Question

2. A manufacturer of replacement bulbs for LCD projectors claims a 1000 hour average life time for the bulbs. This is important, since the bulbs sell for around $300 each. Concerned that the claim may not be valid because of a recently discovered problem in the manufacturing process, the company decides to test it. Twelve bulbs are randomly selected from a production run, each is burned until it fails, and its lifetime recorded. Here are the 12 lifetimes, in hours. 991.989 948 1005 980 957 968 993 991 927 1001 984 What do these 12 lifetimes suggest about the manufacturer's claim? Be specific about assumptions, hypotheses being tested, test statistic used, and the critical region for your test. Interpret the results of your test in regard to the claim of the bulb manufacturer. (You may do this test by hand or get SPSS to do it for you. If you use SPSS, please provide the output it produces along with your interpretation of the results. Note that you have not been asked for nor should you consider a confidence interval in this problem.

Answer 1

Solution:

Here, we have to use one sample t test for the population mean.

H₀: µ = 1000 versus H_a: µ ≠ 1000

This is a two tailed test.

We are given

Level of significance = α = 0.05

From given data, we have

n = 12

Xbar = 977.8333333

S = 23.40098418

df = n – 1 = 11

Critical values = - 2.2010 and 2.2010

(by using t-table)

Critical Region: Reject H₀ if t < - 2.2010 or t > 2.2010

Test statistic is given as below:

t = (Xbar - µ)/[S/sqrt(n)]

t = (977.8333333 – 1000) / [23.40098418/sqrt(12)]

t = -3.2814

P-value = 0.0073

(by using t-table)

P-value < α = 0.05

Test statistic = t = -3.2814 < critical value = - 2.2010

So, we reject the null hypothesis

There is insufficient evidence to conclude that the average life time for the bulbs is 1000 hours.