a)The distribution of employee wages in a certain company is modelled as a normal random variable, and previous years records indicate that the standard deviation is Sh.12000.

i.)Calculate the width of a 95% confidence interval for the mean wage if a sample of size 100 is available. [3 marks]

ii.)Determine the minimum sample size required to ensure that a 95% confidence interval for the mean wage is of width at most . [3 marks]

iii.)Comment briefly on the comparison of the confidence intervals in (i) and (ii) with respect to widths and sample sizes used. [2 mark]

b)Five studentsthe females Ann and Betty and the males Clint, Douglas, and Edwardare rated equally qualified for admission to a certain course in KCA University, ahead of other applicants. However, all but two positions have been filled for the entering class. The admissions committee can admit only two more students, so it decides to randomly select two of these five candidates.For this strategy, let X= number of males admitted.

i.)Construct the probability distribution for X.[4 marks]

ii.)Construct the sampling distribution of the sample proportion of the students selected who are male. [4 marks]

c)Say, in each of the following, whether true or false in regard to the Central Limit Theorem:

i.)All variables have bell-shaped sample data distributions if a random sample contains at least about 30 observations

ii.)Population distributions are normal whenever the population size is large.

iii.)For large random samples, the sampling distribution ofis approximately normal, regardless of the shape of the population distribution

iv.)The sampling distribution looks more like the population distribution as the sample size increases.[4 marks]

d)Jua kali City was designed to attract retired people. Its current population of 50,000 residents has a mean age of 60 years and a standard deviation of 16 years. The distribution of ages is skewed to the left, reflecting the predominance of older individuals. A random sample of 100 residents of Jua kali City has = 58.3 and 15.0.

i.)Find the center and spread of the sampling distribution of for n = 100. What shape does it have and what does it describe? [4 marks]

ii.)Explain why it would not be unusual to observe a person of age 40 in Juakali City, but it would be highly unusual to observe a sample mean of 40, for a random sample size of 100. [6 marks]