OCR S3 (Statistics 3) 2007 January

1 The marks obtained by a randomly chosen student in the two papers of an examination are denoted by the random variables \(X\) and \(Y\), where \(X \sim \mathrm {~N} ( 45,81 )\) and \(Y \sim \mathrm {~N} ( 33,63 )\). The student's overall mark for the examination, \(T\), is given by \(T = X + \lambda Y\), where the constant \(\lambda\) is chosen such that \(\mathrm { E } ( T ) = 100\).

1. Show that \(\lambda = \frac { 5 } { 3 }\).
2. Assuming that \(X\) and \(Y\) are independent, state the distribution of \(T\), giving the values of its parameters.
3. Comment on the assumption of independence.

2 The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only with probability density function f . The graph of \(y = \mathrm { f } ( x )\) consists of the two line segments shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4a6d94a2-66e1-449a-ac0e-1fbada74bb3b-2_524_1287_950_429}

1. Show that \(a = \frac { 2 } { 3 }\).
2. Find the equations of the two line segments.
3. Hence write down the probability density function of \(X\).
4. Find \(\mathrm { E } ( X )\).

3 A new treatment of cotton thread, designed to increase the breaking strength, was tested on a random sample of 6 pieces of a standard length. The breaking strengths, in grams, were as follows. $$\begin{array} { l l l l l l } 17.3 & 18.4 & 18.6 & 17.2 & 17.5 & 19.3 \end{array}$$ The breaking strengths of a random sample of 5 similar pieces of the thread which had not been treated were as follows. \section*{\(\begin{array} { l l l l l } 18.6 & 17.2 & 16.3 & 17.4 & 16.8 \end{array}\)} A test of whether the treatment has been successful is to be carried out.

1. State what distributional assumptions are needed.
2. Carry out the test at the \(10 \%\) significance level.

4 A machine is set to produce metal discs with mean diameter 15.4 mm . In order to test the correctness of the setting, a random sample of 12 discs was selected and the diameters, \(x \mathrm {~mm}\), were measured. The results are summarised by \(\Sigma x = 177.6\) and \(\Sigma x ^ { 2 } = 2640.40\). Diameters may be assumed to be normally distributed with mean \(\mu \mathrm { mm }\).

1. Find a \(95 \%\) confidence interval for \(\mu\).
2. Test, at the \(5 \%\) significance level, the null hypothesis \(\mu = 15.4\) against the alternative hypothesis \(\mu < 15.4\).

5 Each person in a random sample of 1200 people was asked whether he or she approved of certain proposals to reduce atmospheric pollution. It was found that 978 people approved. The proportion of people in the whole population who would approve is denoted by \(p\).

1. Write down an estimate \(\hat { p }\) of \(p\).
2. Find a 90\% confidence interval for \(p\).
3. Explain, in the context of the question, the meaning of a \(90 \%\) confidence interval.
4. Estimate the sample size that would give a value for \(\hat { p }\) that differs from the value of \(p\) by less than 0.01 with probability \(90 \%\).

6 The lifetime of a particular machine, in months, can be modelled by the random variable \(T\) with probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { t ^ { 4 } } & t \geqslant 1 \\ 0 & \text { otherwise. } \end{cases}$$

1. Obtain the (cumulative) distribution function of \(T\).
2. Show that the probability density function of the random variable \(Y\), where \(Y = T ^ { 3 }\), is given by \(\mathrm { g } ( y ) = \frac { 1 } { y ^ { 2 } }\), for \(y \geqslant 1\).
3. Find \(\mathrm { E } ( \sqrt { Y } )\).

7 It is thought that a person's eye colour is related to the reaction of the person's skin to ultra-violet light. As part of a study, a random sample of 140 people were treated with a standard dose of ultra-violet light. The degree of reaction was classified as None, Mild or Strong. The results are given in Table 1. The corresponding expected frequencies for a \(\chi ^ { 2 }\) test of association between eye colour and reaction are shown in Table 2. \begin{table}[h] \end{table} \begin{table}[h] \end{table}

1. (a) State suitable hypotheses for the test.
   (b) Show how the expected frequency of 18.11 in Table 2 is obtained.
   (c) Show that the three cells in the top row together contribute 4.53 to the calculated value of \(\chi ^ { 2 }\), correct to 2 decimal places.
   (d) You are given that the total calculated value of \(\chi ^ { 2 }\) is 12.78 , correct to 2 decimal places. Give the smallest value of \(\alpha\) obtained from the tables for which the null hypothesis would be rejected at the \(\alpha \%\) significance level.
2. Test, at the \(5 \%\) significance level, whether the proportions of people in the whole population with blue eyes, brown eyes and other colours are in the ratios \(2 : 2 : 1\).