OCR S3 (Statistics 3) 2016 June

1 On a motorway, lorries pass an observation point independently and at random times. The mean number of lorries travelling north is 6 per minute and the mean number travelling south is 8 per minute. Find the probability that at least 16 lorries pass the observation point in a given 1 -minute period.

2 A random sample of 200 American voters were asked about which political party they supported and their attitude to a proposed new form of taxation. The voters' responses are summarised in the table. Attitude Carry out a \(\chi ^ { 2 }\) test, at the \(1 \%\) level of significance, to investigate whether there is an association between party supported and attitude to the proposed form of taxation.

3

1. A company packages butter. Of 50 randomly selected packs, 8 were found to have damaged wrappers. Find an approximate \(95 \%\) confidence interval for the proportion of packs with damaged wrappers.
2. The mass of a pack has a normal distribution with standard deviation 8.5 g . In a random sample of 10 packs the masses, in g , are as follows. $$\begin{array} { l l l l l l l l l l } 220 & 225 & 218 & 223 & 224 & 220 & 229 & 228 & 226 & 228 \end{array}$$ Find a 99\% confidence interval for the mean mass of a pack.

4 A group of students were tested in geography before and after a fieldwork course. The marks of 10 randomly selected students are shown in the table.

1. Use a suitable \(t\)-test, at the \(5 \%\) level of significance, to test whether the students' performance has improved.
2. State the necessary assumption in applying the test.

5 The independent random variables \(X\) and \(Y\) have distributions \(\mathrm { N } \left( 30 , \sigma ^ { 2 } \right)\) and \(\mathrm { N } \left( 20 , \sigma ^ { 2 } \right)\) respectively. The random variable \(a X + b Y\), where \(a\) and \(b\) are constants, has the distribution \(\mathrm { N } \left( 410,130 \sigma ^ { 2 } \right)\).

1. Given that \(a\) and \(b\) are integers, find the value of \(a\) and the value of \(b\).
2. Given that \(\mathrm { P } ( X > Y ) = 0.966\), find \(\sigma ^ { 2 }\).

6 The masses at birth, in kg, of random samples of babies were recorded for each of the years 1970 and 2010. The table shows the sample mean and an unbiased estimate of the population variance for each year.

1. A researcher tests the null hypothesis that babies born in 2010 are 0.04 kg heavier, on average, than babies born in 1970, against the alternative hypothesis that they are more than 0.04 kg heavier on average. Show that, at the \(5 \%\) level of significance, the null hypothesis is not rejected.
2. Another researcher chooses samples of equal size, \(n\), for the two years. Using the same hypothesis as in part (i), she finds that the null hypothesis is rejected at the \(5 \%\) level of significance. Assuming that the sample means and unbiased estimates of population variance for the two years are as given in the table, find the smallest possible value of \(n\).

7 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 1
a x ^ { 2 } & 1 < x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac { 12 } { 31 }\).
2. Find \(\mathrm { E } ( X )\). It is thought that the time taken by a student to complete a task can be well modelled by \(X\). The times taken by 992 randomly chosen students are summarised in the table, together with some of the expected frequencies.

   Time	\(0 \leqslant x < 0.5\)	\(0.5 \leqslant x < 1\)	\(1 \leqslant x < 1.5\)	\(1.5 \leqslant x \leqslant 2\)
   Observed frequency	8	92	279	613
   Expected frequency	6	90

3. Find the other expected frequencies and test, at the \(5 \%\) level of significance, whether the data can be well modelled by \(X\).

8 The radius, \(R\), of a sphere is a random variable with a continuous uniform distribution between 0 and 10 .

1. Find the cumulative distribution function and probability density function of \(A\), the surface area of the sphere.
2. Find \(\mathrm { P } ( \mathrm { A } \leqslant 200 \pi )\). \section*{END OF QUESTION PAPER}