Question

Suppose you want to estimate the average number of years employees have worked at your company so far. You take a random sample of 100 workers and you find the average number of years they have worked at your company so far is 10 years. (Assume the standard deviation for number of years worked is known to be 2 years.)

Let X be the number of years an employee has worked at this company. Assuming this company has been around for a long time, you might expect the distribution of X to be skewed to the right, and hence does NOT have a normal distribution. Explain why this might be the case.

Find a 95% confidence interval for the average number of years worked for employees over the whole company.

Explain why you could not do the previous problem (and use a formula involving a Z value) without use of the Central Limit Theorem. Remember, X does not have a normal distribution!

Why are the proper conditions met in order to use the CLT here? Explain.

Answer 1

Part 1

Since the company has been around for a long time, the weightage to the length of employement for the older employees would be very high and so the distribution would tend to be skewed to the right.

Part 2

We have to calculate 95% Confidence Interval

Conceptual Knowledge

To calculate the 95% Confidence Interval we use the t statistic and sample mean (M) to generate an interval estimate of a population mean (μ).

The formula for estimation is:

μ = M ± t*S_M

where:

M = sample mean
t = t statistic determined by confidence level
S_M = standard error = √(s²/n)

Calculation

M = 10
t = t_100-1,0.025 = t_99,0.025= 1.98
S_M = (2²/100)^0.5 = 0.2

μ = M ± t * S_M
μ = 10 ± 1.98*0.2
μ = 10 ± 0.4

Therefore 95% CI [9.6, 10.4]

You can be 95% confident that the population mean (μ) falls between 9.6 and 10.4.

Mean values tend to normal. So without using CLT we could not have completed this problem

Part 3

• For a skewed population, the sample size should be fairly large.
• The data must be sampled randomly
• The sample values must be independent of each other