When making a statistical inference about the mean of a normally distributed population based on a sample drawn from that population, which of the following statements is correct, all else being equal?
The 68% confidence interval is wider than the 90% confidence interval.
The 90% confidence interval is narrower than the 95% confidence interval.
The 90% confidence interval is wider than the 99% confidence interval.
The 99% confidence interval is narrower than the 68% confidence interval.
To answer this question, we need to understand the concept of confidence intervals in statistical inference.
A confidence interval is a range of values within which we expect the true population parameter to fall with a certain level of confidence. The level of confidence is determined by the chosen confidence level, typically expressed as a percentage.
In general, the wider the confidence level, the more likely it is to capture the true population parameter, and the narrower the confidence interval, the more precise the estimate is.
Now, let's analyze the given statements:
1. The 68% confidence interval is wider than the 90% confidence interval.
This statement is incorrect because a higher confidence level results in a wider interval. Thus, the 90% confidence interval is wider than the 68% confidence interval.
2. The 90% confidence interval is narrower than the 95% confidence interval.
This statement is correct. A higher confidence level requires a wider interval, so the 90% confidence interval is narrower than the 95% confidence interval.
3. The 90% confidence interval is wider than the 99% confidence interval.
This statement is incorrect because a higher confidence level requires a wider interval. Therefore, the 90% confidence interval is narrower than the 99% confidence interval.
4. The 99% confidence interval is narrower than the 68% confidence interval.
This statement is correct. A higher confidence level necessitates a wider interval, so the 99% confidence interval is narrower than the 68% confidence interval.
In summary, the correct statement among the given options is:
The 90% confidence interval is narrower than the 95% confidence interval.
The correct statement is:
The 99% confidence interval is narrower than the 68% confidence interval.
In general, as the confidence level increases, the width of the confidence interval increases. This is because a higher confidence level means we want to be more certain that the interval contains the true population mean. Therefore, the 99% confidence interval, which provides a higher level of confidence, will be narrower than the 68% confidence interval.
the wider the interval, the higher the confidence
look at the extreme
... if the sample was the entire population
... you would be 100% confident that the mean matched the population