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

7878

hours. A random sample of

3636

light bulbs indicated a sample mean life of

280280

hours. Complete parts​ (a) through​ (d) below.

a. Construct a

9595​%

confidence interval estimate for the population mean life of light bulbs in this shipment.The

9595​%

confidence interval estimate is from a lower limit of

254.5254.5

hours to an upper limit of

305.5305.5

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

330330

​hours? Explain.Based on the sample​ data, the manufacturer

does not have

the right to state that the lightbulbs have a mean life of

330330

hours. A mean of

330330

hours is

more than 3

standard errors

above

the sample​ mean, so it is

highly unlikely

that the lightbulbs have a mean life of

330330

hours.

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

A.

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

B.

​No, since

sigmaσ

is​ known, the sampling distribution of the mean does not need to be approximately normally distributed.Your answer is not correct.

C.

​No, since

sigmaσ

is known and the sample size is large​ enough, the sampling distribution of the mean is approximately normal by the Central Limit Theorem.This is the correct answer.

D.

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

d. Suppose the standard deviation changes to

6666

hours. What are your answers in​ (a) and​ (b)?The

9595​%

confidence interval estimate would be from a lower limit of

258.4258.4

hours to an upper limit of

301.6301.6

hours.

​(Round to one decimal place as​ needed.)

Based on the sample data and a standard deviation of

6666

​hours, the manufacturer

does not have

the right to state that the lightbulbs have a mean life of

330330

hours. A mean of

330330

hours is

more than 4

standard errors

above

the sample​ mean, so it is

highly unlikely

  that the lightbulbs have a mean life of

330330

hours.

Answer 1

a) 95% CI

= (254.5 , 305.5)

b)

Based on the sample​ data, the manufacturer does not have the right to state that the light bulbs have a mean life of 330

hours. A mean of 330 hours is more than 3 standard errors above the sample​ mean, so it is highly unlikely that the light bulbs have a mean life of 330 hours.

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

d) When

95% CI is

=( 258.4, 301.6)

Based on the sample data and a standard deviation of 66 ​hours, the manufacturer does not have the right to state that the light bulbs have a mean life of 330 hours. A mean of 330 hours is more than 4 standard errors above the sample​ mean, so it is highly unlikely that the light bulbs have a mean life of 330 hours.