Question

21 multiple choice 4 points
select the most appropriate response.
data collected by child development scientists produced the following 90% confidence interval for the average age (in months) at which children say their first word: 10.4 < μ < 13.8
in this example, what does the phrase 90% confident mean?
there is a 90% chance that the sample mean is between 10.4 and 13.8 months
if many samples are drawn, the computed confidence intervals will contain the population mean approximately 90% of the time
there is a 90% chance that the population mean is between 10.4 and 13.8 months
if many samples are drawn, the population mean will be between 10.4 and 13.8 months approximately 90% of the time.

Explanation:

• Confidence level in statistics refers to the long - run frequency.
• A 90% confidence interval means that if we were to take many samples and construct confidence intervals in the same way, approximately 90% of those intervals would contain the population mean.
• The sample mean is a single value (the center of the interval in the formula for a confidence interval \(\bar{x}\pm z\frac{\sigma}{\sqrt{n}}\)), and we are not talking about the probability of the sample mean being in the interval.
• The population mean is a fixed value. We don't say there is a 90% chance that the population mean is in the interval (frequentist interpretation). It either is or isn't. The confidence level is about the method of constructing intervals.

Answer:

If many samples are drawn, the computed confidence intervals will contain the population mean approximately 90% of the time.