Question

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 X. 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

Write down the distribution of X, female height.

The female height follows normal distribution with mean = 70 cm and standard deviation = 25

Write down the distribution of X. Are you relying on any theorems in your answer? If yes explain.

We are using the Central Limit theorem as n > 30

Derive the 95% confidence interval around the sample mean.

We need to construct the 95% confidence interval for the population mean \muμ. The following information is provided:

Sample Mean	70
Population Standard Deviation	25
Sample Size	100

The critical value for α=0.05 is z_c= 1.96. The corresponding confidence interval is computed as shown below:

CI = (65.1, 74.9)

Therefore, based on the data provided, the 95% confidence interval for the population mean is 65.1 < μ < 74.9, which indicates that we are 95% confident that the true population mean \muμ is contained by the interval (65.1, 74.9)