AQA S1 (Statistics 1) 2010 January

1 Draught excluder for doors and windows is sold in rolls of nominal length 10 metres.
The actual length, \(X\) metres, of draught excluder on a roll may be modelled by a normal distribution with mean 10.2 and standard deviation 0.15 .

1. Determine:
   1. \(\mathrm { P } ( X < 10.5 )\);
   2. \(\mathrm { P } ( 10.0 < X < 10.5 )\).
2. A customer randomly selects six 10 -metre rolls of the draught excluder. Calculate the probability that all six rolls selected contain more than 10 metres of draught excluder.

2 Lizzie, the receptionist at a dental practice, was asked to keep a weekly record of the number of patients who failed to turn up for an appointment. Her records for the first 15 weeks were as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 20 & 26 & 32 & a & 37 & 14 & 27 & 34 & 15 & 18 & b & 25 & 37 & 29 & 25 \end{array}$$ Unfortunately, Lizzie forgot to record the actual values for two of the 15 weeks, so she recorded them as \(a\) and \(b\). However, she did remember that \(a < 10\) and that \(b > 40\).

1. Calculate the median and the interquartile range of these 15 values.
2. Give a reason why, for these data:
   1. the mode is not an appropriate measure of average;
   2. the standard deviation cannot be used as a measure of spread.
3. Subsequent investigations revealed that the missing values were 8 and 43 . Calculate the mean and the standard deviation of the 15 values.

3 The table shows, for each of a random sample of 7 weeks, the number of customers, \(x\), who purchased fuel from a filling station, together with the total volume, \(y\) litres, of fuel purchased by these customers.

1. Calculate the equation of the least squares regression line of \(y\) on \(x\).
2. Estimate the volume of fuel sold during a week in which 200 customers purchase fuel.
3. Comment on the likely reliability of your estimate in part (b), given that, for the regression line calculated in part (a), the values of the 7 residuals lie between approximately - 415 litres and + 430 litres.

4 Each school-day morning, three students, Rita, Said and Ting, travel independently from their homes to the same school by one of three methods: walk, cycle or bus. The table shows the probabilities of their independent daily choices.

1. Calculate the probability that, on any given school-day morning:
   1. all 3 students walk to school;
   2. only Rita travels by bus to school;
   3. at least 2 of the 3 students cycle to school.
2. Ursula, a friend of Rita, never travels to school by bus. The probability that: Ursula walks to school when Rita walks to school is 0.9 ; Ursula cycles to school when Rita cycles to school is 0.7 . Calculate the probability that, on any given school-day morning, Rita and Ursula travel to school by:
   1. the same method;
   2. different methods.

5 In a random sample of 12 bags of flour, the weight, in grams, of flour in each bag was recorded as follows.
\(\begin{array} { l l l l l l l l l l l l } 1011 & 995 & 1018 & 1022 & 1014 & 1005 & 1017 & 1015 & 993 & 1018 & 992 & 1020 \end{array}\)

1. It may be assumed that the weight of flour in a bag is normally distributed with a standard deviation of 10.5 grams.
   1. Construct a \(98 \%\) confidence interval for the mean weight, \(\mu\) grams, of flour in a bag, giving the limits to four significant figures.
   2. State why, in constructing your confidence interval, use of the Central Limit Theorem was not necessary.
   3. If the distribution of the weight of flour in a bag was unknown, indicate a minimum number of weights that you would consider necessary for a confidence interval for \(\mu\) to be valid.
2. The statement ' 1 kg ' is printed on each bag. Comment on this statement using both the confidence interval that you constructed in part (a)(i) and the weights of the given sample of 12 bags.
3. Given that \(\mu = 1000\), state the probability that a \(98 \%\) confidence interval for \(\mu\) will not contain 1000.
   (l mark)

6 During the winter, the probability that Barry's cat, Sylvester, chooses to stay outside all night is 0.35 , and the cat's choice is independent from night to night.

1. Determine the probability that, during a period of 2 weeks ( 14 nights) in winter, Sylvester chooses to stay outside:
   1. on at most 7 nights;
   2. on at least 11 nights;
   3. on more than 5 nights but fewer than 10 nights.
2. Calculate the probability that, during a period of \(\mathbf { 3 }\) weeks in winter, Sylvester chooses to stay outside on exactly 4 nights.
3. Barry claims that, during the summer, the number of nights per week, \(S\), on which Sylvester chooses to stay outside can be modelled by a binomial distribution with \(n = 7\) and \(p = \frac { 5 } { 7 }\).
   1. Assuming that Barry's claim is correct, find the mean and the variance of \(S\).
   2. For a period of 13 weeks during the summer, the number of nights per week on which Sylvester chose to stay outside had a mean of 5 and a variance of 1.5 . Comment on Barry's claim.
      (2 marks)

7 [Figure 1, printed on the insert, is provided for use in this question.]
Harold considers himself to be an expert in assessing the auction value of antiques. He regularly visits car boot sales to buy items that he then sells at his local auction rooms. Harold's father, Albert, who is not convinced of his son's expertise, collects the following data from a random sample of 12 items bought by Harold.

1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
2. Interpret your value in the context of this question.
3. 1. On Figure 1, complete the scatter diagram for these data.
   2. Comment on what this reveals.
4. When items J and L are omitted from the data, it is found that $$S _ { x x } = 4854.4 \quad S _ { y y } = 4216.1 \quad S _ { x y } = 4268.8$$
   1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\) for the remaining 10 items.
   2. Hence revise as necessary your interpretation in part (b).