1 The quality control manager at a light bulb factory needs

1. The quality control manager at a light bulb factory needs to estimate the mean life of a large shipment of light bulbs. A random sample of 36 light bulbs indicated a sample mean life of 350 hours. The standard deviation of the sample is 100 hours.

Construct a 99% confidence interval estimate for the population mean life to light bulbs in this shipment.

Step 1: figure out a point estimate:

Step 2: Is (population standard deviation) known? If no, then find sample size nand s/n

---Here s is the sample standard deviation.

Step 3: find the confidence level (1-) and the critical value t_{(1-/2, n-1)}.

Step 4: construct a confidence interval for population mean. Please interpret the confidence interval.

2. If you want to be 95% confident of estimating the population mean to within a sampling error of +1.5 and the standard deviation is assumed to be 10, what sample size is required?

Solution

1.

Step 1: figure out a point estimate:

Xbar = 350 hours. [ANSWER]

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Step 2: Is (population standard deviation) known? If no, then find sample size nand s/n

NO. We only know the sample standard deviation.

n = 36 bulbs.

s/sqrt(n) = 100/sqrt(36) = 16.66666667 [ANSWER]

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Step 3: find the confidence level (1-) and the critical value t(1-/2, n-1).

Confidence level = 0.99 [ANSWER]

Here,
              
alpha/2 = (1 - confidence level)/2 =    0.005      
df = n - 1 =    35          

Thus,
  
t(alpha/2) = critical t for the confidence interval =    2.723805589   [ANSWER]

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Step 4: construct a confidence interval for population mean. Please interpret the confidence interval.

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 =    0.005          
X = sample mean =    350          
t(alpha/2) = critical t for the confidence interval =    2.723805589          
s = sample standard deviation =    100          
n = sample size =    36          
df = n - 1 =    35          
Thus,              
Margin of Error E =    45.39675982          
Lower bound =    304.6032402          
Upper bound =    395.3967598          
              
Thus, the confidence interval is              
              
(   304.6032402   ,   395.3967598   ) [ANSWER]

We are 99% confident that the true population mean life is between 304.603 to 395.397 hours. [ANSWER]

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2.

Note that      
      
n = z(alpha/2)^2 s^2 / E^2      
      
where      
      
alpha/2 = (1 - confidence level)/2 =    0.025  
      
Using a table/technology,      
      
z(alpha/2) =    1.959963985  
      
Also,      
      
s = sample standard deviation =    100  
E = margin of error =    1.5  
      
Thus,      
      
n =    17073.15031  
      
Rounding up,      
      
n =    17074   [ANSWER]

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