CAIE S2 (Statistics 2) 2024 November

1 The heights of a certain species of deer are known to have standard deviation 0.35 m . A zoologist takes a random sample of 150 of these deer and finds that the mean height of the deer in the sample is 1.42 m .

1. Calculate a 96\% confidence interval for the population mean height.
2. Bubay says that \(96 \%\) of deer of this species are likely to have heights that are within this confidence interval. Explain briefly whether Bubay is correct.

2 The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\mathrm { N } ( 16.0,0.4 )\) and \(\mathrm { N } ( 51.0,0.9 )\) respectively. Find the probability that the total mass of 3 randomly chosen small bags is greater than the mass of one randomly chosen large bag.
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3 The times, \(T\) minutes, taken by a random sample of 75 students to complete a test were noted. The results were summarised by \(\Sigma t = 230\) and \(\Sigma t ^ { 2 } = 930\).

1. Calculate unbiased estimates of the population mean and variance of \(T\).
   You should now assume that your estimates from part (a) are the true values of the population mean and variance of \(T\).
2. The times taken by another random sample of 75 students were noted, and the sample mean, \(\bar { T }\), was found. Find the value of \(a\) such that \(P ( \bar { T } > a ) = 0.234\).

4 A random variable \(X\) has probability density function f defined by $$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } - \frac { 18 } { x ^ { 3 } } & 2 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac { 27 } { 2 }\).
2. Show that \(\mathrm { E } ( X ) = \frac { 27 } { 2 } \ln \frac { 3 } { 2 } - 3\).

5 The lengths, in centimetres, of worms of a certain kind are normally distributed with mean \(\mu\) and standard deviation 2.3. An article in a magazine states that the value of \(\mu\) is 12.7 . A scientist wishes to test whether this value is correct. He measures the lengths, \(x \mathrm {~cm}\), of a random sample of 50 worms of this kind and finds that \(\sum x = 597.1\). He plans to carry out a test, at the \(1 \%\) significance level, of whether the true value of \(\mu\) is different from 12.7 .

1. State, with a reason, whether he should use a one-tailed or a two-tailed test.
2. Carry out the test.

6 The numbers of customers arriving at service desks \(A\) and \(B\) during a 10 -minute period have the independent distributions \(\operatorname { Po } ( 1.8 )\) and \(\operatorname { Po } ( 2.1 )\) respectively.

1. Find the probability that during a randomly chosen 15 -minute period more than 2 customers will arrive at \(\operatorname { desk } A\).
2. Find the probability that during a randomly chosen 5-minute period the total number of customers arriving at both desks is less than 4 .
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3. An inspector waits at desk \(B\). She wants to wait long enough to be \(90 \%\) certain of seeing at least one customer arrive at the desk. Find the minimum time for which she should wait, giving your answer correct to the nearest minute.

7 The number of accidents per year on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 3.3 . Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5 \%\) significance level.

1. Calculate the probability of a Type I error.
2. Given that \(X = 2\), carry out the test.
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   \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-11_2716_29_107_22}
3. The council decides to carry out another similar test at the \(5 \%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is 0.6 , calculate the probability of a Type II error.
4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than 10 accidents in 30 years.
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