CAIE S2 (Statistics 2) 2022 June

1

1. A javelin thrower noted the lengths of a random sample of 50 of her throws. The sample mean was 72.3 m and an unbiased estimate of the population variance was \(64.3 \mathrm {~m} ^ { 2 }\). Find a \(92 \%\) confidence interval for the population mean length of throws by this athlete.
2. A discus thrower wishes to calculate a \(92 \%\) confidence interval for the population mean length of his throws. He bases his calculation on his first 50 throws in a week. Comment on this method.

2 In the past, the mean height of plants of a particular species has been 2.3 m . A random sample of 60 plants of this species was treated with fertiliser and the mean height of these 60 plants was found to be 2.4 m . Assume that the standard deviation of the heights of plants treated with fertiliser is 0.4 m . Carry out a test at the \(2.5 \%\) significance level of whether the mean height of plants treated with fertiliser is greater than 2.3 m .

3 It is known that \(1.8 \%\) of children in a certain country have not been vaccinated against measles. A random sample of 200 children in this country is chosen.

1. Use a suitable approximating distribution to find the probability that there are fewer than 3 children in the sample who have not been vaccinated against measles.
2. Justify your approximating distribution.

4 The number of cars arriving at a certain road junction on a weekday morning has a Poisson distribution with mean 4.6 per minute. Traffic lights are installed at the junction and council officer wishes to test at the \(2 \%\) significance level whether there are now fewer cars arriving. He notes the number of cars arriving during a randomly chosen 2 -minute period.

1. State suitable null and alternative hypotheses for the test.
2. Find the critical region for the test.
   The officer notes that, during a randomly chosen 2 -minute period on a weekday morning, exactly 5 cars arrive at the junction.
3. Carry out the test.
4. State, with a reason, whether it is possible that a Type I error has been made in carrying out the test in part (c).
   The number of cars arriving at another junction on a weekday morning also has a Poisson distribution with mean 4.6 per minute.
5. Use a suitable approximating distribution to find the probability that more than 300 cars will arrive at this junction in an hour.

5 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 16 } \left( 4 x - x ^ { 2 } \right) & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$

1. Show that \(\mathrm { E } ( X ) = \frac { 11 } { 4 }\).
2. Find \(\operatorname { Var } ( X )\).
3. Given that the median of \(X\) is \(m\), find \(\mathrm { P } ( m < X < 3 )\).

6 The masses, in kilograms, of large and small sacks of grain have the distributions \(\mathrm { N } ( 53,11 )\) and \(\mathrm { N } ( 14,3 )\) respectively.

1. Find the probability that the mass of a randomly chosen large sack is greater than four times the mass of a randomly chosen small sack.
2. A lift can safely carry a maximum mass of 1000 kg . Find the probability that the lift can safely carry 12 randomly chosen large sacks and 25 randomly chosen small sacks.
   \(7 X\) is a random variable with distribution \(\operatorname { Po } ( 2.90 )\). A random sample of 100 values of \(X\) is taken. Find the probability that the sample mean is less than 2.88 .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.