OCR S3 (Statistics 3) 2010 June

1 The numbers of minor flaws that occur on reels of copper wire and reels of steel wire have Poisson distributions with means 0.21 per metre and 0.24 per metre respectively. One length of 5 m is cut from each reel.

1. Calculate the probability that the total number of flaws on these two lengths of wire is at least 2 .
2. State one assumption needed in the calculation.

2 A coffee machine used in a bar is claimed by the manager to dispense 170 ml of coffee per cup on average. A customer believes that the average amount of coffee dispensed is less than 170 ml . She measures the amount of coffee in 6 randomly chosen cups. The results, in ml , are as follows. $$\begin{array} { l l l l l l } 167 & 171 & 164 & 169 & 168 & 166 \end{array}$$ Assuming a relevant normal distribution, test the manager's claim at the 5\% significance level.

3 The developers of a shopping mall sponsored a study of the shopping habits of its users. Each of a random sample of 100 users was asked whether their weekend shopping was mainly on Saturday or mainly on Sunday. The results, classified according to whether the user lived in the city or the country, are shown in the table.

1. Test, at the \(10 \%\) significance level, whether there is an association between the area in which shoppers live and the day on which they shop at the weekend.
2. State, with a reason, whether the conclusion of the test would be different at the \(3 \%\) significance level.

4 Part of an ecological study involved measuring the heights of trees in a young forest. In order to obtain an estimate of the mean height of all the trees in the forest, a random sample of 70 trees was selected and their heights measured. These heights, \(x\) metres, are summarised by \(\Sigma x = 246.6\) and \(\Sigma x ^ { 2 } = 1183.65\). The mean height of all trees in the forest is denoted by \(\mu\) metres.

1. Calculate a symmetric \(90 \%\) confidence interval for \(\mu\).
2. A student was asked to interpret the interval and said,
   "If 100 independent \(90 \%\) confidence intervals were calculated then 90 of them would contain \(\mu\)." Explain briefly what is wrong with this statement.
3. Four independent \(90 \%\) confidence intervals for \(\mu\) are obtained. Calculate the probability that at least three of the intervals contain \(\mu\).

5 A random variable \(X\) is believed to have (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 ,
1 - \mathrm { e } ^ { - x ^ { 2 } } & x \geqslant 0 . \end{cases}$$ In order to test this, a random sample of 150 observations of \(X\) were taken, and their values are summarised in the following grouped frequency table. The expected frequencies, correct to 1 decimal place, corresponding to the above distribution, are 33.2, 61.6 and 39.4 respectively for the first 3 cells.

1. Find the expected frequencies for the last 2 cells.
2. Carry out a goodness of fit test at the \(2 \frac { 1 } { 2 } \%\) significance level.

6 It has been suggested that people who suffer anxiety when they are about to undergo surgery can have their anxiety reduced by listening to relaxation tapes. A study was carried out on 18 experimental subjects who listened to relaxation tapes, and 13 control subjects who listened to neutral tapes. After listening to the tapes, the subjects were given a test which produced an anxiety score, \(X\). Higher scores indicated higher anxiety. The results are summarised in the table.

1. Use a two-sample \(t\)-test, at the \(5 \%\) significance level, to test whether anxiety is reduced by listening to relaxation tapes. State two necessary assumptions for the validity of your test.
2. State why a test using a normal distribution would not have been appropriate.

7 The employees of a certain company have masses which are normally distributed. Female employees have a mean of 66.7 kg and standard deviation 9.3 kg , and male employees have a mean of 78.3 kg and standard deviation 8.5 kg . It may be assumed that all employees' masses are independent. On the ground floor 6 women and 9 men enter the empty staff lift for which it is stated that the maximum load is 1150 kg .

1. Calculate the probability that the maximum load is exceeded. At the first floor all 15 passengers leave and 6 women, 8 men and an unknown employee enter.
2. Assuming that the unknown employee is equally likely to be a woman or a man, calculate the probability that the maximum load is now exceeded.

8 The continuous random variable \(S\) has probability density function given by $$f ( s ) = \begin{cases} \frac { 8 } { 3 s ^ { 3 } } & 1 \leqslant s \leqslant 2
0 & \text { otherwise } \end{cases}$$ An isosceles triangle has equal sides of length \(S\), and the angle between them is \(30 ^ { \circ }\) (see diagram).

1. Find the (cumulative) distribution function of the area \(X\) of the triangle, and hence show that the probability density function of \(X\) is \(\frac { 1 } { 3 x ^ { 2 } }\) over an interval to be stated.
2. Find the median value of \(X\). {www.ocr.org.uk}) after the live examination series.
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