CAIE S2 (Statistics 2) 2014 June

1 The weights, in grams, of a random sample of 8 packets of cereal are as follows. $$\begin{array} { l l l l l l l l } 250 & 248 & 255 & 244 & 259 & 250 & 242 & 258 \end{array}$$ Calculate unbiased estimates of the population mean and variance.

2 Each day Samuel travels from \(A\) to \(B\) and from \(B\) to \(C\). He then returns directly from \(C\) to \(A\). The times, in minutes, for these three journeys have the independent distributions \(\mathrm { N } \left( 20,2 ^ { 2 } \right) , \mathrm { N } \left( 18,1.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 30,1.8 ^ { 2 } \right)\), respectively. Find the probability that, on a randomly chosen day, the total time for his two journeys from \(A\) to \(B\) and \(B\) to \(C\) is less than the time for his return journey from \(C\) to \(A\). [5]

3 The number of calls per day to an enquiry desk has a Poisson distribution. In the past the mean has been 5 . In order to test whether the mean has changed, the number of calls on a random sample of 10 days was recorded. The total number of calls was found to be 61 . Use an approximate distribution to test at the 10\% significance level whether the mean has changed.

4

1. The random variable \(W\) has the distribution \(\operatorname { Po } ( 1.5 )\). Find the probability that the sum of 3 independent values of \(W\) is greater than 2 .
2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Given that \(\mathrm { P } ( X = 0 ) = 0.523\), find the value of \(\lambda\) correct to 3 significant figures.
3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\), where \(\mu \neq 0\). Given that $$\mathrm { P } ( Y = 3 ) = 24 \times \mathrm { P } ( Y = 1 )$$ find \(\mu\).

5 Mahmoud throws a coin 400 times and finds that it shows heads 184 times. The probability that the coin shows heads on any throw is denoted by \(p\).

1. Calculate an approximate \(95 \%\) confidence interval for \(p\).
2. Mahmoud claims that the coin is not fair. Use your answer to part (i) to comment on this claim.
3. Mahmoud's result of 184 heads in 400 throws gives an \(\alpha \%\) confidence interval for \(p\) with width 0.1 . Calculate the value of \(\alpha\).

6 The time, \(T\) hours, spent by people on a visit to a museum has probability density function $$\mathrm { f } ( t ) = \begin{cases} k t \left( 16 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 64 }\).
2. Calculate the probability that two randomly chosen people each spend less than 1 hour on a visit to the museum.
3. Find the mean time spent on a visit to the museum.

7 A researcher is investigating the actual lengths of time that patients spend with the doctor at their appointments. He plans to choose a sample of 12 appointments on a particular day.

1. Which of the following methods is preferable, and why?
   • Choose the first 12 appointments of the day.
   • Choose 12 appointments evenly spaced throughout the day.
   Appointments are scheduled to last 10 minutes. The actual lengths of time, in minutes, that patients spend with the doctor may be assumed to have a normal distribution with mean \(\mu\) and standard deviation 3.4. The researcher suspects that the actual time spent is more than 10 minutes on average. To test this suspicion, he recorded the actual times spent for a random sample of 12 appointments and carried out a hypothesis test at the 1\% significance level.
2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
3. Given that the total length of time spent for the 12 appointments was 147 minutes, carry out the test.
4. Give a reason why the Central Limit theorem was not needed in part (iii).