AQA S1 (Statistics 1) 2007 January

1 The times, in seconds, taken by 20 people to solve a simple numerical puzzle were

1. Calculate the mean and the standard deviation of these times.
2. In fact, 23 people solved the puzzle. However, 3 of them failed to solve it within the allotted time of 60 seconds. Calculate the median and the interquartile range of the times taken by all 23 people.
   (4 marks)
3. For the times taken by all 23 people, explain why:
   1. the mode is not an appropriate numerical measure;
   2. the range is not an appropriate numerical measure.

2 A hotel has 50 single rooms, 16 of which are on the ground floor. The hotel offers guests a choice of a full English breakfast, a continental breakfast or no breakfast. The probabilities of these choices being made are \(0.45,0.25\) and 0.30 respectively. It may be assumed that the choice of breakfast is independent from guest to guest.

1. On a particular morning there are 16 guests, each occupying a single room on the ground floor. Calculate the probability that exactly 5 of these guests require a full English breakfast.
2. On a particular morning when there are 50 guests, each occupying a single room, determine the probability that:
   1. at most 12 of these guests require a continental breakfast;
   2. more than 10 but fewer than 20 of these guests require no breakfast.
3. When there are 40 guests, each occupying a single room, calculate the mean and the standard deviation for the number of these guests requiring breakfast.

3 Estimate, without undertaking any calculations, the value of the product moment correlation coefficient between the variables \(x\) and \(y\) in each of the three scatter diagrams.

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_631_659_516_301}
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_620_647_525_1119}
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_624_655_1279_303}
   (5 marks)

4 A very popular play has been performed at a London theatre on each of 6 evenings per week for about a year. Over the past 13 weeks ( 78 performances), records have been kept of the proceeds from the sales of programmes at each performance. An analysis of these records has found that the mean was \(\pounds 184\) and the standard deviation was \(\pounds 32\).

1. Assuming that the 78 performances may be considered to be a random sample, construct a \(90 \%\) confidence interval for the mean proceeds from the sales of programmes at an evening performance of this play.
2. Comment on the likely validity of the assumption in part (a) when constructing a confidence interval for the mean proceeds from the sales of programmes at an evening performance of:
   1. this particular play;
   2. any play.

5 Dafydd, Eli and Fabio are members of an amateur cycling club that holds a time trial each Sunday during the summer. The independent probabilities that Dafydd, Eli and Fabio take part in any one of these trials are \(0.6,0.7\) and 0.8 respectively. Find the probability that, on a particular Sunday during the summer:

1. none of the three cyclists takes part;
2. Fabio is the only one of the three cyclists to take part;
3. exactly one of the three cyclists takes part;
4. either one or two of the three cyclists take part.

6 When Monica walks to work from home, she uses either route A or route B.

1. Her journey time, \(X\) minutes, by route A may be assumed to be normally distributed with a mean of 37 and a standard deviation of 8 . Determine:
   1. \(\mathrm { P } ( X < 45 )\);
   2. \(\mathrm { P } ( 30 < X < 45 )\).
2. Her journey time, \(Y\) minutes, by route B may be assumed to be normally distributed with a mean of 40 and a standard deviation of \(\sigma\). Given that \(\mathrm { P } ( Y > 45 ) = 0.12\), calculate the value of \(\sigma\).
3. If Monica leaves home at 8.15 am to walk to work hoping to arrive by 9.00 am , state, with a reason, which route she should take.
4. When Monica travels to work from home by car, her journey time, \(W\) minutes, has a mean of 18 and a standard deviation of 12 . Estimate the probability that, for a random sample of 36 journeys to work from home by car, Monica's mean time is more than 20 minutes.
5. Indicate where, if anywhere, in this question you needed to make use of the Central Limit Theorem.

7 [Figure 1, printed on the insert, is provided for use in this question.]
Stan is a retired academic who supplements his pension by mowing lawns for customers who live nearby. As part of a review of his charges for this work, he measures the areas, \(x \mathrm {~m} ^ { 2 }\), of a random sample of eight of his customers' lawns and notes the times, \(y\) minutes, that it takes him to mow these lawns. His results are shown in the table.

1. On Figure 1, plot a scatter diagram of these data.
2. Calculate the equation of the least squares regression line of \(y\) on \(x\). Draw your line on Figure 1.
3. Calculate the value of the residual for Customer H and indicate how your value is confirmed by your scatter diagram.
4. Given that Stan charges \(\pounds 12\) per hour, estimate the charge for mowing a customer's lawn that has an area of \(560 \mathrm {~m} ^ { 2 }\).