Question 859:

Asked on March 18, 2010

Tags: Confidence Intervals

1

Answer:

No answer provided yet.1. The 99% confidence interval is found using the t-distribution since we have a small sample size. The confidence interval is generated by first finding the margin of error.
The margin of error is the standard error times the t-critical value for a 99% confidence level.
The standard error is found as the standard deviation divided by the square root of the sample size = 1.034/SQRT(16) = .2585. Using a t-table gets us the critical value on 15 degrees of freedom as 2.94.

The margin of error is then 2.94*.2585 = .76 making the 99% confidence interval equal to .611 to 2.135. This means we can be 99% confident the true mean weight of dry seeds in crops is between .611  and 2.135 grams.

2. The 95% confidence interval is found using the t-distribution since we have a small sample size. The confidence interval is generated by first finding the margin of error.
The margin of error is the standard error times the t-critical value for a 95% confidence level.
The standard error is found as the standard deviation divided by the square root of the sample size. The standard deviation of this sample is 2, making the standard error = 2/SQRT(3) = 1.15. Using a t-table gets us the critical value on 3 degrees of freedom as 4.30.
a. The 95% confidence interval is between 22.032 and 31.96 ft.
b. Since the lower boundary of the confidence interval does not dip below 22, we can be at least 95% confident the mean shark length around Bermuda is above 22ft.
c. Since the interval dips well below 22, there is not enough evidence to support the claim that the mean is above 22ft.

3. The 99% confidence interval is found using the t-distribution since we have a small sample size. The confidence interval is generated by first finding the margin of error.
The margin of error is the standard error times the t-critical value for a 99% confidence level.
The standard error is found as the standard deviation divided by the square root of the sample size = 9.59/SQRT(20) = 2.144.  Using a t-table gets us the critical value on 19 degrees of freedom as 2.86.
a. The 99% confidence interval is between 47.895 and 60.165, so we can be 99% confident the true mean would fall within this range.
b. By taking a random sample it prevents resents abnormal trends in a given period from biasing the average across the entire 140 year span.
c. There would likely be some years with a lot of snowfall, abnormally high, potentially skewing the data upward. It would be a good idea to graph the data to look for significant departures from normality.