A sample of 106 golfers showed that their average score on a particular golf course was 87.98 with a standard deviation of 5.39.
Answer each of the following (show all work and state the final answer to at least two decimal places.):
(A) Find the 90% confidence interval of the mean score for all 106 golfers.
(B) Find the 90% confidence interval of the mean score for all golfers if this is a sample of 130 golfers instead of a sample of 106.
(C) Which confidence interval is larger and why?
90% = mean ± 1.645 SEm
Use same process indicated in your first post.
(A) To find the 90% confidence interval of the mean score for all 106 golfers, we can use the formula:
Confidence Interval = Sample Mean ± (Z * Standard Error)
where Z is the Z-score corresponding to the desired confidence level, and the Standard Error is calculated as:
Standard Error = Standard Deviation / √(Sample Size)
First, we need to find the Z-score for a 90% confidence level. The Z-score corresponds to the area under the normal distribution curve. Since we want to find the area in the middle (90% confidence interval), we have 5% on each tail, so the remaining 90% is in the middle. Therefore, the Z-score for a 90% confidence level is 1.645.
Now we can find the Standard Error:
Standard Error = 5.39 / √(106) ≈ 0.524
Finally, we can calculate the confidence interval:
Confidence Interval = 87.98 ± (1.645 * 0.524) ≈ 87.98 ± 0.861
Therefore, the 90% confidence interval for the mean score for all 106 golfers is approximately (87.119, 88.841).
(B) To find the 90% confidence interval of the mean score for all golfers if this is a sample of 130 instead of 106, we need to recalculate the Standard Error using the new sample size and repeat the same steps as in part (A).
Standard Error = 5.39 / √(130) ≈ 0.472
Confidence Interval = 87.98 ± (1.645 * 0.472) ≈ 87.98 ± 0.777
Therefore, the 90% confidence interval for the mean score for all 130 golfers is approximately (87.203, 88.757).
(C) The confidence interval in part (B) is smaller than the confidence interval in part (A). This is because as the sample size increases, the Standard Error decreases. A smaller Standard Error means that we have more precise estimates of the population mean, thus resulting in a narrower confidence interval.