CI with known population variance

5 The concentration level of mercury in a large lake is known to have a normal distribution with standard deviation 0.24 in suitable units. At the beginning of June 2008, the mercury level was measured at five randomly chosen places on the lake, and the sample mean is denoted by \(\bar { x } _ { 1 }\). Towards the end of June 2008 there was a spillage in the lake which may have caused the mercury level to rise. Because of this the level was then measured at six randomly chosen points of the lake, and the mean of this sample is denoted by \(\bar { x } _ { 2 }\).

1. State hypotheses for a test based on the two samples for whether, on average, the level of mercury had increased. Define any parameters that you use.
2. Find the set of values of \(\bar { x } _ { 2 } - \bar { x } _ { 1 }\) for which there would be evidence at the 5\% significance level that, on average, the level of mercury had increased.
3. Given that the average level had actually increased by 0.3 units, find the probability of making a Type II error in your test, and comment on its value.

7 The random variable \(X\) has distribution \(\mathrm { N } ( \mu , 1 )\). A random sample of 4 observations of \(X\) is taken. The sample mean is denoted by \(\bar { X }\).

1. Find the value of the constant \(a\) for which ( \(\bar { X } - a , \bar { X } + a\) ) is a \(98 \%\) confidence interval for \(\mu\). The independent random variable \(Y\) has distribution \(\mathrm { N } ( \mu , 9 )\). A random sample of 16 observations of \(Y\) is taken. The sample mean is denoted by \(\bar { Y }\).
2. Write down the distribution of \(\bar { X } - \bar { Y }\).
3. A \(90 \%\) confidence interval for \(\mu\) based on \(\bar { Y }\) is given by ( \(\bar { Y } - 1.234 , \bar { Y } + 1.234\) ). Find the probability that this interval does not overlap with the interval in part (i).

6 The mean Intelligence Quotient (IQ) of a random sample of 15 pupils at \(\operatorname { School } A\) is 109 . The mean IQ of a random sample of 20 pupils at School \(B\) is 112 . You may assume that the IQs for the populations from which these samples are taken are normally distributed, and that both distributions have standard deviation 15. Find a \(90 \%\) confidence interval for \(\mu _ { B } - \mu _ { A }\), where \(\mu _ { A }\) and \(\mu _ { B }\) are the population mean IQs.