Question

Q1.

1. Explain what happens to the 95 % confidence interval as the sample size increases.
2. Explain what happens to the width of the confidence interval for a 99% interval versus a 95%.

Q2. Consider the population of adult females resident in Melbourne. Our focus is on the population mean height. Assume we do not know ? (population standard deviation) or 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 and the sample standard deviation as 25cm.

1. Derive the 95% confidence interval for the population mean
2. Compare your interval here and last week – which is larger and why?

Answer 1

Q1 .

i. We know the margin error for the confidence interval is when population standard deviation is known. Here n is the sample size . O clearly when n increses the margin error part decreases. And the width of the confidence interval also decreases if confidence level is fixed.

So for 95% confidence interval if sample size increases then the width of the interval will decrease.

ii.

If confidence level increases the critical value for the corresponding confidence level will also increase and then the margin error will also increase . Because more wide confidence interval means we are more confident about the unknown parameter.

So the width of the 99% confidence interval is more than the width of 95% confidence interval.

Q2.

i . Here n= 100 , sample mean M= 70 and sample standard deviation is s= 25

Then 95% confidence interval for population mean is

t = t score for 95% confidence with df = n-1 = 99

= 1.984 (from t table)

So the 95% confidence interval is

= ( 70 - 4.96 , 70 + 4.96 )

= ( 65.04 , 74.96 )