Edexcel S4 (Statistics 4) 2005 June

1. The random variable \(X\) has a \(\chi ^ { 2 }\)-distribution with 9 degrees of freedom.
   1. Find \(\mathrm { P } ( 2.088 < X < 19.023 )\).
   The random variable \(Y\) follows an \(F\)-distribution with 12 and 5 degrees of freedom.
2. Find the upper and lower \(5 \%\) critical values for \(Y\).
   (3)
   (Total 6 marks)

2. The standard deviation of the length of a random sample of 8 fence posts produced by a timber yard was 8 mm . A second timber yard produced a random sample of 13 fence posts with a standard deviation of 14 mm .

1. Test, at the \(10 \%\) significance level, whether or not there is evidence that the lengths of fence posts produced by these timber yards differ in variability. State your hypotheses clearly.
2. State an assumption you have made in order to carry out the test in part (a).

3. A machine is set to fill bags with flour such that the mean weight is 1010 grams. To check that the machine is working properly, a random sample of 8 bags is selected. The weight of flour, in grams, in each bag is as follows. $$\begin{array} { l l l l l l l l } 1010 & 1015 & 1005 & 1000 & 998 & 1008 & 1012 & 1007 \end{array}$$ Carry out a suitable test, at the \(5 \%\) significance level, to test whether or not the mean weight of flour in the bags is less than 1010 grams. (You may assume that the weight of flour delivered by the machine is normally distributed.)
(Total 8 marks)

4. A farmer set up a trial to assess the effect of two different diets on the increase in the weight of his lambs. He randomly selected 20 lambs. Ten of the lambs were given \(\operatorname { diet } A\) and the other 10 lambs were given diet \(B\). The gain in weight, in kg , of each lamb over the period of the trial was recorded.

1. State why a paired \(t\)-test is not suitable for use with these data.
2. Suggest an alternative method for selecting the sample which would make the use of a paired \(t\)-test valid.
3. Suggest two other factors that the farmer might consider when selecting the sample. The following paired data were collected.

   Diet \(A\)	5	6	7	4.6	6.1	5.7	6.2	7.4	5	3
   Diet \(B\)	7	7.2	8	6.4	5.1	7.9	8.2	6.2	6.1	5.8

4. Using a paired \(t\)-test, at the \(5 \%\) significance level, test whether or not there is evidence of a difference in the weight gained by the lambs using \(\operatorname { diet } A\) compared with those using \(\operatorname { diet } B\).
5. State, giving a reason, which diet you would recommend the farmer to use for his lambs.
   (Total 13 marks)

5. Define

1. a Type I error,
2. the size of a test. Jane claims that she can read Alan's mind. To test this claim Alan randomly chooses a card with one of 4 symbols on it. He then concentrates on the symbol. Jane then attempts to read Alan's mind by stating what symbol she thinks is on the card. The experiment is carried out 8 times and the number of times \(X\) that Jane is correct is recorded. The probability of Jane stating the correct symbol is denoted by \(p\).
   To test the hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\) against \(\mathrm { H } _ { 1 } : p > 0.25\), a critical region of \(X > 6\) is used.
3. Find the size of this test.
4. Show that the power function of this test is \(8 p ^ { 7 } - 7 p ^ { 8 }\). Given that \(p = 0.3\), calculate
5. the power of this test,
6. the probability of a Type II error.
7. Suggest two ways in which you might reduce the probability of a Type II error.

6. Brickland and Goodbrick are two manufacturers of bricks. The lengths of the bricks produced by each manufacturer can be assumed to be normally distributed. A random sample of 20 bricks is taken from Brickland and the length, \(x \mathrm {~mm}\), of each brick is recorded. The mean of this sample is 207.1 mm and the variance is \(3.2 \mathrm {~mm} ^ { 2 }\).

1. Calculate the \(98 \%\) confidence interval for the mean length of brick from Brickland. A random sample of 10 bricks is selected from those manufactured by Goodbrick. The length of each brick, \(y \mathrm {~mm}\), is recorded. The results are summarised as follows. $$\sum y = 2046.2 \quad \sum y ^ { 2 } = 418785.4$$ The variances of the length of brick for each manufacturer are assumed to be the same.
2. Find a \(90 \%\) confidence interval for the value by which the mean length of brick made by Brickland exceeds the mean length of brick made by Goodbrick.

7. A bag contains marbles of which an unknown proportion \(p\) is red. A random sample of \(n\) marbles is drawn, with replacement, from the bag. The number \(X\) of red marbles drawn is noted. A second random sample of \(m\) marbles is drawn, with replacement. The number \(Y\) of red marbles drawn is noted. Given that \(p _ { 1 } = \frac { a X } { n } + \frac { b Y } { m }\) is an unbiased estimator of \(p\),

1. show that \(a + b = 1\). Given that \(p _ { 2 } = \frac { ( X + Y ) } { n + m }\),
2. show that \(p _ { 2 }\) is an unbiased estimator for \(p\).
3. Show that the variance of \(p _ { 1 }\) is \(p ( 1 - p ) \left( \frac { a ^ { 2 } } { n } + \frac { b ^ { 2 } } { m } \right)\).
4. Find the variance of \(p _ { 2 }\).
5. Given that \(a = 0.4 , m = 10\) and \(n = 20\), explain which estimator \(p _ { 1 }\) or \(p _ { 2 }\) you should use.