Answer to Question #146635 in Statistics and Probability for ADNILLLL

Question

The quality control manager at a lightbulb factory needs to estimate the mean life of a large shipment of lightbulbs. The process standard deviation is known to be 100 hours. A random sample of 64 lightbulbs indicate a sample mean life of 350 hours.
a. Set up 95% confidence interval estimate of the true population mean life of lightbulbs in this shipment.
b. Do you think that manufacturer has the right to state that the lightbulbs last an average of 400 hours? Explain.
c. Does the population of lightbulbs life have to be normally distributed here? Explain.
d. Explain why an observed value of 320 hours is not unusual, even though it is outside the confidence interval you calculated.
e. Suppose that the process standard deviation changed to 80 hours. What would be your answers in (a) and (b)?

Answer 1

$a.\ \overline{x}=350\\ \sigma=100\\ N=64\\ z_{0.05}=1.64\\ \text{We have:}\\ (350-(1.64)\frac{100}{\sqrt{64}},350+(1.64)\frac{100}{\sqrt{64}})\\ (329.5,370.5)\\ b.\ \text{The answer is no because 400 does not fall into the confidence interval.}\\ \text{So we reject } H_0:a=a_0=400\text{ and accept } H_1:a\neq a_0=400\\ \text{at a significance level }\alpha=0.05.\\ c.\ \text{Yes, we build the confidence interval on the assumption}\\ \text{that life of lightbulbs has normal distribution}.\\ d.\ \text{Our confidence interval covers the population mean}\\ \text{with the probability 0.95. So an observed value of 320 hours}\\ \text{is not unusual}.\\ e.\ \text{We have:}\\ (350-(1.64)\frac{80}{\sqrt{64}},350+(1.64)\frac{80}{\sqrt{64}})\\ (333.6,376.4)\\ \text{The answer is no because 400 does not fall into the confidence interval.}\\ \text{So we reject } H_0:a=a_0=400\text{ and accept } H_1:a\neq a_0=400\\ \text{at a significance level }\alpha=0.05.$