CAIE S2 (Statistics 2) 2007 November

1 Isaac claims that \(30 \%\) of cars in his town are red. His friend Hardip thinks that the proportion is less than \(30 \%\). The boys decided to test Isaac's claim at the \(5 \%\) significance level and found that 2 cars out of a random sample of 18 were red. Carry out the hypothesis test and state your conclusion. [5]

2 In summer the growth rate of grass in a lawn has a normal distribution with mean 3.2 cm per week and standard deviation 1.4 cm per week. A new type of grass is introduced which the manufacturer claims has a slower growth rate. A hypothesis test of this claim at the \(5 \%\) significance level was carried out using a random sample of 10 lawns that had the new grass. It may be assumed that the growth rate of the new grass has a normal distribution with standard deviation 1.4 cm per week.

1. Find the rejection region for the test.
2. The probability of making a Type II error when the actual value of the mean growth rate of the new grass is \(m \mathrm {~cm}\) per week is less than 0.5 . Use your answer to part (i) to write down an inequality for \(m\).

3

1. Explain what is meant by the term 'random sample'. In a random sample of 350 food shops it was found that 130 of them had Special Offers.
2. Calculate an approximate \(95 \%\) confidence interval for the proportion of all food shops with Special Offers.
3. Estimate the size of a random sample required for an approximate \(95 \%\) confidence interval for this proportion to have a width of 0.04 .

4 The cost of electricity for a month in a certain town under scheme \(A\) consists of a fixed charge of 600 cents together with a charge of 5.52 cents per unit of electricity used. Stella uses scheme \(A\). The number of units she uses in a month is normally distributed with mean 500 and variance 50.41.

1. Find the mean and variance of the total cost of Stella's electricity in a randomly chosen month. Under scheme \(B\) there is no fixed charge and the cost in cents for a month is normally distributed with mean 6600 and variance 421. Derek uses scheme \(B\).
2. Find the probability that, in a randomly chosen month, Derek spends more than twice as much as Stella spends.

5 The length, \(X \mathrm {~cm}\), of a piece of wooden planking is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { b } & 0 \leqslant x \leqslant b
0 & \text { otherwise } \end{cases}$$ where \(b\) is a positive constant.

1. Find the mean and variance of \(X\) in terms of \(b\). The lengths of a random sample of 100 pieces were measured and it was found that \(\Sigma x = 950\).
2. Show that the value of \(b\) estimated from this information is 19 . Using this value of \(b\),
3. find the probability that the length of a randomly chosen piece is greater than 11 cm ,
4. find the probability that the mean length of a random sample of 336 pieces is less than 9 cm .

6 The random variable \(X\) denotes the number of worms on a one metre length of a country path after heavy rain. It is given that \(X\) has a Poisson distribution.

1. For one particular path, the probability that \(X = 2\) is three times the probability that \(X = 4\). Find the probability that there are more than 3 worms on a 3.5 metre length of this path.
2. For another path the mean of \(X\) is 1.3.
   (a) On this path the probability that there is at least 1 worm on a length of \(k\) metres is 0.96 . Find \(k\).
   (b) Find the probability that there are more than 1250 worms on a one kilometre length of this path.