CAIE S2 (Statistics 2) 2012 November

1
\includegraphics[max width=\textwidth, alt={}, center]{879cb813-2380-47a7-bd96-cad0a74d0b4d-2_369_531_255_806} The diagram shows the graph of the probability density function, f , of a random variable \(X\). Find the median of \(X\).

2 The heights of a certain type of plant have a normal distribution. When the plants are grown without fertilizer, the population mean and standard deviation are 24.0 cm and 4.8 cm respectively. A gardener wishes to test, at the \(2 \%\) significance level, whether Hiergro fertilizer will increase the mean height. He treats 150 randomly chosen plants with Hiergro and finds that their mean height is 25.0 cm . Assuming that the standard deviation of the heights of plants treated with Hiergro is still 4.8 cm , carry out the test.

3 The cost of hiring a bicycle consists of a fixed charge of 500 cents together with a charge of 3 cents per minute. The number of minutes for which people hire a bicycle has mean 142 and standard deviation 35.

1. Find the mean and standard deviation of the amount people pay when hiring a bicycle.
2. 6 people hire bicycles independently. Find the mean and standard deviation of the total amount paid by all 6 people.

4 A cereal manufacturer claims that \(25 \%\) of cereal packets contain a free gift. Lola suspects that the true proportion is less than \(25 \%\). In order to test the manufacturer's claim at the \(5 \%\) significance level, she checks a random sample of 20 packets.

1. Find the critical region for the test.
2. Hence find the probability of a Type I error. Lola finds that 2 packets in her sample contain a free gift.
3. State, with a reason, the conclusion she should draw.

5 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x - 1 } & 3 \leqslant x \leqslant 5
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { \ln 2 }\).
2. Find \(a\) such that \(\mathrm { P } ( X < a ) = 0.75\).

6 In order to obtain a random sample of people who live in her town, Jane chooses people at random from the telephone directory for her town.

1. Give a reason why Jane's method will not give a random sample of people who live in the town. Jane now uses a valid method to choose a random sample of 200 people from her town and finds that 38 live in apartments.
2. Calculate an approximate \(99 \%\) confidence interval for the proportion of all people in Jane's town who live in apartments.
3. Jane uses the same sample to give a confidence interval of width 0.1 for this proportion. This interval is an \(x \%\) confidence interval. Find the value of \(x\).

7 A random variable \(X\) has the distribution \(\operatorname { Po } ( 1.6 )\).

1. The random variable \(R\) is the sum of three independent values of \(X\). Find \(\mathrm { P } ( R < 4 )\).
2. The random variable \(S\) is the sum of \(n\) independent values of \(X\). It is given that $$\mathrm { P } ( S = 4 ) = \frac { 16 } { 3 } \times \mathrm { P } ( S = 2 )$$ Find \(n\).
3. The random variable \(T\) is the sum of 40 independent values of \(X\). Find \(\mathrm { P } ( T > 75 )\).