Question

The quality control manager at a light bulb factory needs to estimate the mean life of a large shipment of light bulbs. The standard deviation is 126 hours. A random sample of

81 light bulbs indicated a sample mean life of 500 hours. Complete parts? (a) through? (d) below.

a. Construct a 95?% confidence interval estimate for the population mean life of light bulbs in this shipment.

The 95?% confidence interval estimate is from a lower limit of ___ hours to an upper limit of ___ hours. (Round to one decimal place as? needed.)

b. Do you think that the manufacturer has the right to state that the lightbulbs have a mean life of 550 ?hours? Explain.

Based on the sample? data, the manufacturer does not have/has the right to state that the lightbulbs have a mean life of 550 hours. A mean of 550 hours is less than 2/more than 3 standard errors above/below the sample? mean, so it is highly unlikely/likely that the lightbulbs have a mean life of 550 hours.

c. Must you assume that the population light bulb life is normally? distributed? Explain.

A. ?No, since ? is known and the sample size is large? enough, the sampling distribution of the mean is approximately normal by the Central Limit Theorem.

B. Yes, the sample size is too large for the sampling distribution of the mean to be approximately normal by the Central Limit Theorem.

C. No, since ? is? known, the sampling distribution of the mean does not need to be approximately normally distributed.

D. Yes, the sample size is not large enough for the sampling distribution of the mean to be approximately normal by the Central Limit Theorem.

d. Suppose the standard deviation changes to 99 hours. What are your answers in? (a) and? (b)? The 95?% confidence interval estimate would be from a lower limit of ___

hours to an upper limit of ___ hours. ?(Round to one decimal place as? needed.)

Based on the sample data and a standard deviation of 99 hours, the manufacturer has/does not have the right to state that the lightbulbs have a mean life of 550 hours. A mean of 550 hours is more than 4/less than 2 standard errors above/below the sample? mean, so it is highly unlikely/likely that the lightbulbs have a mean life of 550 hours.

Answer 1

(a)

SE = /

= 126/ = 14

= 0.05

From Table, critical values of Z = 1.96

Confidence interval:

Z SE

= 500 (1.96 X 14)

= 500 27.44

=(472.56 , 527.44)

The 95% confidence interval estimate is from lower limit of 472.6 hours to an upper limit of 527.4.

(b)

+ 3 SE = 500 + (3 X 14) = 542

Based on the sample data, the manufacturer does not have the right to state that the lightbulbs have a mean life of 550 hours. A mean of 550 hours is more than 3 standard errors above the sample mean, so it is highly unlikely that the lightbulbs have a mean life of 550 hours.

(c)

Correct option:

A, No, since is known and the sample size is large enough, the sampling distribution of the mean is approximately normal by Central Limit Theorem.

(d)

SE = 99/ = 11

Confidence interval:

500 (1.96 X 11)

= 500 21.56

= ( 478.4, 521.6)

The 95% confidence interval estimate would be from a lower limit of 478.4 hours to an upper limit of 521.6 hours.

+ 4 SE = 500 + (4 X 11) = 544

Based on the sample data and a standard deviation of 99 hours, the manufacturer has the right to state that the lightbulbs have a mean life of 550 hours. A mean of 550 hours is less than 2 standard errors above the sample mean, so it is likely that the lightbulbs have a mean life of 550 hours.