X is a random variable which represents the weight in kg of crates of apples in the local market. Assume it has a normal distribution with a mean of 50 and variance of 49.

In a game you have to select a 3-digit number, you pay 50p to play a game.

(Each selected digit can be any number from 0 to 9 and may be selected more than once.)

If you select the correct number, you win £250

a)    How many different selections are possible?                 (2 marks)
b)    What is the probability of winning?                                 (2 marks)
c)    If you win, what is your profit?                                        (1 mark)
d)    Find the expected value of winning and say whether you would play this game.                                                                                                          (3 marks)

 (Total 8 marks)

 

 

Question 3

The table below summarises the job applications made by some graduates during their last year:

Was the application successful?

Yes                  No

Male graduates                                 263                  148

Female graduates                             178                  122

 

If you were to randomly select a graduate:

 

1. What is the probability that an application was not successful? (2 marks)

 

1. b) What is the probability the application was made by a male graduate or was successful?                                                        (4 marks)

(Total 6 marks)

 

 

Question 4

X is a random variable which represents the weight in kg of crates of apples in the local market. Assume it has a normal distribution with a mean of 50 and variance of 49.

 

a)      What proportion of the crates weigh over 80kg?
	(4 marks)

 

b)      68% of the crates will weigh between [what?, what?].
	(2 marks)

 

c)      What proportion of the crates will weigh between 40kg and 53kg?
	(6 marks)

 

d)      A sample of 25 crates was randomly drawn. What is the probability that the sample mean is between 47kg and 54kg?
	(8 marks)

 

(Total 20 marks)

 

 

 

 

Question 5

 

For a sample of 201 households from Norwich it was found that the average monthly expenditure on food was £500 with sample variance £2,500.

 

1. Using the given formula construct a 95% confidence interval for the average monthly expenditure on food. (4 marks)

 

1. You were informed that, for the population as a whole, the monthly expenditure on food is £510; you believe that this is too high. Conduct a test in order to prove that you are right.                               (6 marks)

 

(Total 10 marks)

 

 

 

Question 6

 

An employer claims that bonuses paid to workers in a factory average £1000.  A random sample of 22 workers gives an average bonus of only £975 with a standard deviation of £100.

 

Conduct a one-tail t test of the employer’s claim against an alternative hypothesis of H₁: µ<1000,  Use a 5% significance level.