QUESTION

(a) A simple random sample of 80 items resulted in a sample of 49. The population standard deviation is =7 .

i. What is the standard error of the mean?

ii. At 99% probability, what is the margin of error?

(b)A hospital employs 250 nurses. In a random sample of 81 of these, the mean number of hours overtime billed in a particular week was 12.3, and the sample standard deviation was 8.4 hours. Find a 98% confidence interval for the mean number of hours overtime billed per nurse in this hospital that week?

(c) In a large hospital an average of 3 out of every 5 patients ask for water with their meal. A random sample of 10 customers is selected. Assuming a binomial distribution, find the probability that

i. Exactly 6 patients ask for water with their meal

ii. At most 4 patients ask for water with their meal

iii. At least 3 patients ask for water with their meal

iv. Find the mean of this distribution

v. Find the standard deviation of this distribution.

(d) The arrival rate of customers arriving at a bank counter follows a Poisson distribution with a mean rate of 4 per 10 minutes interval. Find the probability that

i. Exactly 0 customer will arrive in 10 minutes interval

ii. Exactly 2 customers will arrive in 10 minutes interval

iii. At most 2 customers will arrive in 10 minutes interval

iv. At least 3 customers will arrive in 10 minutes interval

(e) In a mainframe computer centre, execution time of programs follows an exponential distribution. The average execution time of the programs is 5 minutes. Find the probability that the execution time of programs is:

i. Less than 4 minutes

ii. More than 6 minutes

QUESTION

In order to estimate likely expenditure by customers at a new supermarket, a sample of till slips from a similar supermarket describing the weekly amounts spent by 500 randomly selected customers was analysed. These data were found to be approximately normally distributed with a mean of K50 and a standard deviation of K15. Using this knowledge, find

(a) The probability that any shopper selected at random

i. Spends more than K80 per week

ii. Spends less than K50 per week

(b) The percentage of shoppers who are expected to:

i. Spend between K30 and K80 per week

ii. Spends between K55 and K70 per week

(c) The expected number of shoppers who will

i. Spend less than K70 per week

ii. Spend between K37.50 and K57.50 per week

(d) The value below which:

i. 70% of the customers are expected to spend

ii. 45% of the customers are expected to spend

(e) The value expected to be exceeded by:

i. 10% of the customers

ii. 80% of the customers

(f) The value below which:

i. 350 of the shoppers are expected to spend

ii. 100 of the shoppers are expected to spend

QUESTION

(a) Customers at TAB are charged for the amount of salad the take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces. Let = Salad plate filling weight.

i. Find the probability density function of

ii. What is the probability that a customer will take between 12 and 15 ounces of salad?

iii. What is the probability that a customer will take fewer than 5 ounces of salad.

iv. Find ( ) and Var( )

(b) Suppose a company fleet of 20 cars contains 7 cars that do not meet government exhaust emissions standards and are therefore releasing excessive pollution. Moreover, suppose that a traffic policeman randomly inspects 5 cars, what is the probability of no more than 2 polluting cars being selected? (Assuming a hypergeometric distribution).

(c) As part of the underwriting process for insurance, each prospective policyholder is tested for high blood pressure. Let represent the number of tests completed when the first person with high blood pressure is found. The expected value of is 12.5. Calculate the probability the sixth person tested is the first one with high blood pressure assuming a geometric distribution.

QUESTION

(a) The supermarket statistician realized that there was a considerable range in the spending power of its customers. Even though the overall spending seemed to have increased the high spenders still spent more than the low spenders and that the individual increases would show a smaller spread. In other words these two populations, 'before' and 'after', and not independent.

Before the next advertising campaign at the supermarket, he took a random sample of 10 customers, to J , and collected their till slips. After the campaign, slips from the same 10 customers were collected and both sets of data recorded. Using the paired data, has there been any mean change at a 95% confidence level?

A B C D E F G H I J

Before 42.30, 55.76, 32.29 ,10.23 ,15.79 ,46.50 ,32.20 ,78.65 ,32.20 ,15.90

After 43.09, 59.20, 31.76, 20.78 ,19.50, 50.67 ,37.32 ,77.80 ,37.39 , 17.24

(b) A training manager wishes to see if there has been any alteration in the aptitude of his trainees after they have been on a course. He gives each an aptitude test before they start the course and an equivalent one after they have completed it. The scores recorded below:

Trainee A B C D E F G H I

Score before training 74 ,69 ,45 ,67 ,67 ,42 ,54 ,67 ,76

Score after training 69, 76 ,56 ,59 ,78 ,63 ,54 ,76 ,75

i. Using the paired data, has any change taken place at a 5% significance level?

ii. What assumptions are necessary to apply a paired difference analysis to the data?

iii. Construct a 95% confidence interval for ( 1 2) . Interpret the confidence interval.

(c) The following data,recorded in days, represent the length of time to recovery for patients randomly treated with one of two medications to clear up severe bladder infections:

n xS2

Medication 1 1417 1.5

Medication 2 1619 1.8

Find a 99% confidence interval for the difference 1 2 in the mean recovery timefor the two medications, assuming normal populations with equal variances.

(d) A random sample of size n1 =25 , taken from a normal population with a standard deviation 1 =5.2 , has a mean x1 = 81. A second random sample of size n2 =36 , taken from a different normal population with a standard deviation 2 =3.4 , has a mean x2 =76 . Test the hypothesis that 1 =2 against the alternative 1 2 .

(e) A cigarette manufacturing firm distributes two brands of cigarettes. If it is found that 56 of 200 smokers prefer brand A and that 29 of 150 smokers prefer brand B, can we conclude at the 0.06 level of significance that brand A outsells brand B?