CAIE S2 (Statistics 2) 2017 November

1

1. 1. A random variable \(X\) has the distribution \(\mathrm { B } ( 2540,0.001 )\). Use the Poisson approximation to the binomial distribution to find \(\mathrm { P } ( X > 1 )\).
   2. Explain why the Poisson approximation is appropriate in this case.
2. Two independent random variables, \(S\) and \(T\), have distributions \(\operatorname { Po } ( 2.1 )\) and \(\operatorname { Po } ( 3.5 )\) respectively. Find the mean and standard deviation of \(S + T\).

2 The number of words in History essays by students at a certain college has mean \(\mu\) and standard deviation 1420.

1. The mean number of words in a random sample of 125 History essays was found to be 4820 . Calculate a \(98 \%\) confidence interval for \(\mu\).
2. Another random sample of \(n\) History essays was taken. Using this sample, a \(95 \%\) confidence interval for \(\mu\) was found to be 4700 to 4980 , both correct to the nearest integer. Find the value of \(n\).

3 The masses, \(m \mathrm {~kg}\), of packets of flour are normally distributed. The mean mass is supposed to be 1.01 kg . A quality control officer measures the masses of a random sample of 100 packets. The results are summarised below. $$n = 100 \quad \Sigma m = 98.2 \quad \Sigma m ^ { 2 } = 104.52$$

1. Test at the \(5 \%\) significance level whether the population mean mass is less than 1.01 kg .
2. Explain whether it was necessary to use the Central Limit theorem in your answer to part (i).

4 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { \sqrt { } x } & 0 < x \leqslant a
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants. It is given that \(\mathrm { E } ( X ) = 3\).

1. Find the value of \(a\) and show that \(k = \frac { 1 } { 6 }\).
2. Find the median of \(X\).

5 The marks in paper 1 and paper 2 of an examination are denoted by \(X\) and \(Y\) respectively, where \(X\) and \(Y\) have the independent continuous distributions \(\mathrm { N } \left( 56,6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 43,5 ^ { 2 } \right)\) respectively.

1. Find the probability that a randomly chosen paper 1 mark is more than a randomly chosen paper 2 mark.
2. Each candidate's overall mark is \(M\) where \(M = X + 1.5 Y\). The minimum overall mark for grade A is 135 . Find the proportion of students who gain a grade A .

6 In a certain factory the number of items per day found to be defective has had the distribution \(\operatorname { Po } ( 1.03 )\). After the introduction of new quality controls, the management wished to test at the \(10 \%\) significance level whether the mean number of defective items had decreased. They noted the total number of defective items produced in 5 randomly chosen days. It is assumed that defective items occur randomly and that a Poisson model is still appropriate.

1. Given that the total number of defective items produced during the 5 days was 2 , carry out the test.
2. Using another random sample of 5 days the same test is carried out again, with the same significance level. Find the probability of a Type I error.
3. Explain what is meant by a Type I error in this context.