Stat 2000, Section 001, Homework Assignment 13 (Due 12/4/2002 12:00 noon)

• 0) Reading: Section 6.1, 6.2, 6.3
  
  Review of old Homeworks, Lecture Notes, and Handouts for the Final.
  
  
• 1) Please work on the following textbook exercises from Moore/McCabe (3rd Edition):
  
  
  • Exercise 6.2 (2 Points):
    Crop researchers plant 15 plots with a new variety of corn. The yields in bushels per acre are
    138.0, 139.1, 113.0, 132.5, 140.7, 109.7, 118.9, 134.8,
    109.6, 127.3, 115.6, 130.4, 130.2, 111.7, 105.5
    Assume that sigma = 10 bushels per acre.
    (a) Find the 90% confidence interval for the mean yield mu for this variety of corn.
    (b) Find the 95% confidence interval.
    (c) Find the 99% confidence interval.
    (d) How do the margins of error in (a), (b), and (c) change as the confidence level increases?
    
    
  • Exercise 6.5 (2 Points):
    Suppose that the crop researchers in Exercise 6.2 obtained the same value of xbar from a sample of 60 plots rather than 15.
    (a) Compute the 95% confidence interval for the mean yield mu.
    (b) Is the margin of error larger or smaller than the margin of error found for the sample of 15 plots in Exercise 6.2? Explain in plain language why the change occurs.
    (c) Will the 90% and 99% intervals for a sample of size 60 be wider or narrower than those for n = 15? (You need not actually calculate these intervals.)
    
    
  • Exercise 6.6 (2 Points):
    A test for the level of potassium in the blood is not perfectly precise. Moreover, the actual level of potassium in a person's blood varies slightly from day to day. Suppose that repeated measurements for the same person on different days vary normally with sigma = 0.2.
    (a) Julie's potassium level is measured once. The result is x = 3.4. Give a 90% confidence interval for her mean potassium level.
    (b) If three measurements were taken on different days and the mean result is xbar = 3.4, what is a 90% confidence interval for Julie's mean blood potassium level?
    
    
  • Exercise 6.7 (2 Points):
    A study of the career paths of hotel general managers sent questionnaires to an SRS of 160 hotels belonging to major U.S. hotel chains. There were 114 responses. The average time these 114 general managers had spent with their current company was 11.78 years. Give a 99% confidence interval for the mean number of years general managers of major-chain hotels have spent with their current company. (Take it as known that the standard deviation of time with the company for all general managers is 3.2 years.)
    
    
  • Exercise 6.10 (2 Points):
    How large a sample of the hotel managers in Exercise 6.7 would be needed to estimate the mean mu within +/- 1 year with 99% confidence?
    
    
  • Exercise 6.19 (2 Points):
    As we prepare to take a sample and compute a 95% confidence interval, we know that the probability that the interval we compute will cover the parameter is 0.95. That's the meaning of 95% confidence. If we use several such intervals, however, our confidence that all give correct results is less than 95%.

    In an agricultural field trial a corn variety is planted in seven separate locations, which may have different mean yields due to differences in soil and climate. At the end of the experiment, seven independent 95% confidence intervals will be calculated, one for the mean yield at each location.
    (a) What is the probability that every one of the seven intervals covers the true mean yield at its location? This probability (expressed as a percent) is our overall confidence level for the seven simultaneous statements.
    (b) What is the probability that at least six of the seven intervals cover the true mean yields?
    
    

  • Exercise 6.23 (2 Points):
    A student reads that a 95% confidence interval for the mean SAT math score of California high school seniors is 452 to 470. Asked to explain the meaning of this interval, the student says, ``95% of California high school seniors have SAT math scores between 452 and 470.'' Is the student right? Justify your answer.
    
    
  • Exercise 6.26 (1 Point):
    Each of the following situations requires a significance test about a population mean mu. State the appropriate null hypothesis H₀ and alternative hypothesis H_a in each case.
    (a) The mean area of the several thousand apartments in a new development is advertised to be 1250 square feet. A tenant group thinks that the apartments are smaller than advertised. They hire an engineer to measure a sample of apartments to test their suspicion.
    (b) Larry's car averages 32 miles per gallon on the highway. He now switches to a new motor oil that is advertised as increasing gas mileage. After driving 3000 highway miles with the new oil, he wants to determine if his gas mileage actually has increased.
    (c) The diameter of a spindle in a small motor is supposed to be 5 millimeters. If the spindle is either too small or too large, the motor will not perform properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target.
    
