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 six specimens, and the resulting
mean and standard deviation were 306.11 newtons and 41.96 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 (in newtons). (Enter your
answer using interval notation. Round your numerical values to three decimal places.)
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We are given the mean breaking force as 306.11 newtons and the standard deviation as 41.96 newtons. We need to estimate the mean breaking force for acrylic bone cement with a 95% confidence interval.
To find the confidence interval, we can use the formula:
Confidence interval = sample mean ± (critical value * standard error)
Since the sample size is small (n = 6) and we are assuming a normal distribution, we need to use the t-distribution and its corresponding critical value. For a 95% confidence level and a sample size of 6, the critical value is 2.571.
The standard error is calculated by dividing the standard deviation by the square root of the sample size:
Standard error = 41.96 / sqrt(6)
Calculating the standard error gives us:
Standard error ≈ 17.165
Now we can calculate the confidence interval:
Lower bound = 306.11 - (2.571 * 17.165)
Upper bound = 306.11 + (2.571 * 17.165)
Calculating the confidence interval gives us:
Lower bound ≈ 264.392
Upper bound ≈ 347.828
Therefore, the estimated mean breaking force for acrylic bone cement is between approximately 264.392 newtons and 347.828 newtons, with a 95% confidence.
The 95% confidence interval can be written as (264.392, 347.828).
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