Question

A simple random sample of size n is drawn from a population that is normally distributed. The sample​ mean, x overbar​, is found to be 112​, and the sample standard​ deviation, s, is found to be 10. ​(a) Construct a 95​% confidence interval about mu if the sample​ size, n, is 23. ​(b) Construct a 95​% confidence interval about mu if the sample​ size, n, is 16. ​(c) Construct a 90​% confidence interval about mu if the sample​ size, n, is 23. ​(d) Could we have computed the confidence intervals in parts​ (a)-(c) if the population had not been normally​ distributed?

Answer 1

Given the random samples are drawn from a normal distribution and also we have ,

sample mean, and sample standard deviation,

Let denote the population mean.

We know that the statistic,

follows student's t distribution with (n-1) degrees of freedom -------- equation (1)

Hence the % confidence limits for are given by:

, where is the tabulated value of t for (n-1) degrees of freedom at significance level .

Hence the required confidence interval for can be obtained as

--------equation (2)

(a) Qn: Construct a 95​% confidence interval about mu if the sample​ size, n, is 23.

i.e., we have   and   

Using equation (2), the required confidence interval for can be obtained as

, {from t tables , for (n-1)= 22 degrees of freedom}

(b) Qn: Construct a 95​% confidence interval about mu if the sample​ size, n, is 16.

i.e., we have   and   

Using equation (2), the required confidence interval for ​​​​​​​ can be obtained as

, {from t tables ​​​​​​​, for (n-1)= 15 degrees of freedom}

(c) Qn: Construct a 90​% confidence interval about mu if the sample​ size, n, is 23.

i.e., we have   and   

Using equation (2), the required confidence interval for ​​​​​​​ can be obtained as

, {from t tables ​​​​​​​, for (n-1)= 22 degrees of freedom}

(d) Qn: Could we have computed the confidence intervals in parts​ (a)-(c) if the population had not been normally​ distributed?

If the population was not normal, we couldn't have used t-distribution since it uses the assumption that the samples are drawn from normal distribution.

But if the sample size was sufficiently large, we could have used normal tables for estimating the confidence levels.