OCR S3 (Statistics 3) 2013 June

1 The blood-test procedure at a clinic is that a person arrives, takes a numbered ticket and waits for that number to be called. The waiting times between the numbers called have independent normal distributions with mean 3.5 minutes and standard deviation 0.9 minutes. My ticket is number 39 and as I take my ticket number 1 is being called, so that I have to wait for 38 numbers to be called. Find the probability that I will have to wait between 120 minutes and 140 minutes.

2 In order to estimate the total number of rabbits in a certain region, a random sample of 500 rabbits is captured, marked and released. After two days a random sample of 250 rabbits is captured and 24 are found to be marked. It may be assumed that there is no change in the population during the two days.

1. Estimate the total number of rabbits in the region.
2. Calculate an approximate \(95 \%\) confidence interval for the population proportion of marked rabbits.
3. Using your answer to part (ii), estimate a 95\% confidence interval for the total number of rabbits in the region.

3
\includegraphics[max width=\textwidth, alt={}, center]{c4adc528-ae3f-4ea7-9420-d3e1068a85fe-2_524_796_1105_623} The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a x & 0 < x \leqslant 1
b ( 2 - x ) ^ { 2 } & 1 < x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants. The graph is shown in the above diagram.

1. Find the values of \(a\) and \(b\).
2. Find the value of \(\mathrm { E } \left( \frac { 1 } { X } \right)\).

4 A new computer was bought by a local council to search council records and was tested by an employee. She searched a random sample of 500 records and the sample mean search time was found to be 2.18 milliseconds and an unbiased estimate of variance was \(1.58 ^ { 2 }\) milliseconds \({ } ^ { 2 }\).

1. Calculate a \(98 \%\) confidence interval for the population mean search time \(\mu\) milliseconds.
2. It is required to obtain a sample mean time that differs from \(\mu\) by less than 0.05 milliseconds with probability 0.95 . Estimate the sample size required.
3. State why it is unnecessary for the validity of your calculations that search time has a normal distribution.

5 The continuous random variable \(Y\) has probability density function given by $$\mathrm { f } ( y ) = \begin{cases} \ln ( y ) & 1 \leqslant y \leqslant \mathrm { e }
0 & \text { otherwise } \end{cases}$$

1. Verify that this is a valid probability density function.
2. Show that the (cumulative) distribution function of \(Y\) is given by $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 1
   y \ln y - y + 1 & 1 \leqslant y \leqslant \mathrm { e }
   1 & \text { otherwise } \end{cases}$$
3. Verify that the upper quartile of \(Y\) lies in the interval [2.45, 2.46].
4. Find the (cumulative) distribution function of \(X\) where \(X = \ln Y\).

6 A random sample of 80 students who had all studied Biology, Chemistry and Art at a college was each asked which they enjoyed most. The results, classified according to gender, are given in the table. It is required to carry out a test of independence between subject most enjoyed and gender at the \(2 \frac { 1 } { 2 } \%\) significance level.

1. Calculate the expected values for the cells.
2. Explain why it is necessary to combine cells, and choose a suitable combination.
3. Carry out the test.

7 Two machines \(A\) and \(B\) both pack cartons in a factory. The mean packing times are compared by timing the packing of 10 randomly chosen cartons from machine \(A\) and 8 randomly chosen cartons from machine \(B\). The times, \(t\) seconds, taken to pack these cartons are summarised below. The packing times have independent normal distributions.

1. Stating a necessary assumption, carry out a test, at the \(1 \%\) significance level, of whether the population mean packing times differ for the two machines.
2. Find the largest possible value of the constant \(c\) for which there is evidence at the \(1 \%\) significance level that \(\mu _ { B } - \mu _ { A } > c\), where \(\mu _ { B }\) and \(\mu _ { A }\) denote the respective population mean packing times in seconds.