    
  • Exercise 6.29 (1 Point):
    Translate each of the following research questions into appropriate H₀ and H_a.
    (a) Census Bureau data show that the mean household income in the area served by a shopping mall is $52,500 per year. A market research firm questions shoppers at the mall to find out whether the mean household income of mall shoppers is higher than that of the general population.
    (b) Last year, your company's service technicians took an average of 2.6 hours to respond to trouble calls from business customers who had purchased service contracts. Do this year's data show a different average response time?
    
    
  • Exercise 6.34 (1 Point):
    Statistics can help decide the authorship of literary works. Sonnets by an Elizabethan poet are known to contain an average of mu = 6.9 new words (words not used in the poet's other works). The standard deviation of the number of new words is sigma = 2.7. Now a manuscript with 5 new sonnets has come to light, and scholars are debating whether it is the poet's work. The new sonnets contain an average of xbar = 11.2 words not used in the poet's known works. We expect poems by another author to contain more new words, so to see if we have evidence that the new sonnets are not by our poet we test
    H₀: mu = 6.9 vs. H_a: mu > 6.9
    Give the z test statistic and its P-value. What do you conclude about the authorship of the new poems?
    
    
  • Exercise 6.37 (2 Points):
    In the past, the mean score of the seniors at South High on the American College Testing (ACT) college entrance examination has been 20. This year a special preparation course is offered, and all 53 seniors planning to take the ACT test enroll in the course. The mean of their 53 ACT scores is 22.1. The principal believes that the new course has improved the students' ACT scores.
    (a) Assume that ACT scores vary normally with standard deviation 6. Is the outcome xbar = 22.1 good evidence that the population mean score is greater than 20? State H₀ and H_a, compute the test statistic and the P-value, and answer the question by interpreting your result.
    (b) The results are in any case inconclusive because of the design of the study. The effects of the new course are confounded with any change from past years, such as other new courses or higher standards. Briefly outline the design of a better study of the effect of the new course on ACT scores.
    
    
  • Exercise 6.41 (2 Points):
    To determine whether the mean nicotine content of a brand of cigarettes is greater than the advertised value of 1.4 milligrams, a health advocacy group tests
    H₀: mu = 1.4 vs. H_a: mu > 1.4
    The calculated value of the test statistic is z = 2.42.
    (a) Is the result significant at the 5% level?
    (b) Is the result significant at the 1% level?
    
    
  • Exercise 6.49 (2 Points):
    You measure the weights of 24 male runners. These runners are not a random sample from a population, but you are willing to assume that their weights represent the weights of similar runners. Here are their weights in kilograms:
    67.8, 61.9, 63.0, 53.1, 62.3, 59.7, 55.4, 58.9,
    60.9, 69.2, 63.7, 68.3, 64.7, 65.6, 56.0, 57.8,
    66.0, 62.9, 53.6, 65.0, 55.8, 60.4, 69.3, 61.7

    Exercise 6.1 asks you to find a 95% confidence interval for the mean weight of the population of all such runners, assuming that the population standard deviation is sigma = 4.5 kg.
    (a) Give the confidence interval from that exercise, or calculate the interval if you did not do the exercise.
    (b) Based on this confidence interval, does a test of

    H₀: mu = 61.3 kg vs. H_a: mu not equal to 61.3 kg
    reject H₀ at the 5% significance level?
    (c) Would H₀: mu = 63 kg be rejected at the 5% level if tested against a two-sided alternative?
    
    
• 2) Work with the Confidence Intervals applet at
  http://www.ruf.rice.edu/~lane/stat_sim/conf_interval/index.html
  
  Experiment with the applet and explain in 2 or 3 sentences what it does. Then, explain to someone without any knowledge in statistics what a confidence interval is, using your own words. (2 Points)
  
  If you like this applet and also the Normal Approximation to Binomial applet from Assignment 12, then you can experiment with many similar applets that can be accessed through the following Web page:
  http://www.ruf.rice.edu/~lane/rvls.html