17. Explaining confidence A $95\%$confidence interval for the mean body mass index $\left(BMI\right)$of young American women is $26.8\pm 0.6$. Discuss whether each of the following explanations is correct.
(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 $26.2$to $27.4$is believable as the true mean $BMI$ of young American women.
(d) In $95\%$of all possible samples, the population mean $BMI$will be between $26.2$and $27.4$.
(e) The mean $BMI$of young American women cannot be $28$.

To determine whether the statement that $95\%$of all young women have $BMI$between$26.2$and $27.4$is correct or not.

Let, the Confidence interval is $95\%$
And the body mass index as:

$\left(BMI\right)=26\pm 0.6$
Also, $26.2$and $27.4$
The confidence interval is only stated for young women in this observation, not for individual women.
As a result, explanation (a) is incorrect.

To determine whether the statement that $95\%$confident that future samples of young women will have mean $BMI$between $26.2$and $27.4$is correct or not.

Let, the confidence interval is $95\%$.

And the body mass index is $\left(\mathrm{BMI}\right)=26\pm 0.6$
Also,

$26.2$ and $27.4$.
Because the confidence intervals for future samples may differ.
It's impossible to predict where the true mean $BMI$will fall between $26.2$ and $27.4$.
As a result, explanation (b) is incorrect

To determine whether the statement that any value from $26.2$to $27.4$is believable as the true mean $BMI$of young American women is correct or not.

Let the confidence interval is $95\%$.

And the body mass index is $\left(BMI\right)=26\pm 0.6$
Also,$26.2$and $27.4$.
The true mean $BMI$ranges from $26.2$to $27.4$.
As a result, any value between $26.2$and $27.4$might be considered reasonable as the genuine mean $BMI$.
As a result, the explanation is correct.

To determine whether the statement that in $95\%$of all possible samples, the population mean $BMI$will be between $26.2$and $27.4$is correct or not.

Let, the confidence interval is$95\%$.

And the body mass index is $\left(\mathrm{BMI}\right)=26\pm 0.6$
Also,

$26.2$and $27.4$.

The true mean $BMI$lies between the intervals $26.2$and $27.4$in the $95$ percent confidence interval.
The genuine mean $BMI$is not present in the remaining $5$ percent samples.
As a result, explanation is incorrect.

To determine whether the statement that the mean $BMI$of young American women cannot be $28$is correct or not.

Let, the confidence interval is $95\%$.

And the Body mass index is $\left(\mathrm{BMI}\right)=26\pm 0.6$
$26.2$and $27.4$
The population mean is unknown in this explanation. As a result, the average $BMI$ may not be equal to $28$.
As a result, explanation (e) is incorrect.