Answer to Question #235508 in Statistics and Probability for blossomqts

Question

1.    The diameter of holes for a cable harness is known to have a normal distribution with a standard deviation of 0.01 inch. A random sample of size 10 yields an average diameter of 1.5045 inch. 

a.    Find a 99% two-sided confidence interval on the mean hole diameter.

b.    Find a 90% two-sided confidence interval on the mean hole diameter.

c.     Explain the difference of the interval between the two confidence levels.

Answer 1

$n=10 \\ M = 1.5045 \\ \sigma=0.01 \\ CI = (M - \frac{Z_c \times \sigma}{\sqrt{n}}, M + \frac{Z_c \times \sigma}{\sqrt{n}})$

a.

$Z_c = 2.576 \\ CI = (1.5045 - \frac{2.576 \times 0.01}{\sqrt{10}}, 1.5045 + \frac{2.576 \times 0.01}{\sqrt{10}}) \\ =(1.5045 - 0.008146, 1.5045 + 0.008146) \\ =(1.496354, 1.512746)$

b.

$Z_c = 1.645 \\ CI = (1.5045 - \frac{1.645 \times 0.01}{\sqrt{10}}, 1.5045 + \frac{1.645 \times 0.01}{\sqrt{10}}) \\ =(1.5045 - 0.005202, 1.5045 + 0.005202) \\ =(1.499298, 1.509702)$

c. A 99 percent confidence interval is wider than a 90 percent confidence interval because to be more confident that the true population value falls within the interval we will need to allow more potential values within the interval.