Edexcel S3 (Statistics 3) Specimen

1. The 240 members of a bowling club are listed alphabetically in the club's membership book. The committee wishes to select a sample of 30 members to fill in a questionnaire about the facilities the club offers.
   1. Explain how the committee could use a table of random numbers to take a systematic sample.
   2. Give one advantage of this method over taking a simple random sample.
   3. The weights of pears, \(P\) grams, are normally distributed with a mean of 110 and a standard deviation of 8 . Geoff buys a bag of 16 pears.
   4. Write down the distribution of \(\bar { P }\), the mean weight of the 16 pears.
   5. Find \(\mathrm { P } ( 110 < \bar { P } < 113 )\).
   6. The three tasks most frequently carried out in a garage are \(A , B\) and \(C\). For each of the tasks the times, in minutes, taken by the garage mechanics are assumed to be normally distributed with means and standard deviations given in the following table.

   Task	Mean	Standard deviation
   \(A\)	225	38
   \(B\)	165	23
   \(C\)	185	27

   Assuming that the times for the three tasks are independent, calculate the probability that
2. the total time taken by a single randomly chosen mechanic to carry out all three tasks lies between 533 and 655 minutes,
3. a randomly chosen mechanic takes longer to carry out task \(B\) than task \(C\).

4. At the end of a season a league of eight ice hockey clubs produced the following table showing the position of each club in the league and the average attendances (in hundreds) at home matches.

1. Calculate the Spearman rank correlation coefficient between position in the league and average home attendance.
2. Stating clearly your hypotheses and using a \(5 \%\) two-tailed test, interpret your rank correlation coefficient. Many sets of data include tied ranks.
3. Explain briefly how tied ranks can be dealt with.

5. For a six-sided die it is assumed that each of the sides has an equal chance of landing uppermost when the die is rolled.

1. Write down the probability function for the random variable \(X\), the number showing on the uppermost side after the die has been rolled.
2. State the name of the distribution. A student wishing to check the above assumption rolled the die 300 times and for the sides 1 to 6 , obtained the frequencies \(41,49,52,58,37\) and 63 respectively.
3. Analyse these data and comment on whether or not the assumption is valid for this die. Use a \(5 \%\) level of significance and state your hypotheses clearly.
   (8)

6. A sociologist was studying the smoking habits of adults. A random sample of 300 adult smokers from a low income group and an independent random sample of 400 adult smokers from a high income group were asked what their weekly expenditure on tobacco was. The results are summarised below.

1. Using a \(5 \%\) significance level, test whether or not the two groups differ in the mean amounts spent on tobacco.
2. Explain briefly the importance of the central limit theorem in this example.
   (2)

7. A survey in a college was commissioned to investigate whether or not there was any association between gender and passing a driving test. A group of 50 male and 50 female students were asked whether they passed or failed their driving test at the first attempt. All the students asked had taken the test. The results were as follows. Stating your hypotheses clearly test, at the \(10 \%\) level, whether or not there is any evidence of an association between gender and passing a driving test at the first attempt.

8. Observations have been made over many years of \(T\), the noon temperature in \({ } ^ { \circ } \mathrm { C }\), on 21 st March at Sunnymere. The records for a random sample of 12 years are given below.
5.2, 3.1, 10.6, 12.4, 4.6, 8.7, 2.5, 15.3, -1.5, 1.8, 13.2, 9.3.

1. Find unbiased estimates of the mean and variance of \(T\). Over the years, the standard deviation of \(T\) has been found to be 5.1.
2. Assuming a normal distribution find a \(90 \%\) confidence interval for the mean of \(T\).
   (5) A meteorologist claims that the mean temperature at noon in Sunnymere on 21st March is \(4 ^ { \circ } \mathrm { C }\).
3. Use your interval to comment on the meteorologist's claim.