Homework SourseThe director of quality at a lightbulb factory needs to esti
The director of quality at a light-bulb factory needs to estimate the average life of a large shipment of light bulbs. The process standard deviation is known to be 100 hours. A random sample of 64 light bulbs indicated a sample average life of 350 hours.
a) Set up a 95% confidence interval estimate of the true average life of light bulbs in this shipment.
b) Do you think that the manufacturer has the right to state that the light bulbs last an average of 400 hours? Explain
c) Does the population of the light-bulb life have to be normally distributed here for the interval to be valid? Explain
d) Suppose that the process is improved so that the standard deviation is reduced to 80 hours. What would be your answers in (a) and (b) ?
Solution
a)
Note that
Margin of Error E = z(alpha/2) * s / sqrt(n)
Lower Bound = X - z(alpha/2) * s / sqrt(n)
Upper Bound = X + z(alpha/2) * s / sqrt(n)
where
alpha/2 = (1 - confidence level)/2 = 0.025
X = sample mean = 350
z(alpha/2) = critical z for the confidence interval = 1.959963985
s = sample standard deviation = 100
n = sample size = 64
Thus,
Margin of Error E = 24.49954981
Lower bound = 325.5004502
Upper bound = 374.4995498
Thus, the confidence interval is
( 325.5004502 , 374.4995498 ) [ANSWER]
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b)
NO. The whole confidence interval is under 400 hrs.
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c)
NO, the central limit theorem states that for any distribution, the distribution of the sample means are approximately normal.
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d)
The confidence interval will then be narrower, and all the more that the upper bound of the distribution is farther from 400. THe manufacturer still has no right to say the light bulbs last at least 400 hrs.
