AQA S1 (Statistics 1) 2013 June

1 The average maximum monthly temperatures, \(u\) degrees Fahrenheit, and the average minimum monthly temperatures, \(v\) degrees Fahrenheit, in New York City are as follows.

1. 1. Calculate, to one decimal place, the mean and the standard deviation of the 12 values of the average maximum monthly temperature.
   2. For comparative purposes with a UK city, it was necessary to convert the temperatures from degrees Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ) to degrees Celsius ( \({ } ^ { \circ } \mathrm { C }\) ). The formula used to convert \(f ^ { \circ } \mathrm { F }\) to \(c ^ { \circ } \mathrm { C }\) is: $$c = \frac { 5 } { 9 } ( f - 32 )$$ Use this formula and your answers in part (a)(i) to calculate, in \({ } ^ { \circ } \mathbf { C }\), the mean and the standard deviation of the 12 values of the average maximum monthly temperature.
      (3 marks)
2. The value of the product moment correlation coefficient, \(r _ { u v }\), between the above 12 values of \(u\) and \(v\) is 0.997 , correct to three decimal places. State, giving a reason, the corresponding value of \(r _ { x y }\), where \(x\) and \(y\) are the exact equivalent temperatures in \({ } ^ { \circ } \mathrm { C }\) of \(u\) and \(v\) respectively.
   (2 marks)

2 The weight, \(X\) grams, of the contents of a tin of baked beans can be modelled by a normal random variable with a mean of 421 and a standard deviation of 2.5.

1. Find:
   1. \(\mathrm { P } ( X = 421 )\);
   2. \(\mathrm { P } ( X < 425 )\);
   3. \(\mathrm { P } ( 418 < X < 424 )\).
2. Determine the value of \(x\) such that \(\mathrm { P } ( X < x ) = 0.98\).
3. The weight, \(Y\) grams, of the contents of a tin of ravioli can be modelled by a normal random variable with a mean of \(\mu\) and a standard deviation of 3.0 . Find the value of \(\mu\) such that \(\mathrm { P } ( Y < 410 ) = 0.01\).

3 An auction house offers items of jewellery for sale at its public auctions. Each item has a reserve price which is less than the lower price estimate which, in turn, is less than the upper price estimate. The outcome for any item is independent of the outcomes for all other items. The auction house has found, from past records, the following probabilities for the outcomes of items of jewellery offered for sale.

4 The girth, \(g\) metres, the length, \(l\) metres, and the weight, \(y\) kilograms, of each of a sample of 20 pigs were measured. The data collected is summarised as follows. $$S _ { g g } = 0.1196 \quad S _ { l l } = 0.0436 \quad S _ { y y } = 5880 \quad S _ { g y } = 24.15 \quad S _ { l y } = 10.25$$

1. Calculate the value of the product moment correlation coefficient between:
   1. girth and weight;
   2. length and weight.
2. Interpret, in context, each of the values that you obtained in part (a).
3. Weighing pigs requires expensive equipment, whereas measuring their girths and lengths simply requires a tape measure. With this in mind, the following formula is proposed to make an estimate of a pig's weight, \(x\) kilograms, from its girth and length. $$x = 69.3 \times g ^ { 2 } \times l$$ Applying this formula to the relevant data on the 20 pigs resulted in $$S _ { x x } = 5656.15 \quad S _ { x y } = 5662.97$$
   1. By calculating a third value of the product moment correlation coefficient, state which of \(g , l\) or \(x\) is the most strongly correlated with \(y\), the weight.
   2. Estimate the weight of a pig that has a girth of 1.25 metres and a length of 1.15 metres.
   3. Given the additional information that \(\bar { x } = 115.4\) and \(\bar { y } = 116.0\), calculate the equation of the least squares regression line of \(y\) on \(x\), in the form \(y = a + b x\).
   4. Comment on the likely accuracy of the estimated weight found in part (c)(ii). Your answer should make reference to the value of the product moment correlation coefficient found in part (c)(i) and to the values of \(b\) and \(a\) found in part (c)(iii).
      (4 marks)

5 Alison is a member of a tenpin bowling club which meets at a bowling alley on Wednesday and Thursday evenings. The probability that she bowls on a Wednesday evening is 0.90 . Independently, the probability that she bowls on a Thursday evening is 0.95 .

1. Calculate the probability that, during a particular week, Alison bowls on:
   1. two evenings;
   2. exactly one evening.
2. David, a friend of Alison, is a member of the same club. The probability that he bowls on a Wednesday evening, given that Alison bowls on that evening, is 0.80 . The probability that he bowls on a Wednesday evening, given that Alison does not bowl on that evening, is 0.15 . The probability that he bowls on a Thursday evening, given that Alison bowls on that evening, is 1 . The probability that he bowls on a Thursday evening, given that Alison does not bowl on that evening, is 0 . Calculate the probability that, during a particular week:
   1. Alison and David bowl on a Wednesday evening;
   2. Alison and David bowl on both evenings;
   3. Alison, but not David, bowls on a Thursday evening;
   4. neither bowls on either evening.

6 The weight, \(X\) kilograms, of sand in a bag can be modelled by a normal random variable with unknown mean \(\mu\) and known standard deviation 0.4 .

1. The sand in each of a random sample of 25 bags from a large batch is weighed. The total weight of sand in these 25 bags is found to be 497.5 kg .
   1. Construct a 98\% confidence interval for the mean weight of sand in bags in the batch.
   2. Hence comment on the claim that bags in the batch contain an average of 20 kg of sand.
   3. State why use of the Central Limit Theorem is not required in answering part (a)(i).
2. The weight, \(Y\) kilograms, of cement in a bag can be modelled by a normal random variable with mean 25.25 and standard deviation 0.35. A firm purchases 10 such bags. These bags may be considered to be a random sample.
   1. Determine the probability that the mean weight of cement in the 10 bags is less than 25 kg .
   2. Calculate the probability that the weight of cement in each of the 10 bags is more than 25 kg .
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