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 )

Step-by-step 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 )