STAT 330: 95-3

Midterm Solutions

  1. Q1

    1. Let p be the proportion of all adult Quebec voters who would say they would vote no if asked. You are to decide between p > 0.5 and p < 0.5. To collect evidence in favour of the referendum failing we must test the opposite hypothesis and assess the strength of evidence against that null. Namely, we define tex2html_wrap_inline48 . The value of tex2html_wrap_inline50 is 465/900 and our test statistic is


      The P-value is the area in the right hand tail since we would reject for large z values. We find tex2html_wrap_inline58 and conclude that there is only weak evidence that the referendum will not pass.

    2. Now let tex2html_wrap_inline60 be the p of part (a) and tex2html_wrap_inline64 be the same proportion but one week earlier in time. We are to test tex2html_wrap_inline66 against the alternative tex2html_wrap_inline68 . If the null hypothesis is true we estimate p, the common value of tex2html_wrap_inline64 and tex2html_wrap_inline60 by pooling the two surveys to get tex2html_wrap_inline76 . Our test statistic is


      and the one-tailed P-value is P= 0.0008 so that we conclude that the proportion supporting NO has definitely increased.

    3. Here I was looking for comments on the fact that most surveys have significant non-response rates, and response bias and the fact that most are not simple random samples but cluster samples. I did not want to see `the population is normal' because there is NO continuous variable around which could be normal.

  2. Q2

    1. Let tex2html_wrap_inline84 be the true weight. Then assuming that the measurements have mean tex2html_wrap_inline84 , that measurement errors have a normal distribution and that replicate measurements may be treated as independent we can use a t confidence interval


      which is, in units of micrograms above 10 grams, 496 tex2html_wrap_inline92 .

    2. Now let tex2html_wrap_inline94 be the weight after dropping. We test the hypothesis that tex2html_wrap_inline96 by using a 2 sample t-test. The pooled estimate of the variance is tex2html_wrap_inline100 so the test statistic is


      which leads to a two sided P value of 0.052. (From the tables you can tell that P is very close to 0.05 and a bit bigger than 0.05.) Thus we conclude that there is rather marginal evidence of a change in weight.

    3. This is a sample size calculation. You are told tex2html_wrap_inline108 , tex2html_wrap_inline110 and tex2html_wrap_inline112 . The alternative is two sided and the formula, from the text, for the sample size is


      The questions gives the denominator the value 5 so we get



  3. Q3

    1. tex2html_wrap_inline118
    2. tex2html_wrap_inline120 so that the bias is tex2html_wrap_inline122 .
    3. The basic principle is that the mle of a function of a parameter is that function applied to the parameter so that tex2html_wrap_inline124 where of course tex2html_wrap_inline126 .

Richard Lockhart
Wed Feb 11 09:09:27 PST 1998