OCR S1 (Statistics 1) 2013 June

1 The lengths, in centimetres, of 18 snakes are given below. $$\begin{array} { l l l l l l l l l l l l l l l l l l } 24 & 62 & 20 & 65 & 27 & 67 & 69 & 32 & 40 & 53 & 55 & 47 & 33 & 45 & 55 & 56 & 49 & 58 \end{array}$$

1. Draw an ordered stem-and-leaf diagram for the data.
2. Find the mean and median of the lengths of the snakes.
3. It was found that one of the lengths had been measured incorrectly. After this length was corrected, the median increased by 1 cm . Give two possibilities for the incorrect length and give a corrected value in each case.

2

1. The table shows the times, in minutes, spent by five students revising for a test, and the grades that they achieved in the test.

   Student	Ann	Bill	Caz	Den	Ed
   Time revising	0	60	35	100	45
   Grade	C	D	E	B	A

   Calculate Spearman's rank correlation coefficient.
2. The table below shows the ranks given by two judges to four competitors.

   Competitor	P	Q	R	S
   Judge 1 rank	1	2	3	4
   Judge 2 rank	3	2	1	4

   Spearman's rank correlation coefficient for these ranks is denoted by \(r _ { s }\). With the same set of ranks for Judge 1, write down a different set of ranks for Judge 2 which gives the same value of \(r _ { s }\). There is no need to find the value of \(r _ { s }\).

3 The probability distribution of a random variable \(X\) is shown.

1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
2. Three independent values of \(X\), denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), are chosen. Given that \(X _ { 1 } + X _ { 2 } + X _ { 3 } = 19\), write down all the possible sets of values for \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) and hence find \(\mathrm { P } \left( X _ { 1 } = 7 \right)\).
3. 11 independent values of \(X\) are chosen. Use an appropriate formula to find the probability that exactly 4 of these values are 5 s .

4 At a stall in a fair, contestants have to estimate the mass of a cake. A group of 10 people made estimates, \(m \mathrm {~kg}\), and for each person the value of \(( m - 5 )\) was recorded. The mean and standard deviation of \(( m - 5 )\) were found to be 0.74 and 0.13 respectively.

1. Write down the mean and standard deviation of \(m\). The mean and standard deviation of the estimates made by another group of 15 people were found to be 5.6 kg and 0.19 kg respectively.
2. Calculate the mean of all 25 estimates.
3. Fiona claims that if a group's estimates are more consistent, they are likely to be more accurate. Given that the true mass of the cake is 5.65 kg , comment on this claim.

5 The table shows some of the values of the seasonally adjusted Unemployment Rate (UR), \(x \%\), and the Consumer Price Index (CPI), \(y \%\), in the United Kingdom from April 2008 to July 2010. These data are summarised below. $$n = 10 \quad \sum x = 70.3 \quad \sum x ^ { 2 } = 503.45 \quad \sum y = 30.8 \quad \sum y ^ { 2 } = 103.94 \quad \sum x y = 211.9$$

1. Calculate the product moment correlation coefficient, \(r\), for the data, showing that \(- 0.6 < r < - 0.5\).
2. Karen says "The negative value of \(r\) shows that when the Unemployment Rate increases, it causes the Consumer Price Index to decrease." Give a criticism of this statement.
3. (a) Calculate the equation of the regression line of \(x\) on \(y\).
   (b) Use your equation to estimate the value of the Unemployment Rate in a month when the Consumer Price Index is 4.0\%.

6 The diagram shows five cards, each with a letter on it.
\includegraphics[max width=\textwidth, alt={}, center]{d06430a6-7957-4313-beea-bb320fadb282-4_113_743_315_662} The letters A and E are vowels; the letters B, C and D are consonants.

1. Two of the five cards are chosen at random, without replacement. Find the probability that they both have vowels on them.
2. The two cards are replaced. Now three of the five cards are chosen at random, without replacement. Find the probability that they include exactly one card with a vowel on it.
3. The three cards are replaced. Now four of the five cards are chosen at random without replacement. Find the probability that they include the card with the letter B on it.

7 In a factory, an inspector checks a random sample of 30 mugs from a large batch and notes the number, \(X\), which are defective. He then deals with the batch as follows.

• If \(X < 2\), the batch is accepted.
• If \(X > 2\), the batch is rejected.
• If \(X = 2\), the inspector selects another random sample of only 15 mugs from the batch. If this second sample contains 1 or more defective mugs, the batch is rejected. Otherwise the batch is accepted.

It is given that \(5 \%\) of mugs are defective.

1. (a) Find the probability that the batch is rejected after just the first sample is checked.
   (b) Show that the probability that the batch is rejected is 0.327 , correct to 3 significant figures.
2. Batches are checked one after another. Find the probability that the first batch to be rejected is either the 4th or the 5th batch that is checked.
3. A bag contains 12 black discs, 10 white discs and 5 green discs. Three discs are drawn at random from the bag, without replacement. Find the probability that all three discs are of different colours.
4. A bag contains 30 red discs and 20 blue discs. A second bag contains 50 discs, each of which is either red or blue. A disc is drawn at random from each bag. The probability that these two discs are of different colours is 0.54 . Find the number of red discs that were in the second bag at the start.

9 A game is played with a token on a board with a grid printed on it. The token starts at the point \(( 0,0 )\) and moves in steps. Each step is either 1 unit in the positive \(x\)-direction with probability 0.8 , or 1 unit in the positive \(y\)-direction with probability 0.2 . The token stops when it reaches a point with a \(y\)-coordinate of 1 . It is given that the token stops at \(( X , 1 )\).

1. (a) Find the probability that \(X = 10\).
   (b) Find the probability that \(X < 10\).
2. Find the expected number of steps taken by the token.
3. Hence, write down the value of \(\mathrm { E } ( X )\).