Edexcel S3 (Statistics 3) 2011 June

1. Explain what you understand by the Central Limit Theorem.
   
2. A county councillor is investigating the level of hardship, \(h\), of a town and the number of calls per 100 people to the emergency services, \(c\). He collects data for 7 randomly selected towns in the county. The results are shown in the table below.

1. Calculate the Spearman's rank correlation coefficient between \(h\) and \(c\). After collecting the data, the councillor thinks there is no correlation between hardship and the number of calls to the emergency services.
2. Test, at the \(5 \%\) level of significance, the councillor's claim. State your hypotheses clearly.

3. A factory manufactures batches of an electronic component. Each component is manufactured in one of three shifts. A component may have one of two types of defect, \(D _ { 1 }\) or \(D _ { 2 }\), at the end of the manufacturing process. A production manager believes that the type of defect is dependent upon the shift that manufactured the component. He examines 200 randomly selected defective components and classifies them by defect type and shift. The results are shown in the table below. Stating your hypotheses, test, at the \(10 \%\) level of significance, whether or not there is evidence to support the manager's belief. Show your working clearly.

1. A shop manager wants to find out if customers spend more money when music is playing in the shop. The amount of money spent by a customer in the shop is \(\pounds x\). A random sample of 80 customers, who were shopping without music playing, and an independent random sample of 60 customers, who were shopping with music playing, were surveyed. The results of both samples are summarised in the table below.

1. Find the values of \(\bar { x }\) and \(s ^ { 2 }\).
2. Test, at the \(5 \%\) level of significance, whether or not the mean money spent is greater when music is playing in the shop. State your hypotheses clearly.

1. The number of hurricanes per year in a particular region was recorded over 80 years. The results are summarised in Table 1 below.

\begin{table}[h] \end{table}

1. Write down two assumptions that will support modelling the number of hurricanes per year by a Poisson distribution.
2. Show that the mean number of hurricanes per year from Table 1 is 4.4875
3. Use the answer in part (b) to calculate the expected frequencies \(r\) and \(s\) given in Table 2 below to 2 decimal places. \begin{table}[h]

   \(h\)	0	1	2	3	4	5	6	7 or more
   Expected frequency	0.90	4.04	\(r\)	13.55	\(s\)	13.65	10.21	13.39

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
4. Test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution. State your hypotheses clearly.

1. The lifetimes of batteries from manufacturer \(A\) are normally distributed with mean 20 hours and standard deviation 5 hours when used in a camera.
   1. Find the mean and standard deviation of the total lifetime of a pack of 6 batteries from manufacturer \(A\).
   Judy uses a camera that takes one battery at a time. She takes a pack of 6 batteries from manufacturer \(A\) to use in her camera on holiday.
2. Find the probability that the batteries will last for more than 110 hours on her holiday. The lifetimes of batteries from manufacturer \(B\) are normally distributed with mean 35 hours and standard deviation 8 hours when used in a camera.
3. Find the probability that the total lifetime of a pack of 6 batteries from manufacturer \(A\) is more than 4 times the lifetime of a single battery from manufacturer \(B\) when used in a camera.

1. Roastie's Coffee is sold in packets with a stated weight of 250 g . A supermarket manager claims that the mean weight of the packets is less than the stated weight. She weighs a random sample of 90 packets from their stock and finds that their weights have a mean of 248 g and a standard deviation of 5.4 g .
   1. Using a \(5 \%\) level of significance, test whether or not the manager's claim is justified. State your hypotheses clearly.
   2. Find the \(98 \%\) confidence interval for the mean weight of a packet of coffee in the supermarket's stock.
   3. State, with a reason, the action you would recommend the manager to take over the weight of a packet of Roastie's Coffee.
   Roastie's Coffee company increase the mean weight of their packets to \(\mu \mathrm { g }\) and reduce the standard deviation to 3 g . The manager takes a sample of size \(n\) from these new packets. She uses the sample mean \(\bar { X }\) as an estimator of \(\mu\).
2. Find the minimum value of \(n\) such that \(\mathrm { P } ( | \bar { X } - \mu ! | < 1 ) \geqslant 0.98\)