Validity and assumptions questions

5 Each person in a random sample of 15 men and 17 women from a university campus was asked how many days in a month they took exercise. The numbers of days for men and women, \(x _ { M }\) and \(x _ { W }\) respectively, are summarised by $$\Sigma x _ { M } = 221 , \quad \Sigma x _ { M } ^ { 2 } = 3992 , \quad \Sigma x _ { W } = 276 , \quad \Sigma x _ { W } ^ { 2 } = 5538 .$$

1. State conditions for the validity of a suitable test of the difference in the mean numbers of days for men and women on the campus.
2. Given that these conditions hold, carry out the test at the \(5 \%\) significance level.
3. If in fact the random sample was drawn entirely from the university Mathematics Department, state with a reason whether the validity of the test is in doubt.

A particle of mass \(m\) is attached to one end \(A\) of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and the particle hangs in equilibrium under gravity. The particle is projected horizontally so that it starts to move in a vertical circle. The string slackens after turning through an angle of \(120°\). Show that the speed of the particle is then \(\sqrt{\left(\frac{4}{3}ga\right)}\) and find the initial speed of projection. [5]

It is suggested that a Poisson distribution with parameter \(\lambda\) can model the number of currants in a currant bun. A random bun is selected in order to test the hypotheses H₀: \(\lambda = 8\) against H₁: \(\lambda \neq 8\), using a 10\% level of significance.

1. Find the critical region for this test, such that the probability in each tail is as close as possible to 5\%. [5]
2. Given that \(\lambda = 10\), find
   1. the probability of a type II error,
   2. the power of the test. [4]

1. Explain briefly what you understand by
   1. an unbiased estimator,
   2. a consistent estimator.

of an unknown population parameter \(\theta\) [3] From a binomial population, in which the proportion of successes is \(p\), 3 samples of size \(n\) are taken. The number of successes \(X_1, X_2\), and \(X_3\) are recorded and used to estimate \(p\).

1. [(b)] Determine the bias, if any, of each of the following estimators of \(p\). \(\hat{p}_1 = \frac{X_1 + X_2 + X_3}{3n}\), \(\hat{p}_2 = \frac{X_1 + 3X_2 + X_3}{6n}\), \(\hat{p}_3 = \frac{2X_1 + 3X_2 + X_3}{6n}\) [4]
2. Find the variance of each of these estimators. [4]
3. State, giving a reason, which of the three estimators for \(p\) is
   1. the best estimator,
   2. the worst estimator. [4]

A continuous uniform distribution on the interval \([0, k]\) has mean \(\frac{k}{2}\) and variance \(\frac{k^2}{12}\). A random sample of three independent variables \(X_1\), \(X_2\) and \(X_3\) is taken from this distribution.

1. Show that \(\frac{2}{3}X_1 + \frac{1}{2}X_2 + \frac{5}{6}X_3\) is an unbiased estimator for \(k\). [3]

An unbiased estimator for \(k\) is given by \(\hat{k} = aX_1 + bX_2\) where \(a\) and \(b\) are constants.

1. Show that Var(\(\hat{k}\)) = \((a^2 - 2a + 2) \frac{k^2}{6}\) [6]
2. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat{k}\) has minimum variance, and calculate this minimum variance. [6]