A random sample of n 25 observations is drawn from a popula

A random sample of n = 25 observations is drawn from a population that is approximately normally distributed. The sample mean is x = 6 and the variance of the sample is S^2 = 4. Find a 95% confidence interval for mu. i. (4.98,7.02) ii. (5.02,6.98) iii. (5.17,6.83) Interpret the confidence interval. i. We are 95% confident that mu is in the interval found in part a. ii. The probability that mu is in the interval found in part a. is 0.95. iii. Both i. and ii. are true. Given the sample, do you think that the true population mean, mu, is greater than 5? i. Yes, the confidence interval in a. is entirely above 5. ii. Yes, the confidence interval in a. contains 5. iii. No, the confidence interval in a. contains 5. iv. No, the confidence interval in a. is entirely above 5.

Solution

a)

Note that              
Margin of Error E = t(alpha/2) * s / sqrt(n)              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    6          
t(alpha/2) = critical t for the confidence interval =    2.063898562          
s = sample standard deviation =    2          
n = sample size =    25          
df = n - 1 =    24          
Thus,              
Margin of Error E =    0.825559425          
Lower bound =    5.174440575          
Upper bound =    6.825559425          
              
Thus, the confidence interval is              
              
(   5.174440575   ,   6.825559425   ) [ANSWER, III]

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b)

OPTION I: We are 95% confident that u is in the interval found in part a. [ANSWER]

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c)

OPTION I: Yes, the confidence interval in a is entirely above 5. [ANSWER]