Suppose (35,40) is a 95% confidence interval estimate for a population mean 𝜇. Which of the following are true statements?I. There is a .95 probability that 𝜇 is between 35 and 40.II. If 100 random samples of the given size are picked and a 95% confidence interval is calculated from each, then 𝜇 will be in 95 of the resulting intervals.III. If 95% confidence intervals are calculated from all possible samples of the given size, 𝜇 will be in 95% of these intervals.I and III and IIIII and IIII, II, and IIINone of the above gives the complete set of true responses.
Question
Suppose (35,40) is a 95% confidence interval estimate for a population mean 𝜇. Which of the following are true statements?I. There is a .95 probability that 𝜇 is between 35 and 40.II. If 100 random samples of the given size are picked and a 95% confidence interval is calculated from each, then 𝜇 will be in 95 of the resulting intervals.III. If 95% confidence intervals are calculated from all possible samples of the given size, 𝜇 will be in 95% of these intervals.I and III and IIIII and IIII, II, and IIINone of the above gives the complete set of true responses.
Solution
Let's analyze each statement step by step:
I. There is a .95 probability that 𝜇 is between 35 and 40.
- This statement is not true. The correct interpretation of a 95% confidence interval is that if we were to take many samples and build a confidence interval from each sample, we would expect 95% of those intervals to contain the true population mean 𝜇. It does not mean that there is a 95% probability that 𝜇 lies within any one specific interval.
II. If 100 random samples of the given size are picked and a 95% confidence interval is calculated from each, then 𝜇 will be in 95 of the resulting intervals.
- This statement is true. By definition, a 95% confidence interval means that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect about 95 of those intervals to contain the true population mean 𝜇.
III. If 95% confidence intervals are calculated from all possible samples of the given size, 𝜇 will be in 95% of these intervals.
- This statement is true. This is essentially the definition of a 95% confidence interval. If we could calculate the confidence interval for every possible sample of the given size, 95% of those intervals would contain the true population mean 𝜇.
Based on the analysis:
- Statement I is false.
- Statement II is true.
- Statement III is true.
Therefore, the correct answer is:
II and III
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