in your district follow a Normal distribution with standard deviation σ = 55. In 2013 the mean score for eighth-graders in your district was 255. You plan to give the reading test to a ran¬dom sample of 25 eighth-graders in your district this year to test whether the mean score μ, for all eighth-graders in your district is still above the basic level. You will therefore test
Ho:μ = 243
Ha: μ > 243
If the true mean score is again 255, on average students are performing above the basic level. You learn that the power of your test at the 5% significance level against the alternative μ = 255 is 0.29.
(a) Explain in simple language what "power = 0.29" means.
(b) Explain why the test you plan will not adequately protect you against deciding that average reading scores in your district are not above
basic level.
4, What's Your Weight? A 2013 Gallup Poll asked a national random sample of 477 adult women to state their current weight. The mean weight in the sample was x-= 157. We will treat these data as an SRS from a Normally distributed population with standard deviation σ = 35.
(a) Give a 95% confidence interval for the mean weight of adult women based on these data.
(b) Do you trust the interval you computed in part (a) as a 95% confidence interval for the mean weight of all U.S. adult women? Why or why not?
5. Too Early. The examinations in a large multi-section statistics class are scaled after grading so that the mean score is 75. The professor thinks that students in the 8:00 A.M. class have trouble paying attention because they are sleepy and suspects that these students have a lower mean score than the class as a whole. The students in the 8:00 A.M. class this semester can be considered a sample from the population of all students in the course, so the professor compares their mean score with 75. State the hypotheses Ho and Ha.
6. Women's Incomes. The average income of American women who work full-time and have only a high school degree is $33,230. You wonder whether the mean income of female graduates from your local high school who work full-time but have only a high school degree is different from the national average. You obtain income information from an SRS of 62 female gradu¬ates who work full-time and have only a high school degree and find that x- = $32,052. What are your null and alternative hypotheses?