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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 25 pieces of the imported wood were tested and yielded the statistics x-75.2 and s 8.6 on breaking strength (MPa). Estimate the true mean breaking strength of the white wood with a 95% confidence interval. Interpret the result. Find the 95% confidence interval for the true mean breaking strength of the white wood (Round to two decimal places as needed.) Interpret this confidence interval. Choose the correct answer below 0 A. The architects can be 5% confident that the mean breaking strength of the white wood is 75.2 O B. The architects can be 95% confident that the mean breaking strength of the white wood is within this interval 0 C. The architects are confident that 95% of the white wood is described by this interval 0 D. The architects can be 97.5% confident that the mean breaking strength of the white wood is outside this interval 0 E. The architects can be 5% confident that the mean breaking strength of the white wood is within this interval

Answer :

95% confidence interval for the true mean breaking strength of the white wood is (71.66 MPA, 78.74 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.

  • Confidence Interval:

The definition says, "A confidence interval is a range of values ​​derived from a sample statistic that likely includes the value of an unknown population parameter."

To calculate the confidence interval, we need to know three parameters of the sample: the mean (average value) μ, the standard deviation σ, and the sample size n (number of measurements). Then you can calculate standard error and tolerance using the following formulas:

                      standard error = σ/√n

Given in the question:

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

Mean = 75.2 MPA

Standard deviation = 8.6 MPA

Sample size, n = 25

Now,

Degree of freedom = n - 1 = 25 - 1 = 24

Confidence level (C) = 95% = 0.95

Significance level = 1 - C

                              = 1 - 0.95

                              = 0.05 = 5%

Therefore, t-value corresponding to 24 degrees of freedom and 5% significance level is 2.06

Again,

E = t × sd/√n

  = 2.06 × 8.6/√25

  = 2.06 × 1.72

  = 3.54 MPA

Lower limit = Mean - E

                  = 75.7 - 5.81

                  = 69.89 MPA

Upper limit = Mean + E

                   = 75.7 + 5.8

                   = 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.

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