CAIE S2 (Statistics 2) 2017 March

1 In a survey, 36 out of 120 randomly selected voters in Hungton said that if there were an election next week they would vote for the Alpha party. Calculate an approximate \(90 \%\) confidence interval for the proportion of voters in Hungton who would vote for the Alpha party.

2 Karim has noted the lifespans, in weeks, of a large random sample of certain insects. He carries out a test, at the \(1 \%\) significance level, for the population mean, \(\mu\). Karim's null hypothesis is \(\mu = 6.4\).

1. Given that Karim's test is two-tail, state the alternative hypothesis.
   Karim finds that the value of the test statistic is \(z = 2.43\).
2. Explain what conclusion he should draw.
3. Explain briefly when a one-tail test is appropriate, rather than a two-tail test.

3 The length, in centimetres, of a certain type of snake is modelled by the random variable \(X\) with mean 52 and standard deviation 6.1. A random sample of 75 snakes is selected, and the sample mean, \(\bar { X }\), is found.

1. Find \(\mathrm { P } ( 51 < \bar { X } < 53 )\).
2. Explain why it was necessary to use the Central Limit theorem in the solution to part (i).

4 At a doctors' surgery, the number of missed appointments per day has a Poisson distribution. In the past the mean number of missed appointments per day has been 0.9 . Following some publicity, the manager carries out a hypothesis test to determine whether this mean has decreased. If there are fewer than 3 missed appointments in a randomly chosen 5-day period, she will conclude that the mean has decreased.

1. Find the probability of a Type I error.
2. State what is meant by a Type I error in this context.
3. Find the probability of a Type II error if the mean number of missed appointments per day is 0.2 .

5

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_292_517_264_338}
   \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_289_518_264_858}
   \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_273_510_365_1377} The diagram shows the graphs of three functions, \(f _ { 1 } , f _ { 2 }\) and \(f _ { 3 }\). The function \(f _ { 1 }\) is a probability density function.
   1. State the value of \(k\).
   2. For each of the functions \(\mathrm { f } _ { 2 }\) and \(\mathrm { f } _ { 3 }\), state why it cannot be a probability density function.
2. The probability density function g is defined by $$g ( x ) = \begin{cases} 6 \left( a ^ { 2 } - x ^ { 2 } \right) & - a \leqslant x \leqslant a
   0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
   1. Show that \(a = \frac { 1 } { 2 }\).
   2. State the value of \(\mathrm { E } ( X )\).
   3. Find \(\operatorname { Var } ( X )\).

6 The masses, in kilograms, of cartons of sugar and cartons of flour have the distributions \(\mathrm { N } \left( 78.8,12.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 62.0,10.0 ^ { 2 } \right)\) respectively.

1. The standard load for a certain crane is 8 cartons of sugar and 3 cartons of flour. The maximum load that can be carried safely by the crane is 900 kg . Stating a necessary assumption, find the percentage of standard loads that will exceed the maximum safe load.
2. Find the probability that a randomly chosen carton of sugar has a smaller mass than a randomly chosen carton of flour.

7 The number of planes arriving at an airport every hour during daytime is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 5.2 )\).

1. State two assumptions required for the Poisson model to be valid in this context.
2. (a) Find the probability that the number of planes arriving in a 15 -minute period is greater than 1 and less than 4,
   (b) Find the probability that more than 3 planes will arrive in a 40-minute period.
3. The airport has enough staff to deal with a maximum of 60 planes landing during a 10-hour day. Use a suitable approximation to find the probability that, on a randomly chosen 10-hour day, staff will be able to deal with all the planes that land.