OCR Further Statistics (Further Statistics) 2021 June

Jo can use either of two different routes, A or B, for her journey to school. She believes that route A has shorter journey times. She measures how long her journey takes for 17 journeys by route A and 12 journeys by route B. She ranks the 29 journeys in increasing order of time taken, and she finds that the sum of the ranks of the journeys by route B is 219.

1. Test at the 10\% significance level whether route A has shorter journey times than route B. [8]
2. State an assumption about the 29 journeys which is necessary for the conclusion of the test to be valid. [1]

The random variable \(X\) is equally likely to take any of the \(n\) integer values from \(m + 1\) to \(m + n\) inclusive. It is given that E\((3X) = 30\) and Var\((3X) = 36\). Determine the value of \(m\) and the value of \(n\). [7]

26 cards are each labelled with a different letter of the alphabet, A to Z. The letters A, E, I, O and U are vowels.

1. Five cards are selected at random without replacement. Determine the probability that the letters on at least three of the cards are vowels. [4]
2. All 26 cards are arranged in a line, in random order.
   1. Show that the probability that all the vowels are next to one another is \(\frac{1}{2990}\). [3]
   2. Determine the probability that three of the vowels are next to each other, and the other two vowels are next to each other, but the five vowels are not all next to each other. [4]

A biased spinner has five sides, numbered 1 to 5. Elmer spins the spinner repeatedly and counts the number of spins, \(X\), up to and including the first time that the number 2 appears. He carries out this experiment 100 times and records the frequency \(f\) with which each value of \(X\) is obtained. His results are shown in Table 1, together with the values of \(xf\). Table 1

1. State an appropriate distribution with which to model \(X\), determining the value(s) of any parameter(s). [3]

Elmer carries out a goodness-of-fit test, at the 5\% level, for the distribution in part (a). Table 2 gives some of his calculations, in which numbers that are not exact have been rounded to 3 decimal places. Table 2

1. Show how the expected frequency corresponding to \(x \geq 7\) was obtained. [2]
2. Carry out the test. [5]