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 . The value of 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 and conclude that there is only weak evidence that the referendum will not pass.

2. Now let be the p of part (a) and be the same proportion but one week earlier in time. We are to test against the alternative . If the null hypothesis is true we estimate p, the common value of and by pooling the two surveys to get . 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 be the true weight. Then assuming that the measurements have mean , 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 .

2. Now let be the weight after dropping. We test the hypothesis that by using a 2 sample t-test. The pooled estimate of the variance is 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 , and . 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

roughly.

3. Q3

1. so that the bias is .
2. The basic principle is that the mle of a function of a parameter is that function applied to the parameter so that where of course .

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