Question

a. A CI is desired for the true average stray-load loss μ (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that stray-load loss is normally distributed with σ = 3.6. Provide a 95% CI for μ when n = 25 and x-bar = 58.6.

.b. Considering the information provided in problem #1, how large must n be if the width of the 99% interval for μ is to be 1.4?

Answer 1

a) Since population standard deviation is known, we use the z-distribution      
95% confidence interval for population mean      
Given      
X̅ = 58.6  Sample Mean    
σ = 3.6  Sample Standard Deviation    
n = 25  Sample Size    
Confidence interval for mean is given by      
       
       
For 95%, α = 0.05, α/2 = 0.025      
From the z-tables, or Excel function NORM.S.INV(α/2)      
z = NORM.S.INV(0.025) = 1.96   (We take the positive value for calculations)   
       
      
       
= (57.1889, 60.0112)      
95% confidence interval is (57.1889, 60.0112)     
       
d) Confidence level is 99%      
To find n = sample size such that width of the interval is 1.4      
For 99%, α = 0.01, α/2 = 0.005      
From the z-tables, or Excel function NORM.S.INV(α/2)      
z = NORM.S.INV(0.005) = 2.576   (We take the positive value for calculations)   
Width of the interval is given by      
              
n = 175.51      
n = 176 Rounding to the next whole number     
n = 176