Question

The diameters of steel shafts produced by a certain manufacturing process should have a mean diameter of 5.5 inches. The diameter is known to have a standard deviation of 0.9 inches. A random sample of 30 shafts.

(a) Find a 90% confidence interval for the mean diameter to process.

(b) Find a 99% confidence interval for the mean diameter to process.

(c) How does the increasing and decreasing of the significance level affect the confidence interval? Why?

Please explain and show your work. Thanks.

Answer 1

Solution :

Given that,

= 5.5

s = 0.9

n = 30

Degrees of freedom = df = n - 1 = 30 - 1 = 29

a ) At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

t _/2,df = t_0.05,29 =1.699

Margin of error = E = t_/2,df * (s /n)

= 1.699 * (0.9 / 30)

= 0.3

Margin of error = 0.3

The 90% confidence interval estimate of the population mean is,

- E < < + E

5.5 - 0.3 < < 5.5 + 0.3

5.2 < < 5.8

(5.2, 5.8 )

b ) At 99% confidence level the t is ,

= 1 - 99% = 1 - 0.99 = 0.01

/ 2 = 0.01 / 2 = 0.005

t _/2,df = t_0.005,24 =2.756

Margin of error = E = t_/2,df * (s /n)

= 2.756 * (0.9 / 30)

= 0.4

Margin of error = 0.4

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

- E < < + E

5.5 - 0.4 < < 5.5 + 0.4

5.1 < < 5.9

(5.1, 5.9)

c ) The increasing of the significance level affect the confidence interval increasing

because the margin of error increasing.