CAIE S2 (Statistics 2) 2015 June

1 Jyothi wishes to choose a representative sample of 5 students from the 82 members of her school year.

1. She considers going into the canteen and choosing a table with five students from her year sitting at it, and using these five people as her sample. Give two reasons why this method is unsatisfactory.
2. Jyothi decides to use another method. She numbers all the students in her year from 1 to 82 . Then she uses her calculator and generates the following random numbers. $$231492 \quad 762305 \quad 346280$$ From these numbers, she obtains the student numbers \(23,14,76,5,34\) and 62 . Explain how Jyothi obtained these student numbers from the list of random numbers.

2 Marie claims that she can predict the winning horse at the local races. There are 8 horses in each race. Nadine thinks that Marie is just guessing, so she proposes a test. She asks Marie to predict the winners of the next 10 races and, if she is correct in 3 or more races, Nadine will accept Marie's claim.

1. State suitable null and alternative hypotheses.
2. Calculate the probability of a Type I error.
3. State the significance level of the test.

3 A die is biased so that the probability that it shows a six on any throw is \(p\).

1. In an experiment, the die shows a six on 22 out of 100 throws. Find an approximate \(97 \%\) confidence interval for \(p\).
2. The experiment is repeated and another \(97 \%\) confidence interval is found. Find the probability that exactly one of the two confidence intervals includes the true value of \(p\).

4 The marks, \(x\), of a random sample of 50 students in a test were summarised as follows. $$n = 50 \quad \Sigma x = 1508 \quad \Sigma x ^ { 2 } = 51825$$

1. Calculate unbiased estimates of the population mean and variance.
2. Each student's mark is scaled using the formula \(y = 1.5 x + 10\). Find estimates of the population mean and variance of the scaled marks, \(y\).

5 The mean breaking strength of cables made at a certain factory is supposed to be 5 tonnes. The quality control department wishes to test whether the mean breaking strength of cables made by a particular machine is actually less than it should be. They take a random sample of 60 cables. For each cable they find the breaking strength by gradually increasing the tension in the cable and noting the tension when the cable breaks.

1. Give a reason why it is necessary to take a sample rather then testing all the cables produced by the machine.
2. The mean breaking strength of the 60 cables in the sample is found to be 4.95 tonnes. Given that the population standard deviation of breaking strengths is 0.15 tonnes, test at the \(1 \%\) significance level whether the population mean breaking strength is less than it should be.
3. Explain whether it was necessary to use the Central Limit theorem in the solution to part (ii).

6 People arrive at a checkout in a store at random, and at a constant mean rate of 0.7 per minute. Find the probability that

1. exactly 3 people arrive at the checkout during a 5 -minute period,
2. at least 30 people arrive at the checkout during a 1-hour period. People arrive independently at another checkout in the store at random, and at a constant mean rate of 0.5 per minute.
3. Find the probability that a total of more than 3 people arrive at this pair of checkouts during a 2-minute period.

7 The probability density function of the random variable \(X\) is given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 } x ( c - x ) & 0 \leqslant x \leqslant c
0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant.

1. Show that \(c = 2\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\) and state the median of \(X\).
3. Find \(\mathrm { P } ( X < 1.5 )\).
4. Hence write down the value of \(\mathrm { P } ( 0.5 < X < 1 )\).