Edexcel S4 (Statistics 4) 2013 June

1. George owns a garage and he records the mileage of cars, \(x\) thousands of miles, between services. The results from a random sample of 10 cars are summarised below.

$$\sum x = 113.4 \quad \sum x ^ { 2 } = 1414.08$$ The mileage of cars between services is normally distributed and George believes that the standard deviation is 2.4 thousand miles. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not these data support George’s belief.

2. Every 6 months some engineers are tested to see if their times, in minutes, to assemble a particular component have changed. The times taken to assemble the component are normally distributed. A random sample of 8 engineers was chosen and their times to assemble the component were recorded in January and in July. The data are given in the table below.

1. Calculate a \(95 \%\) confidence interval for the mean difference in times.
2. Use your confidence interval to state, giving a reason, whether or not there is evidence of a change in the mean time to assemble a component. State your hypotheses clearly.

3. An archaeologist is studying the compression strength of bricks at some ancient European sites. He took random samples from two sites \(A\) and \(B\) and recorded the compression strength of these bricks in appropriate units. The results are summarised below. It can be assumed that the compression strength of bricks is normally distributed.

1. Test, at the \(2 \%\) level of significance, whether or not there is evidence of a difference in the variances of compression strength of the bricks between these two sites. State your hypotheses clearly.
   (5) Site \(A\) is older than site \(B\) and the archaeologist claims that the mean compression strength of the bricks was greater at the younger site.
2. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test the archaeologist's claim.
3. Explain briefly the importance of the test in part (a) to the test in part (b).

1. A random sample of size \(2 , X _ { 1 }\) and \(X _ { 2 }\), is taken from the random variable \(X\) which has a continuous uniform distribution over the interval \([ - a , 2 a ] , a > 0\)
   1. Show that \(\bar { X } = \frac { X _ { 1 } + X _ { 2 } } { 2 }\) is a biased estimator of \(a\) and find the bias.
   The random variable \(Y = k \bar { X }\) is an unbiased estimator of \(a\).
2. Write down the value of the constant \(k\).
3. Find \(\operatorname { Var } ( Y )\). The random variable \(M\) is the maximum of \(X _ { 1 }\) and \(X _ { 2 }\)
   The probability density function, \(m ( x )\), of \(M\) is given by $$m ( x ) = \left\{ \begin{array} { c l } \frac { 2 ( x + a ) } { 9 a ^ { 2 } } & - a \leqslant x \leqslant 2 a
   0 & \text { otherwise } \end{array} \right.$$
4. Show that \(M\) is an unbiased estimator of \(a\). Given that \(\mathrm { E } \left( M ^ { 2 } \right) = \frac { 3 } { 2 } a ^ { 2 }\)
5. find \(\operatorname { Var } ( M )\).
6. State, giving a reason, whether you would use \(Y\) or \(M\) as an estimator of \(a\). A random sample of two values of \(X\) are 5 and - 1
7. Use your answer to part (f) to estimate \(a\).

5. Water is tested at various stages during a purification process by an environmental scientist. A certain organism occurs randomly in the water at a rate of \(\lambda\) every 10 ml . The scientist selects a random sample of 20 ml of water to check whether there is evidence that \(\lambda\) is greater than 1 . The criterion the scientist uses for rejecting the hypothesis that \(\lambda = 1\) is that there are 4 or more organisms in the sample of 20 ml .

1. Find the size of the test.
2. When \(\lambda = 2.5\) find P (Type II error). A statistician suggests using an alternative test. The statistician's test involves taking a random sample of 10 ml and rejecting the hypothesis that \(\lambda = 1\) if 2 or more organisms are present but accepting the hypothesis if no organisms are in the sample. If only 1 organism is found then a second random sample of 10 ml is taken and the hypothesis is rejected if 2 or more organisms are present, otherwise the hypothesis is accepted.
3. Show that the power of the statistician's test is given by $$1 - \mathrm { e } ^ { - \lambda } - \lambda ( 1 + \lambda ) \mathrm { e } ^ { - 2 \lambda }$$ Table 1 below gives some values, to 2 decimal places, of the power function of the statistician’s test. \begin{table}[h] \end{table} Table 1 Figure 1 shows a graph of the power function for the scientist's test.
4. On the same axes draw the graph of the power function for the statistician’s test. Given that it takes 20 minutes to collect and test a 20 ml sample and 15 minutes to collect and test a 10 ml sample
5. show that the expected time of the statistician's test is slower than the scientist's test for \(\lambda \mathrm { e } ^ { - \lambda } > \frac { 1 } { 3 }\)
6. By considering the times when \(\lambda = 1\) and \(\lambda = 2\) together with the power curves in part (e) suggest, giving a reason, which test you would use.
   (2) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{399f7507-4878-45ad-b77e-02ebd807ed75-10_1185_1157_1452_392} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{399f7507-4878-45ad-b77e-02ebd807ed75-11_81_47_2622_1886}

6. The carbon content, measured in suitable units, of steel is normally distributed. Two independent random samples of steel were taken from a refining plant at different times and their carbon content recorded. The results are given below. Sample A: \(\quad 1.5 \quad 0.9 \quad 1.3 \quad 1.2\)
\(\begin{array} { l l l l l l l } \text { Sample } B : & 0.4 & 0.6 & 0.8 & 0.3 & 0.5 & 0.4 \end{array}\)

1. Stating your hypotheses clearly, carry out a suitable test, at the \(10 \%\) level of significance, to show that both samples can be assumed to have come from populations with a common variance \(\sigma ^ { 2 }\).
2. Showing your working clearly, find the \(99 \%\) confidence interval for \(\sigma ^ { 2 }\) based on both samples.