Edexcel S1 (Statistics 1)

1. Joel buys a box of second-hand Jazz and Blues CDs at a car boot sale.

In the box there are 30 CDs, 8 of which were recorded live. 16 of the CDs are predominantly Jazz and 13 of these were recorded in the studio. This information is shown in the following table.

1. Copy and complete the table above. Joel picks a CD at random to play first.
   Find the probability that it is
2. a Blues CD that was recorded live,
3. a Jazz CD, given that it was recorded in the studio.

2. The discrete random variable \(Q\) has the following probability distribution.

1. Write down the name of this distribution. The discrete random variable \(R\) has the following probability distribution.

   \(r\)	14	24	34	44	54
   \(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)

2. State the relationship between \(R\) and \(Q\) in the form \(R = a Q + b\). Given that \(\mathrm { E } ( Q ) = 3\) and \(\operatorname { Var } ( Q ) = 2\),
3. find \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).

3. The random variable \(X\) is normally distributed with a mean of 42 and a variance of 18 . Find

1. \(\mathrm { P } ( X \leq 45 )\),
2. \(\mathrm { P } ( 32 \leq X \leq 38 )\),
3. the value of \(x\) such that \(\mathrm { P } ( X \leq x ) = 0.95\)

4. The ages of 300 houses in a village are recorded giving the following table of results. Use linear interpolation to estimate for these data

1. the median,
2. the limits between which the middle \(80 \%\) of the ages lie. An estimate of the mean of these data is calculated to be 86.6 years.
3. Explain why the mean and median are so different and hence say which you consider best represents the data.

5. The discrete random variable \(Y\) has the following cumulative distribution function.

1. Write down the probability distribution of \(Y\).
2. Find \(\mathrm { P } ( 1 \leq Y < 3 )\).
3. Show that \(\mathrm { E } ( Y ) = 2.7\)
4. Find \(\mathrm { E } ( 2 Y + 4 )\).
5. Find \(\operatorname { Var } ( Y )\).

6. A software company sets exams for programmers who wish to qualify to use their packages. Past records show that \(55 \%\) of candidates taking the exam for the first time will pass, \(60 \%\) of those taking it for the second time will pass, but only \(40 \%\) of those taking the exam for the third time will pass. Candidates are not allowed to sit the exam more than three times. A programmer decides to keep taking the exam until he passes or is allowed no further attempts. Find the probability that he will

1. pass the exam on his second attempt,
2. pass the exam. Another programmer already has the qualification.
3. Find, correct to 3 significant figures, the probability that she passed first time. At a particular sitting of the exam there are 400 candidates.
   The ratio of those sitting the exam for the first time to those sitting it for the second time to those sitting it for the third time is \(5 : 3 : 2\)
4. How many of the 400 candidates would be expected to pass?

7. A doctor wished to investigate the effects of staying awake for long periods on a person's ability to complete simple tasks. She recorded the number of times, \(n\), that a subject could clinch his or her fist in 30 seconds after being awake for \(h\) hours. The results for one subject were as follows.

1. Plot a scatter diagram of \(n\) against \(h\) for these results. You may use $$\Sigma h = 180 , \quad \Sigma n = 875 , \quad \Sigma h ^ { 2 } = 3660 , \quad \Sigma h n = 17204 .$$
2. Obtain the equation of the regression line of \(n\) on \(h\) in the form \(n = a + b h\).
3. Give a practical interpretation of the constant b.
4. Explain why this regression line would be unlikely to be appropriate for values of \(h\) between 0 and 16 .
   (2 marks)
   Another subject underwent the same tests giving rise to a regression line of \(n = 213.4 - 5.87\) h
5. After how many hours of being awake together would you expect these two subjects to be able to clench their fists the same number of times in 30 seconds?