**STAT 330: 95-3
**

Midterm Solutions

- Q1
- 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 isThe

*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. - 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 isand the one-tailed

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

- Let
- Q2
- 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 intervalwhich is, in units of micrograms above 10 grams, 496 .

- 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 iswhich 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. - 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.

- 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
- Q3
- so that the bias is .
- 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 .

Wed Feb 11 09:09:27 PST 1998