Edexcel S2 (Statistics 2) 2006 June

Before introducing a new rule the secretary of a golf club decided to find out how members might react to this rule.

1. Explain why the secretary decided to take a random sample of club members rather than ask all the members. [1]
2. Suggest a suitable sampling frame. [1]
3. Identify the sampling units. [1]

The continuous random variable \(L\) represents the error, in mm, made when a machine cuts rods to a target length. The distribution of \(L\) is continuous uniform over the interval \([-4.0, 4.0]\). Find

1. P\((L < -2.6)\), [1]
2. P\((L < -3.0 \text{ or } L > 3.0)\). [2]

A random sample of 20 rods cut by the machine was checked.

1. Find the probability that more than half of them were within 3.0 mm of the target length. [4]

An estate agent sells properties at a mean rate of 7 per week.

1. Suggest a suitable model to represent the number of properties sold in a randomly chosen week. Give two reasons to support your model. [3]
2. Find the probability that in any randomly chosen week the estate agent sells exactly 5 properties. [2]
3. Using a suitable approximation find the probability that during a 24 week period the estate agent sells more than 181 properties. [6]

Breakdowns occur on a particular machine at random at a mean rate of 1.25 per week.

1. Find the probability that fewer than 3 breakdowns occurred in a randomly chosen week. [4]

Over a 4 week period the machine was monitored. During this time there were 11 breakdowns.

1. Test, at the 5\% level of significance, whether or not there is evidence that the rate of breakdowns has changed over this period. State your hypotheses clearly. [7]

A manufacturer produces large quantities of coloured mugs. It is known from previous records that 6\% of the production will be green. A random sample of 10 mugs was taken from the production line.

1. Define a suitable distribution to model the number of green mugs in this sample. [1]
2. Find the probability that there were exactly 3 green mugs in the sample. [3]

A random sample of 125 mugs was taken.

1. Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using
   1. a Poisson approximation, [3]
   2. a Normal approximation. [6]

The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{1+x}{k}, & 1 \leqslant x \leqslant 4, \\ 0, & \text{otherwise}. \end{cases}$$

1. Show that \(k = \frac{21}{2}\). [3]
2. Specify fully the cumulative distribution function of \(X\). [5]
3. Calculate E\((X)\). [3]
4. Find the value of the median. [3]
5. Write down the mode. [1]
6. Explain why the distribution is negatively skewed. [1]

It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.

1. Using a 5\% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to 2.5\% as possible. [6]
2. State the actual significance level of the above test. [1]

At a later date, a random sample of 20 bowls was taken and 2 of them were found to have defects.

1. Test, at the 10\% level of significance, whether or not there is evidence that the proportion of bowls with defects has decreased. State your hypotheses clearly. [7]