a) The continuous random variable X is normally distributed with mean μ and variance σ2.
Required:
Sketch the distribution of X and indicate on your sketch the mean, median and mode.                                                    (4 marks)

b) CarJoy Manufacturing Company produces electronic components which have life spans that are normally distributed with mean, μ hours and standard deviation, σ hours. Only 1% of the components have a life span less than 3,500 hours and 2.5% have a life span greater than 5,500 hours.
Required:
i) By standardizing the values of X to the standard normal values, obtain TWO (2) linear equations in μ and σ.           (4 marks)

ii) Using an appropriate method of solving simultaneous equations, determine the value of μ and value of σ in (i) above. (4 marks)

iii) Determine the proportion of electronic components with life spans between 4,120.50 hours and 5,052.45 hours.     (4 marks)

iv) If the company gives warranty of 4,000 hours on the components, find the percentage of components the company is expected to replace under the warranty.                                                                                                                              (4 marks)

a) The normal distribution is a probability distribution which usually applies to continuous variables.
Required:
i) State FOUR (4) properties of the Normal Distribution.                                                                                                  (4 marks)

ii) State TWO (2) examples of phenomena which closely follow a Normal Distribution.                                          (2 marks)

b) In a study commissioned by Ofo Stores, the researcher examined the spending habits of the residents of Kojokrom. He found the spending habits to be normally distributed with mean of GH¢700 and standard deviation of GH¢70.
Required:
a) Determine the probability that a resident selected at random spends:
i) Less than GH¢620                                                                                                                                                      (3 marks)

ii) More than GH¢1,000                                                                                                                                                 (3 marks)

iii) Between GH¢800 and GH¢900                                                                                                                                (4 marks)

b) Calculate the amount:
i) Above which 80% of the residents will spend in a week.                                                                                          (2 marks)

ii) Below which 30% of the residents will spend in a week.                                                                                          (2 marks)

The distribution of the total loan amounts defaulted by customers of a bank annually is approximated by a normal distribution. An average default amount is GH¢1.50 million and the standard deviation is GH¢0.50 million.
Required:
a) Determine the probability that the total loan amount defaulted exceeds GH¢1.50 million.                                           (3 marks)

b) Determine the probability that the total loan amount defaulted is between GH¢0.86 million and GH¢0.90 million.   (4 marks)

c) Determine the probability that the total amount defaulted is at most GH¢2million.                                                      (4 marks)

d) Determine the amount to be allowed per annum for loan defaults if 1% of actual defaults exceed this amount.          (3 marks)

e) Determine the lower quartile of the distribution of total loan amounts defaulted.                                                          (3 marks)

f) Determine the upper quartile of the distribution of total loan amounts defaulted.                                                          (3 marks)

a) Receipts at a particular depot have amounts which follow the Normal distribution with mean of GH¢103.60 and a standard deviation of GH¢8.75.
Required:
i) Determine the percentage of receipts over GH¢ 120.05.                                                                                                 (3 marks)

ii) Determine the percentage of receipts below GH¢ 92.75.                                                                                               (3 marks)

iii) Determine the percentage of receipts between GH¢ 83.65 and GHS 117.60.                                                               (3 marks)

iv) Determine the receipts amount such that approximately 25 percent of receipts are greater amounts.                          (3 marks)

v) Above what amount will 90 percent of receipts lie?                                                                                                       (3 marks)

b) If 10.56 percent of receipts have an amount above 110.05 and 4.01 percent of receipts have an amount above 120.05.
Required:
Calculate, the mean and standard deviation of the receipts assuming that they are normally distributed.                         (5 marks)

A large departmental store, BOBO, has analyzed the monthly amount spent by its credit card customers and found that it is normally distributed with a mean of GHC 100 and a standard deviation of GHC 15.
Required:
Determine, the percentage of people who will spend:

(i) Over GHC 130.                                                                                                                                                            (2 Marks)

(ii) Over GHC 120.                                                                                                                                                            (2 Marks)

(iii) Below GHC 70.                                                                                                                                                           (2 Marks)

(iv) Between GHC 100 and GHC 130.                                                                                                                              (2 Marks)

(v) Between GHC 115 and GHC 130.                                                                                                                              (2 Marks)

Calculate, the minimum amount spent of:
(vi) The top 10%.                                                                                                                                                             (2.5 Marks)

(vii) The top 3% of customers.                                                                                                                                        (2.5 Marks)