# Interpreting confidence intervals

The weight of an SRS of 35 pumpkins is measured. A 95% confidence interval for the mean weight, in pounds, for the pumpkins on this farm is (9,22). Which of the following statements gives a valid interpretation of this interval?

a. 95% of the sample of pumpkins have a weight, in pounds, between 9 and 22.
b. We can be 95% confident that the true mean weight of the pumpkins on the family farm is between 9 and 22 pounds.
c. If the procedure were repeated many times, 95% of the sample means would be between 9 and 22.


I understand that the correct interpretation would be that there is a 95% confindence that the interval (9,22) contains the population parameter; however, none of the choices seem to match this.

Also, is it correct to interpret a 90% confidence interval of, for example, $1\pm0.2$, as saying that any value between $(0.8,1.2)$ is believable as the true mean of the population?

• The answer is b. But what does this really mean? It means that if you were to draw samples and repeatedly construct 95% confidence intervals over and over again for each sample, 95% of those many confidence intervals would contain the population mean. In other words: if you drew 1,000,000 random samples and constructed a confidence interval based on the data in each sample, about 950,000, or 95%, of them would contain the true mean. – symplectomorphic Mar 18 '16 at 23:47

B is the only one that can be considered correct. Your confusion seems to be in not realizing that the "population parameter" here is "the true mean weight of the pumpkins on the family farm". So your understanding is consistent with B.

But both statements depend on what you mean by "95% confident that the true weight is between 9 and 22". The only reasonable definition of "confident" here is that $[9,22]$ is a 95% confidence interval for the true weight, so this is a tautology.

In your second statement, it again depend on your definition of "believable". If you define a believable value to be one you are confident in, then your statement is fine, excepting that you totally lose the quantified part of the statement. You would have the same definition for 95% and 90% confidence intervals, even though there are different degrees of "believability" here. The danger is if you start to interpret the word "believable" in the wrong way. For instance, if you start to think the "believability" in a value within a 90% confidence interval is that there is a 90% chance for the true mean to be in that range, then you are incorrectly interpreting things.