Question

Acrylic bone cement is sometimes used in hip and knee replacements to secure an artificial joint in place. The force required to break an acrylic bone cement bond was measured for seven specimens, and the resulting mean and standard deviation were 306.19 newtons and 41.92 newtons, respectively. Assuming that it is reasonable to believe that breaking force has a distribution that is approximately normal, use a 95% confidence interval to estimate the mean breaking force for acrylic bone cement. (Round your answers to three decimal places.)

Answer 1

The 95% confidence interval for the mean breaking force for acrylic bone cement, based on the provided sample, is approximately 267.677 to 344.703 newtons.

Step-by-step explanation:

To estimate the mean breaking force for acrylic bone cement using a 95% confidence interval, we will apply the concepts of statistics, particularly those related to the normal distribution and confidence intervals. Given that the sample mean is 306.19 newtons and the sample standard deviation is 41.92 newtons for seven specimens, we can use the t-distribution because the sample size is less than 30. The formula for a 95% confidence interval is:

Mean ± (t-value * (Standard Deviation / sqrt(n))),

where n is the sample size, and the t-value is based on the t-distribution with n-1 degrees of freedom. Looking up the t-value for 95% confidence and 6 degrees of freedom, we find it is approximately 2.447. Plugging in the numbers:

306.19 ± (2.447 * (41.92 / sqrt(7)))

This results in a range of:

306.19 ± (2.447 * (41.92 / 2.6457))

306.19 ± (38.513)

Thus, the 95% confidence interval for the mean breaking force is approximately:

267.677 to 344.703 newtons (rounded to three decimal places).

Our confidence interval suggests that we can be 95% confident that the true mean breaking force for acrylic bone cement lies within this range.