Question

Consider a machine that produces metal pieces which are cylindrical in shape. We select
a sample of nine pieces and measure the diameters:
1.01; 0.97; 1.03; 1.04; 0.99; 0.98; 0.99; 1.01; 1.03
The sample and sample standard deviation are x̄ = 1.00556 and s = 0.02455, respectively.
Give a 95% confidence interval for the true mean diameter, assume that the
population is normal.

Answer 1

Solution :

Given that,

$\bar x$ =1.00556

s = 0.02455

n = 9

Degrees of freedom = df = n - 1 = 9 - 1 = 8

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_0.025,8 =2.306

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

= 2.306 * (0.02455 / $\sqrt$ 9)

= 0.01887

Margin of error = 0.01887

The 99% confidence interval estimate of the population mean is,

$\bar x$ - E <$\mu$  < $\bar x$ + E

1.00556 - 0.01887 < $\mu$ <1 .00556 + 0.01887

0.9867 < $\mu$ < 1.0244

(2.70, 3.70 )