Question

Believe it or not, there are some people who don't think that Duke will win the NCAA tournament this year. Let's say that you want to measure the average intelligence of Duke Doubters. Problem is you could only find 28 DDs on this whole campus. The mean of your sample was 90 and the standard deviation was 60. Calculate the 90% Confidence Interval for the IQ of DDs

Answer 1

The 90% confidence interval is
`87.95   <  \mu < 92.05`

Explanation:

From the question we are told that

The sample size is
`n = 28`

The sample mean is
`\= x = 90`

The standard deviation is
`s= 60`

Generally given that the sample size is not large enough i.e n < 28 then we will make use of the t distribution table

Generally the degree of freedom is mathematically represented as

`df = n- 1`

=>
`df = 28 - 1`

=>
`df = 2 7`

From the question we are told the confidence level is 95% , hence the level of significance is

`\alpha = (100 - 95 ) \%`

=>
`\alpha = 0.10`

Generally from the student t distribution table the critical value of
`(\alpha )/(2)` at a degree of freedom of
`df = 2 7` is

`t_{(\alpha )/(2) , 27  } =  2.052`

Generally 95% confidence interval is mathematically represented as

`\= x -E <  \mu <  \=x  +E`

=>
`90  - 2.052  <  \mu < 90  + 2.052`

=>
`87.95   <  \mu < 92.05`