A random sample of thirtyfive 200meter swims has the mean ti

A random sample of thirty-five 200-meter swims has the mean time of 3.122 minutes and a standard deviation of 0.080 minutes. A 95% confidence interval for the population mean time is (3.100,3.144). Construct a 95% confidence interval for the population mean time using a standard deviation of 0.05 minutes. Which confidence interval is wider?

Solution

Here sample size is 35. Sample mean is 3.122 minutes and sample SD is 0.080 minutes.

So 95% confidence interval is 3.122-0.080/35*t_34,0.025,3.122+0.080/35*t_34,0.025. where t_34,0.025 is the upper 0.025 point of t distribution with df 34.

Now this interval is given as (3.100,3.144) so the upper 0.025 point of t distribution with df 34 is (3.122-3.100)/0.080*35=1.627

So 95% confidence interval with SD 0.05 is (3.122-0.05*1.627/35,3.122+0.05*1.627/35)=(3.108,3.136)

For the of first interval the gap is 3.144-3.100=0.044 and for the second interval it is 3.136-3.108=0.028. So the first one is wider. [Answer]