A 95 percent confidence interval of 14 to 17 means which of the following: that 95 percent of intervals calculated like this one will contain the population mean, that 95 percent of the time the sample mean will be between 14 and 17, that 95 percent of the population means lie between 14 and 17, or all of these?

Step-by-step solution

1. 1

   Remember what is random and what is fixed. The population mean $\mu$ is a fixed, unknown constant. The interval is what varies: draw a new sample and you get new endpoints. Every correct interpretation has to put the randomness on the interval, not on $\mu$.

2. 2

   Evaluate the first statement. '95% of the confidence intervals calculated like this one will contain $\mu$' places the randomness on the intervals and describes the long-run behaviour of the procedure. That is precisely the definition of a $95\%$ confidence level, so this statement is correct.

3. 3

   Reject the statement about the sample mean. '$95\%$ of the time $\bar{X}$ will be between $14$ and $17$' confuses the confidence interval with a prediction interval for future sample means. In fact $\bar{X}$ sits at the centre of this particular interval by construction; a fresh sample's mean has no guaranteed $95\%$ chance of landing inside these fixed endpoints.

4. 4

   Reject the statement about many population means. '$95\%$ of the $\mu$'s will be between $14$ and $17$' is doubly wrong: there is only one $\mu$, and it is not a random quantity. Saying 'there is a 95% probability that $\mu$ is between 14 and 17' is the same error dressed differently — once the data are in, $\mu$ either is or is not in $[14,17]$, and the probability is $0$ or $1$, we just do not know which.

5. 5

   Reject 'all of the other answers'. Since two of the three statements are false, this option fails too.

6. 6

   State the answer. Only the first statement is correct: 95% of the confidence intervals calculated like this one will contain $\mu$. The confidence level is a property of the method across repeated sampling, not of the single interval in front of you.