OCR FS1 AS (Further Statistics 1 AS) 2018 March

1 A learner driver keeps taking the driving test until she passes. The number of attempts taken, up to and including the pass, is denoted by \(X\).

1. State two assumptions needed for \(X\) to be well modelled by a geometric distribution. Assume now that \(X \sim \operatorname { Geo } ( 0.4 )\).
2. Find \(\mathrm { P } ( X < 6 )\).
3. Find \(\mathrm { E } ( X )\).
4. Find \(\operatorname { Var } ( X )\).

2 The number of calls received by a customer service department in 30 minutes is denoted by \(W\). It is known that \(\mathrm { E } ( W ) = 6.5\).

1. It is given that \(W\) has a Poisson distribution.
   (a) Write down the standard deviation of \(W\).
   (b) Find the probability that the total number of calls received in a randomly chosen period of 2 hours is less than 30 .
2. It is given instead that \(W\) has a uniform distribution on \([ 1 , N ]\). Calculate the value of \(\mathrm { P } ( W > 3 )\).

3 A pack of 40 cards consists of 10 cards in each of four colours: red, yellow, blue and green. The pack is dealt at random into four "hands", each of 10 cards. The hands are labelled North, South, East and West.

1. Find the probability that West has exactly 3 red cards.
2. Find the probability that West has exactly 3 red cards, given that East and West have between them a total of exactly 5 red cards.
3. South has 5 red cards and 5 blue cards. These cards are placed in a row in a random order. Find the probability that the colour of each card is different from the colour of the preceding card.

4 A spinner has edges numbered \(1,2,3,4\) and 5 . When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table. It is known that \(\mathrm { E } ( N ) = 2.55\).

1. Find \(\operatorname { Var } ( N )\).
2. Find \(\mathrm { E } ( 3 N + 2 )\).
3. Find \(\operatorname { Var } ( 3 N + 2 )\).

5 The speed \(v \mathrm {~ms} ^ { - 1 }\) of a car at time \(t\) seconds after it starts to accelerate was measured at 1 -second intervals. The results are shown in the following diagram.
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1. State whether \(t\) or \(v\) or neither is a controlled variable. The value of the product moment correlation coefficient \(r\) for the data is 0.987 correct to 3 significant figures.
2. The speed of the car is converted to miles per hour and the time to minutes. State the value of \(r\) for the converted data.
3. State the value of Spearman's rank correlation coefficient \(r _ { s }\) for the data.
4. What information does \(r\) give about the data that is not given by \(r _ { s }\) ?

6 The discrete random variable \(R\) has the distribution \(\operatorname { Po } ( \lambda )\).
Use an algebraic method to find the range of values of \(\lambda\) for which the single most likely value of \(R\) is 7. [7]

7 The numbers of students taking A levels in three subjects at a school were classified by the year in which they entered the school as follows. The Head of the school carries out a significance test at the \(10 \%\) level to test whether subjects taken are independent of year of entry.

1. Show that in carrying out the test it is necessary to combine columns.
2. Suggest a reason why it is more sensible to combine the columns for Mathematics and Physics than the columns for Physics and English.
3. Carry out the test.
4. State which cell gives the largest contribution to the test statistic.
5. Interpret your answer to part (iv).

8 In a competition, entrants have to give ranks from 1 to 7 to each of seven resorts. The correct ranks for the resorts are decided by an expert.

1. One competitor chooses his ranks randomly. By considering all the possible rankings, find the probability that the value of Spearman's rank correlation coefficient \(r _ { s }\) between the competitor's ranks and the expert's ranks is at least \(\frac { 27 } { 28 }\).
2. Another competitor ranks the seven resorts. A significance test is carried out to test whether there is evidence that this competitor is merely guessing the rank order of the seven resorts. The critical region is \(r _ { s } \geqslant \frac { 27 } { 28 }\). State the significance level of the test. \section*{END OF QUESTION PAPER}