STAT 330: 98-1: Midterm Solutions

Midterm, 23 February 1998Instructor: Richard Lockhart

  1. The table below shows, in the column labelled BEFORE, 15 measurements of the speed of sound in air. After taking the measurements the machine is recalibrated and a further 15 measurements are taken; these are in the column labelled after. Differences between the before and after measurements are in the last column. The table gives means and standard deviations as well.

    1 331.32331.59 0.27
    2 331.59331.43-0.16
    3 331.01331.85 0.84
    4 331.53331.63 0.10
    5 330.69331.66 0.97
    6 330.95331.21 0.26
    7 331.13331.75 0.62
    8 330.69332.14 1.45
    9 331.84331.36-0.48
    10 330.88331.82 0.94
    11 331.41330.84-0.57
    12 331.09331.80 0.71
    13 331.20331.84 0.64
    14 330.80331.15 0.35
    15 330.53331.40 0.87
    Mean331.111 331.565 0.454
    SD 0.374 0.335 0.564

    1. Give a 95% confidence interval for the true speed of sound in air based on the measurements before recalibration. [ 4 marks]

      Solution: This is a one sample t type confidence interval problem. The assumption is that the measurements are a sample from a normal population whose mean is the true speed of sound in air. The interval is




    2. Has the precision of the measurements been changed by recalibration? [4 marks]

      Solution: The question is whether or not tex2html_wrap_inline105 has changed between the two sets. You have tex2html_wrap_inline107 from a tex2html_wrap_inline109 population (the BEFORE) measurements and tex2html_wrap_inline111 from a tex2html_wrap_inline113 population (the AFTER) measurements. You want to test tex2html_wrap_inline115 against a two sided alternative. The test statistic is


      The critical points for a 10% level test are tex2html_wrap_inline119 and tex2html_wrap_inline121 and we must accept this null hypothesis. There is little evidence of a change in precision following recalibration.

    3. Has the bias of the measurements been changed by recalibration? [4 marks]

      Solution: This question asks for a test of tex2html_wrap_inline123 with a two sided alternative. Since the recalibration happens between the first 15 measurements and the second 15 there is no sense in which the data are really paired. The pooled estimate of tex2html_wrap_inline105 is


      leading to


      This leads to a P value (using t tables with 28 degrees of freedom) of about 0.0016. So there is very clear evidence that the bias has changed after recalibration.

  2. Two independent surveys of 400 male and 300 female voters in Calgary show support for the Reform Party at 49% among men and 37% among women. Is there a difference in support levels between male and female voters in Calgary? [4 marks]

    Solution: This is a two sample test of tex2html_wrap_inline135 against a two sided alternative. The pooled estimate of p under the null hypothesis is


    and then


    which leads to a P value of 2(1-.9992)=0.0016. There is very strong evidence of a difference between men and women in support for Reform in Calgary.

  3. I have two possible routes to work. I intend to carry out the following experiment. I will drive each route n times. On each trip I will record how long the trip takes. I expect the standard deviation for the time taken to be around 4 minutes on either route. I plan to analyze the data using a test carried out with fixed level 0.05. As a time fanatic I want to run a risk of no more that 1% of not detecting a difference in average travel times of 1 minute between the routes.

    1. How many times should I try each route? [2 marks]

      Solution: This a two sample two sided test and we will see that the solution is a large sample size so that the normal distribution formulas will be just fine. On the bottom of page 351 we see that


      or just 589 trips on each route. This sort of accuracy will require 4 or 5 years of experimentation!

    2. Now suppose that the answer to the first part of this question is 20. I am considering three plans for organizing my 40 trips. Plan A has me take route 1 the next 20 times and then route 2 the 20 times after that. Plan B has me alternate: a trip on route 1 then one on route 2 then route 1 again and so on. Plan C has me make up 40 tickets, 20 of which are marked route 1 and each time I go to make a trip selecting another ticket at random. Assume I come to work Monday, Tuesday, Wednesday and Friday each week. Which plan would a statistician recommend and why? Your answer should include a 1 sentence specific criticism of the plans not recommended. Your answer will be about 4 sentences long. [2 marks]

      Solution: Plan A is susceptible to a change in traffic patterns over time. If traffic levels go up over time the experiment will be biased in favour of the route I try first. Plan B is susceptible to day of the week effects - route 1 will be Monday and Wednesday. Since traffic is heaviest on Friday the design would be biased in favour of route 1. The randomization balances out (probably) any other factors which might affect the length of my trip (day of week, time of day, time of year, weather, ...).

  4. You have a sample tex2html_wrap_inline149 from the density tex2html_wrap_inline151 for x > 0 and tex2html_wrap_inline155 for tex2html_wrap_inline157 .
    1. Find the MLE of tex2html_wrap_inline159 . [3 marks]

      Solution: The likelihood is


      which simplifies to


      Take logs to get the log likelihood:


      The derivative with respect to tex2html_wrap_inline159 is


      Set this equal to 0 and solve to get


    2. The expected value of tex2html_wrap_inline173 is


      What integral do you have to do to show this? You need not do the integral! [1 mark]



    3. What is the method of moments estimate of tex2html_wrap_inline159 ? [1 mark]

      Solution: Set tex2html_wrap_inline181 or


      and solve for tex2html_wrap_inline159 to get


Richard Lockhart
Tue Feb 24 22:14:23 PST 1998