Question

8. A random sample of size 100 from a population has a mean I = 3.87 and standard deviation s = 1.25. Construct a 95% confidence interval for the population mean u.

Answer 1

Solution :

Given that,

Point estimate = sample mean = $\bar x$ = 3.87

sample standard deviation = s = 1.25

sample size = n = 100

Degrees of freedom = df = n - 1 = 100 - 1 = 99

At 95% confidence level the t is ,

$\alpha$ = 1 - 95% = 1 - 0.95 = 0.05

$\alpha$ / 2 = 0.05 / 2 = 0.025

t_{$\alpha$ /2,df} = t_0.025,99 = 1.984

Margin of error = E = t_{$\alpha$/2,df} * (s /$\sqrt$n)

= 1.984 * (1.25 / $\sqrt$100)

= 0.25

The 95% confidence interval estimate of the population mean is,

$\bar x$ - E < $\mu$ < $\bar x$ + E

3.87 - 0.25 < $\mu$ < 3.87 + 0.25

3.62 < $\mu$ < 4.12

(3.62 , 4.12)