c Construct a 95 confidence interval for the population mean

c. Construct a 95% confidence interval for the population mean.

Solution

a)
Normal Distribution
Mean ( u ) =8
Standard Deviation ( sd )=36/Sqrt(   25) = 36/5 = 7.2
Number ( n ) = 25
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
b)
P(X > 7.5)         = (7.5-8)/36/ Sqrt ( 25 )
= -0.5/7.2= -0.0694
= P ( Z >-0.0694) From Standard Normal Table
= 0.5277                  
P(X < = 7.5) = (1 - P(X > 7.5)
= 1 - 0.5277 = 0.4723                  
c)
Confidence Interval
CI = x ± Z a/2 * (sd/ Sqrt(n))
Where,
    x    = Mean
    sd   = Standard Deviation
    a    = 1 - (Confidence Level/100)
    Za/2 = Z-table value
    CI   = Confidence Interval
Mean(x)=5.7
Standard deviation( sd )=36
Sample Size(n)=25
Confidence Interval = [ 5.7 ± Z a/2 ( 36/ Sqrt ( 25) ) ]
= [ 5.7 - 1.96 * (7.2) , 5.7 + 1.96 * (7.2) ]
= [ -8.41,19.81 ]