A random sample of fifty-four 200-meter swims has a mean time of 3.125 minutes and a standard deviation of 0.080 minutes. A 95% confidence interval for the population mean time is (3.107,3.143).
Construct a 95% confidence interval for the population mean time using a standard deviation of 0.05 minutes. Which confidence interval is wider? Explain.
The 95% confidence interval is ( ____ , ____ ) (Round to three decimal places as needed.)
Solution :
Given that,
= 3.125
s = 0.05
n = 54
Degrees of freedom = df = n - 1 = 54 - 1 = 53
At 95% confidence level the t is ,
= 1 - 95% = 1 - 0.95 = 0.05
/ 2 = 0.05 / 2 = 0.025
t /2,df = t0.025,54 = 2.006
Margin of error = E = t/2,df * (s /n)
= 2.006 * (0.05 / 54)
= 0.014
The 95% confidence interval estimate of the population mean is,
- E < < + E
3.125 - 0.014 < < 3.125 + 0.014
3.111 < < 3.139
The 95% confidence interval is (3.111, 3.139)
The confidence interval for standard deviation of 0.08 is wider because difference between interval is greater than standard deviation of 0.05
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