Could you help me with this? It's due by tonight and I'm having some trouble understand it and I'm looking for a reference to make sure I'm doing it right. If you could solve these, it would be greatly appreciated!
25 students were chosen randomly from a statistics class to solve some problems on confidence intervals. The time required to complete all problems was recorded for every student. The sample mean time was 11 minutes with a sample standard deviation of 5 minutes.
1. Construct a 95% confidence interval for mu, the mean time for the population of students in that statistics class, to solve those problems.
2. Interpret your answer to question 1, i.e., state your conclusion.
3. Describe how the sample size n affects the width of the confidence interval on mu.
4. Describe how the variability of the data affects the width of the confidence interval on mu.
5. Describe how the confidence level affects the width of the confidence interval on mu.
Answers & Comments
Verified answer
Since the population standard deviation is not known, we need to use the t-distribution.
For a 95% confidence interval, the t-critical value with 24 degrees of freedom (n-1) is 2.064
1)
Sample mean = 11
Standard deviation = 5
Standard error of mean = σ / √ n
Standard error of mean = 5 / √ 25
SE = 5/5
Standard error of mean 1
Confidence interval 11-(1)(2.064) and 11+(1)(2.064)
95% confidence interval is (8.94, 13.06)
2)
If repeated samples of size 25 were taken and a confidence interval computed for each sample, 95% of the time the true mean time will be expected to lie between 8.94 and 13.06.
3)
Increasing the sample size will make the interval narrower (shorter).
4)
More variability will make the confidence interval wider.
5)
More the level of confidence, wider the interval. For example, a 99% confidence interval would produce a wider interval than a 95% confidence interval.