AQA S1 (Statistics 1) 2015 June

1
The table shows the annual gas consumption, \(x \mathrm { kWh }\), and the annual electricity consumption, \(y \mathrm { kWh }\), for a sample of 10 bungalows of similar size and occupancy. $$S _ { x x } = 76581640 \quad S _ { y y } = 694250 \quad S _ { x y } = 3629670$$

1. Calculate the value of \(r _ { x y }\), the product moment correlation coefficient between \(x\) and \(y\).
2. Interpret your value of \(r _ { x y }\) in the context of this question.

2 The table summarises the diameters, \(d\) millimetres, of a random sample of 60 new cricket balls to be used in junior cricket.

3 A ferry sails once each day from port D to port A. The ferry departs from D on time or late but never early. However, the ferry can arrive at A early, on time or late. The probabilities for some combined events of departing from \(D\) and arriving at \(A\) are shown in the table below.

1. Complete the table.
2. Write down the probability that, on a particular day, the ferry:
   1. both departs and arrives on time;
   2. departs late.
3. Find the probability that, on a particular day, the ferry:
   1. arrives late, given that it departed late;
   2. does not arrive late, given that it departed on time.
4. On three particular days, the ferry departs from port D on time. Find the probability that, on these three days, the ferry arrives at port A early once, on time once and late once. Give your answer to three decimal places.
   [0pt] [4 marks]
5. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Answer space for question 3}

   \multirow{2}{*}{}		Arrive at A
   		Early	On time	Late	Total
   \multirow{2}{*}{Depart from D}	On time	0.16	0.56	0.08
   	Late
   	Total	0.22	0.65		1.00

   \end{table}

4 Stephan is a roofing contractor who is often required to replace loose ridge tiles on house roofs. In order to help him to quote more accurately the prices for such jobs in the future, he records, for each of 11 recently repaired roofs, the number of ridge tiles replaced, \(x _ { i }\), and the time taken, \(y _ { i }\) hours. His results are shown in the table.

1. The pairs of data values for roofs 1 to 7 are plotted on the scatter diagram shown on the opposite page. Plot the 4 pairs of data values for roofs 8 to 11 on the scatter diagram.
2. 1. Calculate the equation of the least squares regression line of \(y _ { i }\) on \(x _ { i }\), and draw your line on the scatter diagram.
   2. Interpret your values for the gradient and for the intercept of this regression line.
3. Estimate the time that it would take Stephan to replace 15 loose ridge tiles on a house roof.
4. Given that \(r _ { i }\) denotes the residual for the point representing roof \(i\) :
   1. calculate the value of \(r _ { 6 }\);
   2. state why the value of \(\sum _ { i = 1 } ^ { 11 } r _ { i }\) gives no useful information about the connection between the number of ridge tiles replaced and the time taken.
      [0pt] [1 mark]
      \section*{Answer space for question 4}
      \includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-11_2385_1714_322_155}

5

1. Wooden lawn edging is supplied in 1.8 m length rolls. The actual length, \(X\) metres, of a roll may be modelled by a normal distribution with mean 1.81 and standard deviation 0.08 . Determine the probability that a randomly selected roll has length:
   1. less than 1.90 m ;
   2. greater than 1.85 m ;
   3. between 1.81 m and 1.85 m .
2. Plastic lawn edging is supplied in 9 m length rolls. The actual length, \(Y\) metres, of a roll may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). An analysis of a batch of rolls, selected at random, showed that $$\mathrm { P } ( Y < 9.25 ) = 0.88$$
   1. Use this probability to find the value of \(z\) such that $$9.25 - \mu = z \times \sigma$$ where \(z\) is a value of \(Z \sim \mathrm {~N} ( 0,1 )\).
   2. Given also that $$\mathrm { P } ( Y > 8.75 ) = 0.975$$ find values for \(\mu\) and \(\sigma\).

6

1. In a particular country, 35 per cent of the population is estimated to have at least one mobile phone. A sample of 40 people is selected from the population.
   Use the distribution \(\mathrm { B } ( 40,0.35 )\) to estimate the probability that the number of people in the sample that have at least one mobile phone is:
   1. at most 15 ;
   2. more than 10 ;
   3. more than 12 but fewer than 18 ;
   4. exactly equal to the mean of the distribution.
2. In the same country, 70 per cent of households have a landline telephone connection. A sample of 50 households is selected from all households in the country.
   Stating a necessary condition regarding this selection, estimate the probability that fewer than 30 households have a landline telephone connection.
   [0pt] [4 marks]

7

1. The weight of a sack of mixed dog biscuits can be modelled by a normal distribution with a mean of 10.15 kg and a standard deviation of 0.3 kg . A pet shop purchases 12 such sacks that can be considered to be a random sample.
   Calculate the probability that the mean weight of the 12 sacks is less than 10 kg .
2. The weight of dry cat food in a pouch can also be modelled by a normal distribution. The contents, \(x\) grams, of each of a random sample of 40 pouches were weighed. Subsequent analysis of these weights gave $$\bar { x } = 304.6 \quad \text { and } \quad s = 5.37$$
   1. Construct a \(99 \%\) confidence interval for the mean weight of dry cat food in a pouch. Give the limits to one decimal place.
   2. Comment, with justification, on each of the following two claims. Claim 1: The mean weight of dry cat food in a pouch is more than 300 grams.
      Claim 2: All pouches contain more than 300 grams of dry cat food.
      [0pt] [4 marks]
      \includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-24_2288_1705_221_155}