CAIE S2 (Statistics 2) 2014 November

1 A researcher wishes to investigate whether the mean height of a certain type of plant in one region is different from the mean height of this type of plant everywhere else. He takes a large random sample of plants from the region and finds the sample mean. He calculates the value of the test statistic, \(z\), and finds that \(z = 1.91\).

1. Explain briefly why the researcher should use a two-tail test.
2. Carry out the test at the \(4 \%\) significance level.

2
\includegraphics[max width=\textwidth, alt={}, center]{323cf83a-e23b-494e-a911-856d8f1c92fd-2_483_791_708_676} The diagram shows the graph of the probability density function, f , of a random variable \(X\).

1. Find the value of the constant \(c\).
2. Find the value of \(a\) such that \(\mathrm { P } ( a < X < 1 ) = 0.1\).
3. Find \(\mathrm { E } ( X )\).

3 The times, in minutes, taken by people to complete a walk are normally distributed with mean \(\mu\). The times, \(t\) minutes, for a random sample of 80 people were summarised as follows. $$\Sigma t = 7220 \quad \Sigma t ^ { 2 } = 656060$$

1. Calculate a \(97 \%\) confidence interval for \(\mu\).
2. Explain whether it was necessary to use the Central Limit theorem in part (i).

4 The masses, in grams, of tomatoes of type \(A\) and type \(B\) have the distributions \(\mathrm { N } \left( 125,30 ^ { 2 } \right)\) and \(\mathrm { N } \left( 130,32 ^ { 2 } \right)\) respectively.

1. Find the probability that the total mass of 4 randomly chosen tomatoes of type \(A\) and 6 randomly chosen tomatoes of type \(B\) is less than 1.5 kg .
2. Find the probability that a randomly chosen tomato of type \(A\) has a mass that is at least \(90 \%\) of the mass of a randomly chosen tomato of type \(B\).

5 It is known that when seeds of a certain type are planted, on average \(10 \%\) of the resulting plants reach a height of 1 metre. A gardener wishes to investigate whether a new fertiliser will increase this proportion. He plants a random sample of 18 seeds of this type, using the fertiliser, and notes how many of the resulting plants reach a height of 1 metre.

1. In fact 4 of the 18 plants reach a height of 1 metre. Carry out a hypothesis test at the \(8 \%\) significance level.
2. Explain which of the errors, Type I or Type II, might have been made in part (i). Later, the gardener plants another random sample of 18 seeds of this type, using the fertiliser, and again carries out a hypothesis test at the \(8 \%\) significance level.
3. Find the probability of a Type I error.

6 The number of calls received at a small call centre has a Poisson distribution with mean 2.4 calls per 5 -minute period. Find the probability of

1. exactly 4 calls in an 8 -minute period,
2. at least 3 calls in a 3-minute period. The number of calls received at a large call centre has a Poisson distribution with mean 41 calls per 5-minute period.
3. Use an approximating distribution to find the probability that the number of calls received in a 5 -minute period is between 41 and 59 inclusive.