Question

Explaining confidence A $95\%$confidence interval for the mean body mass index (BMI) of young American women is $26.8\pm 0.6$. Discuss whether each of the following explanations is correct, based on that information.

a. We are confident that $95\%$of all young women have BMI between $26.2$and $27.4$.

b. We are $95\%$confident that future samples of young women will have mean BMI

between $26.2$and$27.4$.

c. Any value from $26.2\mathrm{to}27.4$is believable as the true mean BMI of young American women.

d. If we take many samples, the population mean BMI will be between $26.2$and $27.4$in about $95\%$of those samples.

e. The mean BMI of young American women cannot be $28$.

Short answer

a. Statement a is incorrect.

b. Statement b is incorrect.

c. Statement c is correct.

d. Statement d is incorrect.

e. Statement e is incorrect.

Step-by-step solution

Given Information

It is given that $95\%$confidence interval for mean body index is$(26.2,27.4)$

To Explain statement a is correct or not on the basis of given information.

From given data, it is incorrect because confidence interval is about BMI of young women, not about BMI of individual women.

To Explain statement b is correct or not on the basis of given information.

The statement is incorrect because it is not confirmed that true population mean is between $26.2\mathrm{and}27.4$.

The mean may or may not lie in confidence interval for further samples.

The statement is incorrect.

To Explain statement c is correct or not on the basis of given information.

The statement is correct because there is $95\%$confidence that population mean lie between $26.2\text{and}27.4$.

The exact value is not known, so it can be any value between $26.2\text{and}27.4$

The statement is correct.

To Explain statement d is correct or not on the basis of given information.

There will be no change in population mean. I will be value between $26.2\text{and}27.4$.

That is, $100\%$or $0\%$of time.

So, the statement is incorrect.

To Explain statement e is correct or not on the basis of given information. 

It is incorrect as there are $95\%$of samples has $95\%$confidence interval having true population mean.

So,$(26.2,27.4)$may be among remaining$5\%$samples not having true population mean.