All right, let's tackle these questions one by one.
a) Calculate a 95% confidence interval to estimate the true mean BMI of young American women. Show your work.
To calculate a 95% confidence interval given the sample mean and margin of error, we use the formula:
\(\[ \text{Confidence Interval} = \text{Sample Mean} \pm \text{Margin of Error} \]\)
Given that the sample mean is 26.8 and the margin of error is 0.6, the confidence interval is computed as follows:
\(\[ \text{Lower Bound} = 26.8 - 0.6 = 26.2 \]\)
\(\[ \text{Upper Bound} = 26.8 + 0.6 = 27.4 \]\)
So the 95% confidence interval is (26.2, 27.4).
b) What was the sample mean (point estimate) and margin of error used to create this confidence interval?
The sample mean, which is the point estimate of the true mean BMI, is 26.8. The margin of error used to create the confidence interval is 0.6.
c) Interpret your 95% confidence interval from part a).
The interpretation of the 95% confidence interval is that we are 95% confident that the true mean BMI of young American women lies between 26.2 and 27.4. This means that if we were to take many samples and construct a confidence interval in the same way from each sample, we would expect about 95% of those confidence intervals to contain the true mean BMI.
d) A doctor's office claims the BMI should be 24. Based on the interval in part a) is there convincing evidence his claim is incorrect? Explain.
The doctor's claim is that the BMI should be 24, which falls outside of our 95% confidence interval of (26.2, 27.4). Since 24 is not within this range, the data provides convincing evidence that the claim made by the doctor's office is inconsistent with the sample evidence suggesting the true mean BMI of young American women. Thus, based on this confidence interval, there is convincing evidence that the doctor's claim is incorrect because the entire interval is above the claimed value of 24.
