Edexcel S1 (Statistics 1)

1. A shop recorded the number of pairs of gloves, \(n\), that it sold and the average daytime temperature, \(T ^ { \circ } \mathrm { C }\), for each month over a 12-month period.

The data was then summarised as follows: $$\Sigma T = 124 , \quad \Sigma n = 384 , \quad \Sigma T ^ { 2 } = 1802 , \quad \Sigma n ^ { 2 } = 18518 , \quad \Sigma T n = 2583 .$$

1. Calculate the product moment correlation coefficient for these data.
2. Comment on what your value shows and suggest a reason for this.

2. Events \(A\) and \(B\) are independent. Given also that $$\mathrm { P } ( A ) = \frac { 3 } { 4 } \quad \text { and } \quad \mathrm { P } \left( A \cap B ^ { \prime } \right) = \frac { 1 } { 4 }$$ Find

1. \(\mathrm { P } ( A \cap B )\),
2. \(\mathrm { P } ( B )\),
3. \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)\).

3. The random variable \(X\) is such that $$\mathrm { E } ( X ) = a \text { and } \operatorname { Var } ( X ) = b$$ Find expressions in terms of \(a\) and \(b\) for

1. \(\mathrm { E } ( 2 X + 3 )\),
2. \(\quad \operatorname { Var } ( 2 X + 3 )\),
3. \(\mathrm { E } \left( X ^ { 2 } \right)\).
4. Show that $$\mathrm { E } \left[ ( X + 1 ) ^ { 2 } \right] = ( a + 1 ) ^ { 2 } + b$$

1. An engineer tested a new material under extreme conditions in a wind tunnel. He recorded the number of microfractures, \(n\), that formed and the wind speed, \(v\) metres per second, for 8 different values of \(v\) with all other conditions remaining constant. He then coded the data using \(x = v - 700\) and \(y = n - 20\) and calculated the following summary statistics.

$$\Sigma x = 100 , \quad \Sigma y = 23 , \quad \Sigma x ^ { 2 } = 215000 , \quad \Sigma x y = 11600 .$$

1. Find an equation of the regression line of \(y\) on \(x\).
2. Hence, find an equation of the regression line of \(n\) on \(v\).
3. Use your regression line to estimate the number of microfractures that would be formed if the material was tested in a wind speed of 900 metres per second with all other conditions remaining constant.
   (2 marks)

5. An antiques shop recorded the value of items stolen to the nearest pound during each week for a year giving the data in the table below. Letting \(x\) represent the mid-point of each group and using the coding \(y = \frac { x - 699.5 } { 200 }\),

1. find \(\sum\) fy.
2. estimate to the nearest pound the mean and standard deviation of the value of the goods stolen each week using your value for \(\sum f y\) and \(\sum f y ^ { 2 } = 424\).
   (6 marks)
   The median for these data is \(\pounds 82\).
3. Explain why the manager of the shop might be reluctant to use either the mean or the median in summarising these data.
   (3 marks)

6. At the start of a gameshow there are 10 contestants of which 6 are female. In each round of the game, one contestant is eliminated. All of the contestants have the same chance of progressing to the next round each time.

1. Show that the probability that the first two contestants to be eliminated are both male is \(\frac { 2 } { 15 }\).
2. Find the probability that more females than males are eliminated in the first three rounds of the game.
3. Given that the first contestant to be eliminated is male, find the probability that the next two contestants to be eliminated are both female.
   (3 marks)

7. A cyber-cafe recorded how long each user stayed during one day giving the following results.

1. Use linear interpolation to estimate the median and quartiles of these data. The results of a previous study had led to the suggestion that the length of time each user stays can be modelled by a normal distribution with a mean of 72 minutes and a standard deviation of 48 minutes.
2. Find the median and quartiles that this model would predict.
3. Comment on the suitability of the suggested model in the light of the new results.