Question

A sample of 13 joint specimens of a particular type gave a sample mean proportional limit stress of 8.59MPa and a sample standard deviation of 0.74 MPa. (a) Calculate and interpret a 95% lower confidence bound for the true average proportional limit stress of all such joints. (Round your answer to two decimal places.) MPa Interpret this bound. With 95% confidence, we can say that the value of the true mean proportional limit stress of all such joints is centered around this value. With 95% confidence, we can say that the value of the true mean proportional limit stress of all such joints is greater than this value. With 95% confidence, we can say that the value of the true mean proportional limit stress of all such joints is less than this value. What, if any, assumptions did you make about the distribution of proportional limit stress? We do not need to make any assumptions. We must assume that the sample observations were taken from a uniformly distributed population. We must assume that the sample observations were taken from a normally distributed population. We must assume that the sample observations were taken from a chi-square distributed population. MPa Interpret this bound. You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

To calculate the 95% lower confidence bound for the true average proportional limit stress of all joints, subtract (1.96 * (sample standard deviation / square root of sample size)) from the sample mean. The lower bound is 8.22 MPa. This means, with 95% confidence, the true mean proportional limit stress of all such joints is greater than 8.22 MPa.

Step-by-step explanation:

To calculate the 95% lower confidence bound for the true average proportional limit stress of all joints, we can use the formula:

Lower Bound = Sample Mean - (Z * (Sample Standard Deviation / Square Root of Sample Size))

Substituting the given values:

Lower Bound = 8.59 - (1.96 * (0.74 / sqrt(13)))

Calculating this equation gives a lower bound of 8.22 MPa. Interpretation: With 95% confidence, we can say that the value of the true mean proportional limit stress of all such joints is greater than 8.22 MPa.

The assumption made about the distribution of proportional limit stress is that the sample observations were taken from a normally distributed population.