# STAT 2606–Sections A and B–Assignment #4

Carleton UniversitySchool of Mathematics and StatisticsSTAT 2606 – Sections A and B – Assignment #4 – Fall 2016Due on Monday, November 21 by 3pmINSTRUCTIONS:I. Assignments are to be uploaded to the course website on CULEARN as a single legible PDF file by the abovedue date and time. No late assignments will be accepted without sufficient advanced notice and a legitimate,documented reason. Technical issues are not considered a valid excuse.II. For full marks, you must show and explain all of your work. You must also justify the use of any formulaor approximation procedure that you use.III. Do not use MINITAB for any part of a question unless it specifically says to do so. For questions thatrequire MINITAB, you must include all relevant output with your assignment. The lab for this assignmentwill take place during the week of November 14.IV. This assignment is intended to represent your individual knowledge. It is not a group assignment.1. Weekly demand at a grocery store for a brand of breakfast cereal is normally distributed with a mean of800 boxes and a standard deviation of 75 boxes.(a) The store manager plans to select a random sample of 16 weeks and measures the demand in each week.What is the sampling distribution of the sample mean? Explain.(b) What is the probability that in a sample of 16 randomly chosen weeks, the average demand will be nomore than 750 boxes?2. Let X represent the number of days per week that your statistics professor goes to the gym. Assume thatthe number of days your professor goes to the gym during one week is independent of the number of days hegoes during each other week.P(X = x)x10.1020.3530.55(a) Consider a sample of two weeks. What is the sampling distribution of the average number of days per weekthat your professor goes to the gym during those two weeks? That is, what is the sampling distribution of Xbased on n = 2? (HINT: Set up a probability table for X .)(b) Consider an entire year (i.e. 52 weeks). What is the approximate sampling distribution of X ? Include thevalues of any relevant parameters.(c) Approximate the probability that over the course of an entire year, your professor goes to the gym an averageof at least 2.3 days per week.3. A consulting company wishes to study the monthly health insurance cost per worker for companies thatoffer paid health insurance. A random sample of 25 such companies produced a monthly mean cost perworker of $135 with a standard deviation of $28. Construct a 99% confidence interval for the mean monthlyhealth insurance cost per worker for all small businesses. Interpret the result. Under what conditions is thisinterpretation valid? 4. A manufacturer of LED 3D televisions claims that only 8% of all its televisions require service during theone-year warranty period. In order to investigate this claim, a consumer protection agency takes a randomsample of 200 households that purchased one of the company’s televisions.(a) Let Pˆ represent the sample proportion of televisions that require service during the warranty period.Assuming that the company’s claim is true, what is the approximate sampling distribution of Pˆ ?(b) Approximate the probability that at least 10% of televisions in the sample will require service during thewarranty period.(c) The sample actually revealed that 24 of the 200 households required their television to be serviced during thewarranty period. Construct a 95% confidence interval for the proportion of all LED 3D televisions manufacturedby this company that are serviced under warranty. Interpret the result.(d) Suppose the consumer protection agency wishes the confidence interval in part (c) to be accurate to within2%. What sample size would you recommend they use? You may use the information in part (c) to estimate thepopulation proportion, p.5. Suppose that weekly change in the price of a stock is normally distributed with a mean of $2.55 and astandard deviation of $0.40. Use MINITAB to generate 50 different random samples from this population,each of size n = 4.•••• Click on CALC ? RANDOM DATA ? NORMAL.In the box for “Number of rows of data to generate”, enter the sample size (in this case, 4).In the box for “Store in columns”, enter “C1-C50”. (without the quotes)In the boxes for mean and standard deviation, enter the values of 2.55 and 0.4, respectively. ClickOK. Each of the first 50 columns should now contain a random sample of 4 observations from the populationunder consideration. Based on each sample, we will then use MINITAB to compute an 90% confidenceinterval for the population mean.••••• Click on STAT ? BASIC STATISTICS ? 1-SAMPLE Z.Click the “Options” button. Enter a value of 90 for the “Confidence level”. Click OK.Ensure that the drop box at the top of the dialogue box says “One or more samples, each in acolumn”.In the next box, specify the columns by entering “C1-C50”. (without the quotes)In the box for “Known standard deviation” enter a value of 0.4. Click OK. You must include theresulting session window output in your assignment submission. Do NOT include thespreadsheet. Based on the 50 confidence intervals you obtained, answer the following questions.(a) Prior to generating the random samples, how many of these 50 intervals would you expect to contain thetrue population mean value of $2.55? Explain.(b) How many of these 50 intervals actually contain the true population mean value of $2.55? Should thisvalue equal the value you gave in part (a)? Why or why not?(c) Are all 50 confidence intervals of exactly the same length? Would this still be the case if the populationstandard deviation was unknown? Explain.(d) Suppose we wanted each confidence interval to have a margin of error of no more than $0.10. Whatsample size would be required to achieve this goal? 6. In an introductory statistics class, the same final exam was administered to 2000 different students. Theirgrades, as a percentage, are recorded in GRADES.MTW. Open this worksheet in MINITAB.(a) Use MINITAB to generate a histogram for this population of measurements.•• Click on GRAPH ? HISTOGRAM. Select “Simple” and click OK. Under “Graph variables”,enter C1. Click OK.Double click on the horizontal axis. Select the “Binning” tab. Under “Interval Type”, select“Cutpoint”. Under “Interval Definition”, enter 20 as the “Number of intervals”. Describe the shape of this histogram. Include this histogram with your assignment.(b) Use MINITAB to generate 1000 independent samples of size n = 30. Compute the sample mean for eachsample You can do this as follows:••• Download and save the file “clt.max.txt” to the P drive.Click on EDITOR ? ENABLE COMMANDS.At the command prompt (MTB >), type the following exactly without the quotes: “%P:clt.mac.txt”.If the drive letter you have saved the file to is not P, then change the letter in this command to thecorrect drive letter. Hit the enter key. You should now have 1000 observed values of the sample mean stored in column C3. (You do NOT need toinclude the spreadsheet output in your assignment submission.)Next, use MINITAB to create a histogram of these 1000 values. (Feel free to play around with the binningagain to create a more visually appealing histogram.) Include this histogram with your assignment.Describe the shape of the histogram, and compare it to the shape of the histogram of the 2000 grades in thepopulation. Explain the discrepancy between the shapes of these two histograms.(c) Use MINITAB to generate descriptive statistics for both the 2000 individual exam scores and, separately,for the 1000 observed sample means.•• Click on STAT ? BASIC STATISTICS ? DISPLAY DESCRIPTIVE STATISTICS. A dialoguebox will appear. Under “Variables” you can enter C1 and C3, with a space between them. Click OK.Click on FILE ? PRINT SESSION WINDOW and print a copy to include with your assignmentsubmission. Highlight the “Mean” and “StDev” for both data sets. Compare the means for the two data sets and explain the reasons behind what you observe. Do the same forthe standard deviations. Use as much detail as possible in your explanation.