Edexcel S1 (Statistics 1) 2002 January

1. (a) Explain briefly what you understand by
   1. a statistical experiment,
   2. an event.
      (b) State one advantage and one disadvantage of a statistical model.
   3. A meteorologist measured the number of hours of sunshine, to the nearest hour, each day for 100 days. The results are summarised in the table below.

   Hours of sunshine	Days
   1	16
   \(2 - 4\)	32
   \(5 - 6\)	28
   7	12
   8	9
   \(9 - 11\)	2
   12	1

   (a) On graph paper, draw a histogram to represent these data.
   (b) Calculate an estimate of the number of days that had between 6 and 9 hours of sunshine.

3. A discrete random variable \(X\) has the probability function shown in the table below.

1. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 6 }\), find \(a\).
2. Find the exact value of Var ( \(X\) ).
3. Find the exact value of \(\mathrm { P } ( X \leq 15 )\).

4. A contractor bids for two building projects. He estimates that the probability of winning the first project is 0.5 , the probability of winning the second is 0.3 and the probability of winning both projects is 0.2 .

1. Find the probability that he does not win either project.
2. Find the probability that he wins exactly one project.
3. Given that he does not win the first project, find the probability that he wins the second.
4. By calculation, determine whether or not winning the first contract and winning the second contract are independent events.

5. The duration of the pregnancy of a certain breed of cow is normally distributed with mean \(\mu\) days and standard deviation \(\sigma\) days. Only \(2.5 \%\) of all pregnancies are shorter than 235 days and \(15 \%\) are longer than 286 days.

1. Show that \(\mu - 235 = 1.96 \sigma\).
2. Obtain a second equation in \(\mu\) and \(\sigma\).
3. Find the value of \(\mu\) and the value of \(\sigma\).
4. Find the values between which the middle \(68.3 \%\) of pregnancies lie.

6. Hospital records show the number of babies born in a year. The number of babies delivered by 15 male doctors is summarised by the stem and leaf diagram below.

1. Find the median and inter-quartile range of these data.
2. Given that there are no outliers, draw a box plot on graph paper to represent these data. Start your scale at the origin.
3. Calculate the mean and standard deviation of these data. The records also contain the number of babies delivered by 10 female doctors.

   34	30	20	15	6
   32	26	19	11	4

   The quartiles are 11, 19.5 and 30 .
4. Using the same scale as in part (b) and on the same graph paper draw a box plot for the data for the 10 female doctors.
5. Compare and contrast the box plots for the data for male and female doctors.

7. A number of people were asked to guess the calorific content of 10 foods. The
mean \(s\) of the guesses for each food and the true calorific content \(t\) are given in the table below. [You may assume that \(\Sigma t = 1340 , \Sigma s = 2310 , \Sigma t s = 396775 , \Sigma t ^ { 2 } = 246050 , \Sigma s ^ { 2 } = 694650\).]

1. Draw a scatter diagram, indicating clearly which is the explanatory (independent) and which is the response (dependent) variable.
2. Calculate, to 3 significant figures, the product moment correlation coefficient for the above data.
3. State, with a reason, whether or not the value of the product moment correlation coefficient changes if all the guesses are 50 calories higher than the values in the table. The mean of the guesses for the portion of rice pudding and for the packet of biscuits are outside the linear relation of the other eight foods.
4. Find the equation of the regression line of \(s\) on \(t\) excluding the values for rice pudding and biscuits.
   [0pt] [You may now assume that \(S _ { t s } = 72587 , S _ { t t } = 63671.875 , \bar { t } = 125.625 , \bar { s } = 187.5\).]
5. Draw the regression line on your scatter diagram.
6. State, with a reason, what the effect would be on the regression line of including the values for a portion of rice pudding and a packet of biscuits. \section*{END}