Edexcel S4 (Statistics 4) 2016 June

1. A new diet has been designed. Its designers claim that following the diet for a month will result in a mean weight loss of more than 2 kg . In a trial, a random sample of 10 people followed the new diet for a month. Their weights, in kg, before starting the diet and their weights after following the diet for a month were recorded. The results are given in the table below.

1. Using a suitable \(t\)-test, at the \(5 \%\) level of significance, state whether or not the trial supports the designers’ claim. State your hypotheses and show your working clearly.
2. State an assumption necessary for the test in part (a).

2. The weights of piglets at birth, \(M \mathrm {~kg}\), are normally distributed \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) A random sample of 9 piglets is taken and their weights at birth, \(m \mathrm {~kg}\), are recorded. The results are summarised as $$\sum m = 11.6 \quad \sum m ^ { 2 } = 15.2$$ Stating your hypotheses clearly, test at the 5\% level of significance

1. whether or not the mean weight of piglets at birth is greater than 1.2 kg ,
2. whether or not the standard deviation of the weights of piglets at birth is different from 0.3 kg .

3. A jar contains a large number of sweets which have either soft centres or hard centres. The jar is thought to contain equal proportions of sweets with soft centres and sweets with hard centres. A random sample of 20 sweets is taken from the jar and the number of sweets with hard centres is recorded.

1. Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the hypothesis that there are equal proportions of sweets with soft centres and sweets with hard centres in the jar.
2. Calculate the probability of a Type I error for this test. Given that there are 3 times as many sweets with soft centres as there are sweets with hard centres,
3. calculate the probability of a Type II error for this test.

1. A manufacturer produces boxes of screws containing short screws and long screws. The manufacturer claims that the probability, \(p\), of a randomly selected screw being long, is 0.5

A shopkeeper does not believe the manufacturer's claim. He designs two tests, \(A\) and \(B\), to test the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) and \(\mathrm { H } _ { 1 } : p < 0.5\) In test \(A\), a random sample of 10 screws is taken from a box of screws and \(\mathrm { H } _ { 0 }\) is rejected if there are fewer than 3 long screws. In test \(B\), a random sample of 5 screws is taken from a box of screws and \(\mathrm { H } _ { 0 }\) is rejected if there are no long screws, otherwise a second random sample of 5 screws is taken from a box of screws. If there are no long screws in this second sample \(\mathrm { H } _ { 0 }\) is rejected, otherwise it is accepted.

1. Find the size of test \(A\).
2. Find the size of test \(B\).
3. Find an expression for the power function of test \(B\) in terms of \(p\). Some values, to 2 decimal places, of the power function for test \(A\) and the power function for test \(B\) are given in the table below.

   \(p\)	0.1	0.2	0.3	0.4
   Power test \(A\)	0.93	\(r\)	0.38	0.17
   Power test \(B\)	0.83	0.55	0.31	0.15

4. Find the value of \(r\). The shopkeeper believes that the value of \(p\) is less than 0.4
5. Suggest which of the tests the shopkeeper should use. Give a reason for your answer.

5. Fire brigades in cities \(X\) and \(Y\) are in similar locations. The response times, in minutes, during a particular month, for randomly selected calls are summarised in the table below. You may assume that the response times are from independent normal distributions.
Stating your hypotheses and showing your working clearly

1. test, at the \(10 \%\) level of significance, whether or not the variances of the populations from which the response times are drawn are the same,
   (5)
2. test, at the \(5 \%\) level of significance, whether or not the mean response time for the fire brigade in city \(X\) is more than 5 minutes longer than the mean response time for the fire brigade in city \(Y\).
3. Explain why your result in part (a) enables you to carry out the test in part (b).

6. A random sample of size \(n\) is taken from the random variable \(X\), which has a continuous uniform distribution over the interval [ \(0 , a\) ], \(a > 0\) The sample mean is denoted by \(\bar { X }\)

1. Show that \(Y = 2 \bar { X }\) is an unbiased estimator of \(a\) The maximum value, \(M\), in the sample has probability density function $$f ( m ) = \left\{ \begin{array} { c c } \frac { n m ^ { n - 1 } } { a ^ { n } } & 0 \leqslant m \leqslant a
   0 & \text { otherwise } \end{array} \right.$$
2. Find E(M)
3. Show that \(\operatorname { Var } ( M ) = \frac { n a ^ { 2 } } { ( n + 2 ) ( n + 1 ) ^ { 2 } }\) The estimator \(S\) is defined by \(S = \frac { n + 1 } { n } M\)
   Given that \(n > 1\)
4. state which of \(Y\) or \(S\) is the better estimator for \(a\). Give a reason for your answer.

7. The times taken to travel to school by sixth form students are normally distributed. A head teacher records the times taken to travel to school, in minutes, of a random sample of 10 sixth form students from her school. Based on this sample, the \(95 \%\) confidence interval for the mean time taken to travel to school for sixth form students from her school is
[0pt] [28.5, 48.7] Calculate a 99\% confidence interval for the variance of the time taken to travel to school for sixth form students from her school.
(9)