Question

The white wood material used for the roof of an ancient temple is imported from a certain country. The wooden roof must withstand as much as 100 centimeters of snow in the winter. Architects at a university conducted a study to estimate the mean bending strength of the white wood roof. A sample of 16 pieces of the imported wood were tested and yielded the statistics x overbar equals 75.7 and s equals 10.9 on breaking strength​ (MPa). Estimate the true mean breaking strength of the white wood with a 95 % confidence interval. Interpret the result.

Answer 1

95% confidence interval for the true mean breaking strength of the white wood is (69.89 MPa, 81.51 MPa)

The result above means that the lower limit of the true mean breaking strength of the white wood is 69.89 MPa and the upper limit is 81.51 MPa.

Explanation:

Confidence interval = mean + or - Error margin (E)

mean = 75.7 MPa

sd = 10.9 MPa

n = 16

degree of freedom = n - 1 = 16 - 1 = 15

Confidence level (C) = 95% = 0.95

Significance level = 1 - C = 1 - 0.95 = 0.05 = 5%

t-value corresponding to 15 degrees of freedom and 5% significance level is 2.131

E = t × sd/√n = 2.131 × 10.9/√16 = 5.81 MPa

Lower limit = mean - E = 75.7 - 5.81 = 69.89 MPa

Upper limit = mean + E = 75.7 + 5.81 = 81.51 MPa

95% confidence interval is (69.89 MPa, 81.51 MPa)

The result means that the true mean breaking strength of the white wood is between 69.89 and 81.51 MPa.

Answer 2

The architects can be 95% confident that the mean breaking strength of the white wood is within this interval (149.736 , 1.664)

Explanation:

Given

Sample = n = 16

x = 75.7

s = 10.9

First we Calculate the degree of freedom.

Degree of freedom is calculated as

df = (n - 1)t

df = (16 - 1)t

df = 15t

The we check the t-table corresponding value for 0.05

t 0.05= 1.75377.

0.05 is gotten from

(100% - 95%) = 5%= 5/100 = 0.05

Calculating the True mean breaking strength through the confidence interval

Solving the confidence interval.

This is given by:

x ± ts√df

Confidence Interval = 75.7 ± 1.75377 * 10.9 * √15

Confidence Interval = 75.7 ± 74.036

Confidence Interval = 75.7 + 74.036 , 75.7 - 74.036

Confidence interval = 149.736 , 1.664

Interpreting the above;

The architects can be 95% confident that the mean breaking strength of the white wood is within this interval (149.736 , 1.664)