CAIE S2 (Statistics 2) 2011 November

Test scores, \(X\), have mean 54 and variance 144. The scores are scaled using the formula \(Y = a + bX\), where \(a\) and \(b\) are constants and \(b > 0\). The scaled scores, \(Y\), have mean 50 and variance 100. Find the values of \(a\) and \(b\). [4]

35% of a random sample of \(n\) students walk to college. This result is used to construct an approximate 98% confidence interval for the population proportion of students who walk to college. Given that the width of this confidence interval is 0.157, correct to 3 significant figures, find \(n\). [5]

Jack has to choose a random sample of 8 people from the 750 members of a sports club.

1. Explain fully how he can use random numbers to choose the sample. [3]

Jack asks each person in the sample how much they spent last week in the club café. The results, in dollars, were as follows. 15 \quad 25 \quad 30 \quad 8 \quad 12 \quad 18 \quad 27 \quad 25

1. Find unbiased estimates of the population mean and variance. [3]
2. Explain briefly what is meant by 'population' in this question. [1]

The random variable \(X\) has probability density function given by $$f(x) = \begin{cases} ke^{-x} & 0 \leqslant x \leqslant 1, \\ 0 & \text{otherwise}. \end{cases}$$

1. Show that \(k = \frac{e}{e-1}\). [3]
2. Find E(\(X\)) in terms of \(e\). [4]

Records show that the distance driven by a bus driver in a week is normally distributed with mean 1150 km and standard deviation 105 km. New driving regulations are introduced and in the next 20 weeks he drives a total of 21 800 km.

1. Stating any assumption(s), test, at the 1% significance level, whether his mean weekly driving distance has decreased. [6]
2. A similar test at the 1% significance level was carried out using the data from another 20 weeks. State the probability of a Type I error and describe what is meant by a Type I error in this context. [2]

Ranjit goes to mathematics lectures and physics lectures. The length, in minutes, of a mathematics lecture is modelled by the variable \(X\) with distribution N(36, 3.5²). The length, in minutes, of a physics lecture is modelled by the independent variable \(Y\) with distribution N(55, 5.2²).

1. Find the probability that the total length of two mathematics lectures and one physics lecture is less than 140 minutes. [4]
2. Ranjit calculates how long he will need to spend revising the content of each lecture as follows. Each minute of a mathematics lecture requires 1 minute of revision and each minute of a physics lecture requires 1½ minutes of revision. Find the probability that the total revision time required for one mathematics lecture and one physics lecture is more than 100 minutes. [4]

The numbers of men and women who visit a clinic each hour are independent Poisson variables with means 2.4 and 2.8 respectively.

1. Find the probability that, in a half-hour period,
   1. 2 or more men and 1 or more women will visit the clinic, [4]
   2. a total of 3 or more people will visit the clinic. [3]
2. Find the probability that, in a 10-hour period, a total of more than 60 people will visit the clinic. [4]