Answer to Question #147333 in Statistics and Probability for Areej

Question

A manufacturer claims that his light bulbs have an average life time of 1500 hours. A purchaser decides to check this claim and finds that for six bulbs the lifetimes are 1472,1486,1401,1350,1610,1590, hours. Does this evidence support the manufacturer’s claim? Assume that the lifetimes of the light bulbs are normally distributed.

Answer 1

$H_0: \mu=1500$

$H_a: \mu\neq1500$

It is two-tailed test.

Let's test the hypothesis on 5% significance level.

Computing test statistic:

Since the sample is small (n=6 is less than 30), we use t-test with 6–1=5 degrees of freedom.

$\bar x=\frac{1472+1486+1401+1350+1610+1590}{6}=1484.83$

$s=\sqrt{\frac{(1472-1484.83)^2+...+(1590-1484.83)^2}{6-1}}=102.08$

$$t=\frac{\bar x-\mu}{\frac{s}{\sqrt n}}=\frac{1484.83-1500}{\frac{102.08}{\sqrt 6}}=-0.364$$

According to t-table with 5% significance level and 5 df the critical value is $\pm2.571$

Since test-statistic does not fall into the critical region,

–2.571 < –0.364 < 2.571, we accept the null hypothesis.

At the 5% significance level the data provide sufficient evidence to conclude that the population mean is 1500. We are 95% confident to conclude that the manufacturer’s claim was correct and the light bulbs have an average life time of 1500 hours.