A factory manufacturing light-emitting diode (LED) bulbs claims that their light bulb lastfor 50 000 hours on the average. To confirm its this claim is valid, a quality control manager got a sample of 50 LED bulbs and obtained a life span of 40 000 hours. The standard deviation of the manufacturing process is 1 000 hours. Do you think the claim of the manufacturer is valid at the 5% level of significance?

$\bar{x}=40000 \\ s=1000 \\ n=50 \\ df=n-1=49 \\ α=0.05 \\$

The hypotheses tested are,

$H_0: \mu=50000 \\vs\\ H_1: \mu ≠50000 \\$

The critical value is,

$t_{{\alpha\over 2},df}=t_{0.025,49}= 2.009575$

The rejection region for this two-tailed test is R={t:|t|>2.009575}

Test-statistic:

$t = \frac{\bar{x}- \mu}{s\over \sqrt{n}} \\ t = \frac{40000-50000}{1000 \over \sqrt{50}} = -70.711 \\ |t|=70.711>2.009575$

The null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean is different than 50000, at the 0.05 significance level.