A manufacturer has developed a new process for making lightb

A manufacturer has developed a new process for making lightbulbs. A random sample of 23 of these lightbulbs is selected and the lifetime of each is determined. The sample mean of the lifetimes was found to be X = 738.4 hours and the sample standard deviation was s = 38.2 hours. A histogram of the data revealed a symmetric, bell-shaped distribution with no outhers. Find a 95% confidence interval for mu, the population mean lifetime for these bulbs. Give an interpretation of your interval. What assumptions are required for your interval to be valid? The manufacturer would like to advertise the bulbs as having a mean lifetime of 750 hours. Can he ethically do this (with a confidence level of 95%)? Explain.

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 =    738.4          
t(alpha/2) = critical t for the confidence interval =    2.073873068          
s = sample standard deviation =    38.2          
n = sample size =    23          
df = n - 1 =    22          
Thus,              
Margin of Error E =    16.51891873          
Lower bound =    721.8810813          
Upper bound =    754.9189187          
              
Thus, the confidence interval is              
              
(   721.8810813   ,   754.9189187   ) [ANSWER]

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b)

We are 95% confident that the true mean is between 721.8810813 and 754.9189187. [ANSWER]

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c)

That the disitribution of the sample means is approximately normal.

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d)

Yes, because 750 is within the confidence interval.