Question

Consider the population of adult females resident in Melbourne. Our focus is on the population mean height. Assume we know ? (population standard deviation) is 25, but we do not know the population mean, µ. We take a sample of adult females resident in Melbourne (n=100) and calculate the sample mean height as 70 cm.

1. Write down the distribution of X, female height.
2. Write down the distribution of . Are you relying on any theorems in your answer? If yes explain.
3. Derive the 95% confidence interval around the sample mean.
4. In your answer to iii) will the specific confidence interval you have derived contain the population mean? Explain carefully.

Answer 1

(I)

The distribution of X, female height is unknown. Only statistic we know is that the distribution of X, female height has mean = 70 cm.

(II)

The distribution of , the sample mean is Normal Distribution with mean equal to population mean and standard deviation given by:

SE = / = 25/ = 2.5

We are relying on Central Limit Theorem, according to which the sampling distribution of sample statistic is Normal distribution forlarge sample size irresptive of the shape of the population distribution.

(III)

= 70

= 25

n = 100

= 0.05

From Table, critical values of Z = 1.96

Confidence Interval:

So,

Answer is:

(65.1, 74.9)

(IV)

The 95% Confidence Interval (65.1, 74.9) is a range of values we are 95% confident will contain the true unknown population mean height of adult females resident in Melbourne.