Question

In a manufacturing process a random sample of 9 bolts manufactured has a mean length of 3 inches with a standard deviation of .3 inches. What is the 95% confidence interval for the true mean length of the bolt?
A. 2.804 to 3.196
B. 2.308 to 3.692
C. 2.769 to 3.231
D. 2.412 to 3.588
E. 2.814 to 3.186

Answer 1

Answer:

The correct option is (C) (2.769, 3.231).

Step-by-step explanation:

The confidence interval for mean when the standard deviation is not known is:

$CI=\bar x\pm t_{\alpha/2, (n-1)}\frac{s}{\sqrt{n}}$

Given:

$\bar x = 3\\s=0.3\\n=9\\\alpha =1-0.95=0.05$

Compute the critical value as follows:

$t_{\alpha/2, (n-1)}=t_{0.05/2, (9-1)}=t_{0.025, 8}=2.31$

**Use a t-table.

The 95% confidence interval for true mean length of the bolt is:

$CI=\bar x\pm t_{\alpha/2, (n-1)}\frac{s}{\sqrt{n}}\\=3\pm 2.31\times \frac{0.30}{\sqrt{9}}\\ =3\pm 0.231\\=(2.769, 3.231)$

Thus, the 95% confidence interval for true mean length of the bolt is (2.769, 3.231).

The correct option is (C).