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Chapter 9: Problem 44

Acrylic bone cement is sometimes used in hip and knee replacements to fix an artificial joint in place. The force required to break an acrylic bone cement bond was measured for six specimens under specified conditions, and the resulting mean and standard deviation were \(306.09\) Newtons and \(41.97\) Newtons, respectively. Assuming that it is reasonable to believe that breaking force under these conditions has a distribution that is approximately normal, estimate the mean breaking force for acrylic bone cement under the specified conditions using a \(95 \%\) confidence interval.

Short Answer

Expert verified
The 95% confidence interval for the mean breaking force for acrylic bone cement under the specified conditions is estimated to be between the calculated lower and upper bounds from Step 3.

Step by step solution

01

Calculate the Standard Error

The standard error (SE) of the mean is calculated by dividing the standard deviation by the square root of the sample size. Here, the standard deviation is \(41.97\) Newtons and the sample size is \(6\). So, the standard error is computed as \(SE = \frac{{41.97}}{{\sqrt{6}}}\).
02

Determine the z-value for 95% confidence

The z-value for a 95% confidence interval is about \(1.960\), based on standard normal distribution tables. This value represents the number of standard errors to add and subtract from the mean to obtain the confidence interval.
03

Calculate the Confidence Interval

The confidence interval is calculated by subtracting and adding the product of the z-value and the standard error from the mean. The lower and upper bounds of the interval are given by \(\mu - z \cdot SE\) and \(\mu + z \cdot SE\), respectively. Here, \(\mu = 306.09\) Newtons and \(z \cdot SE = 1.960 \cdot \frac{41.97}{\sqrt{6}}\) Newtons. Calculating gives the confidence interval.

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