Question

In a 21 element sample, the sample mean is equal tox-bar=10; the standard error of the mean is equal to 5. If the two-tailed combined proportions for the 90% confidence interval for the true mean is either t0.10,20 = 1.725 or t0.05,20 = 2.086, what are the limits of the confidence interval?

Answer 1

Answer: (1.375, 18.625 )

Explanation:

The confidence interval for population mean is given by :-

`\overline{x}\pm t^*SE` (1)

, where
`\overline{x}` = sample mean

t* = critical value.

SE = standard error

As per given , we have

n= 21

Degree of freedom : df = n-1=21-1=20

`\overline{x}=10`

Significance level :
`\alpha=1-0.90=0.10`

Standard error : SE= 5

The critical t-value for confidence interval is a two-tailed value with respect to the given degree of freedom i.e.
`t^*=t_((\alpha/2,df))` .

Using student's t-distribution table ,

`t_((\alpha/2,df))=t_((0.10/2, 20))=t_((0.05, 20))=\pm1.725`

The 90% confidence interval will be :

`10\pm (1.725) (5)` [by formula in (1)]

`10\pm 8.625`

`\approx(10- 8.625,\ 10+ 8.625)=(1.375,\ 18.625 )`

Hence, the limits of the confidence interval=(1.375, 18.625 )