The average one-way commute time for a random sample of 256 area residents of a city is 90 minutes, with a standard deviation of 24 minutes.
Construct a 90% confidence interval for the mean commute time of all residents.
Construct a 98% confidence interval for the mean commute time of all residents.
Which interval is larger? Why?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (for two-tailed, .05 and .01) and get the respective Z scores.
90% = Mean ± Z(SD)
98% = mean ± Z(SD)
After the calculations, you should be able to answer why.
To construct a confidence interval for the mean commute time of all residents, we use the formula:
Confidence interval = sample mean ± (critical value) * (standard deviation / √n),
where the critical value is determined based on the desired level of confidence, and n is the sample size.
For a 90% confidence interval:
Sample mean = 90 minutes
Standard deviation = 24 minutes
Sample size (n) = 256
Critical value (z) for a 90% confidence level = 1.645 (obtained from a standard normal distribution table or calculator)
Calculating the confidence interval:
Confidence interval = 90 ± (1.645) * (24 / √256)
Confidence interval = 90 ± (1.645) * (24 / 16)
Confidence interval = 90 ± (1.645) * (1.5)
Confidence interval = 90 ± 2.468
Confidence interval = (87.532, 92.468)
Therefore, the 90% confidence interval for the mean commute time of all residents is (87.532, 92.468) minutes.
For a 98% confidence interval:
Sample mean, standard deviation, and sample size remain the same.
Critical value (z) for a 98% confidence level = 2.326 (again, obtained from a standard normal distribution table or calculator)
Calculating the confidence interval:
Confidence interval = 90 ± (2.326) * (24 / √256)
Confidence interval = 90 ± (2.326) * (24 / 16)
Confidence interval = 90 ± (2.326) * (1.5)
Confidence interval = 90 ± 3.489
Confidence interval = (86.511, 93.489)
Therefore, the 98% confidence interval for the mean commute time of all residents is (86.511, 93.489) minutes.
The 98% confidence interval is larger than the 90% confidence interval. This is because as we increase the desired level of confidence, we need to account for a larger margin of error. The critical value for a higher confidence level (98%) is larger than the critical value for a lower confidence level (90%), resulting in a wider confidence interval.