- 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. - 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.
- 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
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
. - 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.
- 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.
-
-
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 
.