Question 1

Part A

When the weather is stormy there is a 40% probability that the lifeboat service of a small coastal town is called out. When the weather isn't stormy the probability is 10%. It is only stormy about 1 night in 20 throughout the year. What is the probability that it is stormy and the lifeboat service is called out on a particular night?

Part B

A private equity company is bidding to purchase two businesses. The chance that the first bid will be successful is reckoned by analysts to be ¼, while the chance that the second is successful is estimated at 1/5. What is the probability that neither bid is successful?

Part C

In a certain firm 40% of the workforce are women. Only 25% of the female workforce are management grade, whereas for male workers the figure is 30%. What is the probability that a worker selected at random is both female and management grade?

Question 2

Assuming that an individual's IQ score has a normal distribution with mean 100 and standard deviation 15, what is the probability that an individual's IQ score

a) Is more than 125?

b) Is between 91 and 121?

Question 3

Part A

A JCB is rented out by a company. The duration in weeks of each rental is assumed to have a normal distribution. A random sample of the duration of 14 rentals gives =2.14 and standard deviation 1.29. Calculate a 99% confidence interval for the mean rental duration.

Part B

From past analyses it is known that daily attendance at a swimming pool is normally distributed with variance 225. During a random sample of five days the following numbers of people used the pool each day:

220 196 210 186 222

Give a 95% confidence interval for the average number of people who use the pool in a day and explain what it means.

Question 4

This data shows the electricity bill of 7 customers and the time they took to pay the bills.

a) Work out the Pearson correlation coefficient.

b) Determine the least squares regression equation that can be used to predict how long a bill of a given size will take to pay.

c) Using your equation predict how long it will take a customer to pay:

(i) a bill of £125
(ii) a bill of £1000

Question 5

The data below shows the percentage of rent collected door-to-door by a council and the % of total rent in arrears.

Calculate the Spearman's correlation coefficient.

Question 6

A company is accused of selling underweight products. This product is supposed to weight 500g: a sample of 6 was weighed and the results were:

495, 512, 480, 505, 490, 502

Is there any evidence that the mean weight is less than 500g? (The weight of the product is known to be normally distributed).

Question 7

A company has analysed the time its customers take to pay an invoice over the past few years and has found that the distribution is normal, with a mean of 5.8 weeks and a standard deviation of 2.3 weeks. In order to speed up payment the company threatened to charge interest if bills were not paid within 3 weeks. A sample of 10 customers was then analysed and the mean time to pay had been reduced to 4.9 weeks. Is there evidence that the mean time for all its customers had actually been reduced?