Overview
Reply to two peers resonses.
My response:
Confidence Interval Calculation
The Z-distribution is suitable in this case since the population standard deviation (σ=100 days) is known and the sample size is relatively large (n = 45), and therefore the sampling distribution of the sample mean is approximately normal according to the Central Limit Theorem.
To calculate the 90% CI:
Standard Error (SE) = = = ≈14.905
Critical z for 90% confidence = 1.645
Margin of error = 1.645 × 14.905 = ≈24.51
Confidence interval = 285 ± 24.51 = (260.49, 309.51) days
The mean of the ACL program (320 days) is greater than the upper limit (309.51) of the 90 percent interval, thus, 320 does not fall within our interval. Since the hypothesized 320 is not within the 90% confidence interval, we have evidence (at the 10% significance level of a two-sided test) that the true mean of the new program is not equal to 320, that is, it is less, which is an indication of improvement. To provide additional background, a 95% CI (z = 1.96) would be 285+-29.22=255.78, 314.22, and this would also rule out 320; again, this would support the conclusion that the new program is likely to lead to a much shorter recovery time than 320 days.
References
The Organic Chemistry Tutor. (2019). How To Find The Z Score, Confidence Interval, and Margin of Error for a Population Mean.
https://www.youtube.com/watch?v=DT-fPG0Hff8Links to an external site.
Peer #1:
1-The Z-distribution will be used to create the confidence interval for the mean. Why can we use this distribution for this problem?
For this problem, we can apply the Z-distribution because the population standard deviation is known (100 days), which is essential for using a Z-based confidence interval. Additionally, our sample of 45 patients is sufficiently large for the Central Limit Theorem to ensure that the sampling distribution of the means is roughly normal, even if the actual recovery times are not perfectly normally distributed.
2. Create your confidence interval and report its value.
Using a sample mean of 285 days, σ = 100, and n = 45, I found the standard error to be about 14.91 days. For a 90% confidence level, the critical z-value is 1.645, resulting in a margin of error of approximately 24.5 days. Therefore, the 90% confidence interval for the true mean recovery time with the new program is roughly (260.5 days, 309.5 days).
3-The current ACL recovery program averages 320 days to fully recover. Based on the confidence interval you constructed, where does this value fall?
The current ACL recovery time averages 320 days, which exceeds the upper limit of this interval. Since 320 is outside our estimated range for the new program’s average, it indicates that patients in the new program are recovering faster on average.
4-Based on the confidence interval, do you think we have enough evidence from a statistical standpoint to say the new procedure is significantly better than the current procedure? Why or why not?
Statistically, this provides evidence that the new ACL recovery program may be significantly more effective. The entire confidence interval lies below the current 320-day average, pointing to a notable improvement in recovery time. Although practical decisions should also consider clinical significance and study design, the statistical findings strongly support the benefits of the new procedure.
Peer#
21 - We can use the Z distribution here because the population standard deviation
σ is assumed known (100 days). In addition, the sample size n=45 is large enough that the sampling distribution of the sample mean is approximately normal by the Central Limit Theorem. Those two facts (known σ and approximately normal sampling distribution) justify using a Z-based confidence interval.
2 - 90% CI is about 260.48, 309.52
3 - The current program average of 320 days lies above the upper bound of the confidence interval (309.52). So, 320 is not contained in the 90% CI.
4 - Statistically speaking, because 320 is beyond the 90% CI, this implies that for 10% significance, the new program's mean recovery time is shorter than 320. So yes, at 90% confidence/0.10 significance level, one does have evidence of shorter recovery times being provided by the new program than at present (320 days).