CAIE S2 (Statistics 2) 2014 June

1 On average 1 in 25000 people have a rare blood condition. Use a suitable approximating distribution to find the probability that fewer than 2 people in a random sample of 100000 have the condition.

2
\includegraphics[max width=\textwidth, alt={}, center]{43b2498f-73e2-4d33-adaf-fc3e460fa36a-2_358_1093_495_520} A random variable \(X\) takes values between 0 and 4 only and has probability density function as shown in the diagram. Calculate the median of \(X\).

3 A die is thrown 100 times and shows an odd number on 56 throws. Calculate an approximate \(97 \%\) confidence interval for the probability that the die shows an odd number on one throw.

4 The weights, \(X\) kilograms, of rabbits in a certain area have population mean \(\mu \mathrm { kg }\). A random sample of 100 rabbits from this area was taken and the weights are summarised by $$\Sigma x = 165 , \quad \Sigma x ^ { 2 } = 276.25 .$$ Test at the \(5 \%\) significance level the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 1.6\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 1.6\).

5 The lifetime, \(X\) years, of a certain type of battery has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 1 \leqslant x \leqslant a
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are positive constants.

1. State what the value of \(a\) represents in this context.
2. Show that \(k = \frac { a } { a - 1 }\).
3. Experience has shown that the longest that any battery of this type lasts is 2.5 years. Find the mean lifetime of batteries of this type.

6 A machine is designed to generate random digits between 1 and 5 inclusive. Each digit is supposed to appear with the same probability as the others, but Max claims that the digit 5 is appearing less often than it should. In order to test this claim the manufacturer uses the machine to generate 25 digits and finds that exactly 1 of these digits is a 5 .

1. Carry out a test of Max's claim at the \(2.5 \%\) significance level.
2. Max carried out a similar hypothesis test by generating 1000 digits between 1 and 5 inclusive. The digit 5 appeared 180 times. Without carrying out the test, state the distribution that Max should use, including the values of any parameters.
3. State what is meant by a Type II error in this context.

7 A Lost Property office is open 7 days a week. It may be assumed that items are handed in to the office randomly, singly and independently.

1. State another condition for the number of items handed in to have a Poisson distribution. It is now given that the number of items handed in per week has the distribution \(\operatorname { Po } ( 4.0 )\).
2. Find the probability that exactly 2 items are handed in on a particular day.
3. Find the probability that at least 4 items are handed in during a 10-day period.
4. Find the probability that, during a certain week, 5 items are handed in altogether, but no items are handed in on the first day of the week.

8 In an examination, the marks in the theory paper and the marks in the practical paper are denoted by the random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} ( 57,13 )\) and \(Y \sim \mathrm {~N} ( 28,5 )\). You may assume that each candidate's marks in the two papers are independent. The final score of each candidate is found by calculating \(X + 2.5 Y\). A candidate is chosen at random. Without using a continuity correction, find the probability that this candidate

1. has a final score that is greater than 140 ,
2. obtains at least 20 more marks in the theory paper than in the practical paper.