Edexcel S1 (Statistics 1) 2004 June

1. A fair die has six faces numbered \(1,2,2,3,3\) and 3 . The die is rolled twice and the number showing on the uppermost face is recorded each time.

Find the probability that the sum of the two numbers recorded is at least 5 .
(5)

2. A researcher thinks there is a link between a person's height and level of confidence. She measured the height \(h\), to the nearest cm , of a random sample of 9 people. She also devised a test to measure the level of confidence \(c\) of each person. The data are shown in the table below. [You may use \(\Sigma h ^ { 2 } = 272094 , \Sigma c ^ { 2 } = 2878966 , \Sigma h c = 884484\) ]

1. Draw a scatter diagram to illustrate these data.
2. Find exact values of \(S _ { h c } S _ { h h }\) and \(S _ { c c }\).
3. Calculate the value of the product moment correlation coefficient for these data.
4. Give an interpretation of your correlation coefficient.
5. Calculate the equation of the regression line of \(c\) on \(h\) in the form \(c = a + b h\).
6. Estimate the level of confidence of a person of height 180 cm .
7. State the range of values of \(h\) for which estimates of \(c\) are reliable.

3. A discrete random variable \(X\) has a probability function as shown in the table below, where \(a\) and \(b\) are constants. Given that \(\mathrm { E } ( X ) = 1.7\),

1. find the value of \(a\) and the value of \(b\). Find
2. \(\mathrm { P } ( 0 < X < 1.5 )\),
3. \(\mathrm { E } ( 2 X - 3 )\).
4. Show that \(\operatorname { Var } ( X ) = 1.41\).
5. Evaluate \(\operatorname { Var } ( 2 X - 3 )\).

4. The attendance at college of a group of 18 students was recorded for a 4-week period. The number of students actually attending each of 16 classes are shown below.

1. 1. Calculate the mean and the standard deviation of the number of students attending these classes.
   2. Express the mean as a percentage of the 18 students in the group. In the same 4-week period, the attendance of a different group of 20, students is shown below.

      20	16	18	19
      15	14	14	15
      18	15	16	17
      16	18	15	14

2. Construct a back-to-back stem and leaf diagram to represent the attendance in both groups.
3. Find the mode, median and inter-quartile range for each group of students. The mean percentage attendance and standard deviation for the second group of students are 81.25 and 1.82 respectively.
4. Compare and contrast the attendance of these 2 groups of students.

5. A health club lets members use, on each visit, its facilities for as long as they wish. The club's records suggest that the length of a visit can be modelled by a normal distribution with mean 90 minutes. Only \(20 \%\) of members stay for more than 125 minutes.

1. Find the standard deviation of the normal distribution.
2. Find the probability that a visit lasts less than 25 minutes. The club introduce a closing time of 10:00 pm. Tara arrives at the club at 8:00 pm.
3. Explain whether or not this normal distribution is still a suitable model for the length of her visit.

6. Three events \(A , B\) and \(C\) are defined in the sample space \(S\). The events \(A\) and \(B\) are mutually exclusive and \(A\) and \(C\) are independent.

1. Draw a Venn diagram to illustrate the relationships between the 3 events and the sample space. Given that \(\mathrm { P } ( A ) = 0.2 , \mathrm { P } ( B ) = 0.4\) and \(\mathrm { P } ( A \cup C ) = 0.7\), find
2. \(\mathrm { P } ( A C )\),
3. \(\mathrm { P } ( A \cup B )\),
4. \(\mathrm { P } ( C )\). END