Explaining confidence. A student reads that a 95% confidence interval for the mean ideal weight given by adult American women is 140 ± 1.4 pounds. Asked to explain the meaning of this interval, the student says, “95% of all adult American women would say that their ideal weight is between 138.6 and 141.4 pounds.” Is the student right? Explain your answer.

We have a population of all adult women. We take a random sample from this population and create a confidence interval. If we take many, many, many different samples from this population we will obtain many, many, many different sample means and confidence intervals created using the data from our samples. If we created many, many, many different confidence intervals in this manner 95% of them would capture the true mean weight of the population of all women.

No, the student's explanation is not entirely correct. The 95% confidence interval does not provide information about what individual women would say about their ideal weight. Instead, it tells us that if we were to take many samples of adult American women and calculate their mean ideal weight, 95% of those sample means would fall within the range of 138.6 to 141.4 pounds.

In other words, the interval 140 ± 1.4 pounds means that we are 95% confident that the true mean ideal weight of adult American women falls within this range. It does not make a statement about the individual responses of women or their actual ideal weights, but rather provides an estimate of the range within which the population mean is likely to fall.

To determine if the student is correct, we need to understand the concept of a confidence interval.

A confidence interval is a range of values that is used to estimate an unknown population parameter, such as the mean ideal weight of adult American women. In this case, the 95% confidence interval suggests that if we were to repeat the study many times and construct a confidence interval each time, approximately 95% of these intervals would contain the true population mean.

The student's response, "95% of all adult American women would say that their ideal weight is between 138.6 and 141.4 pounds," is not entirely accurate.

The confidence interval of 140 ± 1.4 pounds indicates that the estimated mean ideal weight for adult American women falls between 138.6 and 141.4 pounds. However, it does not mean that individual women will specifically state their ideal weight falls within this range.

The correct interpretation of the confidence interval is that we are 95% confident that the true mean ideal weight of adult American women lies within the range of 138.6 to 141.4 pounds. This means that if we were to repeat the study many times, we would expect our calculated intervals to capture the true population mean 95% of the time.

So, overall, the student's interpretation is not accurate. The confidence interval provides information about our level of confidence in estimating the true population mean, rather than the specific ideal weight of individual women.