A sample of size n 5 from a large population that is though

A sample of size n = 5 from a large population that is thought to be normal gave the following data:

7.5, 9.3, 6.2, 8.4, 9.8

(a) Create a 90% CI for the population mean µ.

(b) If a 95% CI were to be constructed using the same data, would it be larger or smaller than the one created in (a)? Explain your answer theoretically, and not by actually creating the interval.

Solution

a)

Note that              
              
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 =    0.05          
X = sample mean =    8.24          
t(alpha/2) = critical t for the confidence interval =    2.131846786          
s = sample standard deviation =    1.439791652          
n = sample size =    5          
df = n - 1 =    4          
Thus,              
              
Lower bound =    6.86731579          
Upper bound =    9.61268421          
              
Thus, the confidence interval is              
              
(   6.86731579   ,   9.61268421   ) [ANSWER]

*********************

B)

It would be a larger interval, because we need a larger interval so that we would be \"more confident\" that we have the true mean inside our interval. [ANSWER]