Question

We have measured the lifetime (in days) of 55 expensive lamps, which led to a sample mean and a standard deviation of 460 and 23, respectively. We have also measured the lifetime of 55 cheap lamps, which led to a sample mean and a standard deviation of 400 and 15, respectively. Carry out a test of hypotheses at significance level of 0.01 to decide whether the true mean of of the lifetime of the cheap lamps is less than 50 days than that of the expensive lamps.

Answer 1

Solution:

Ho:

Ha:

alpha=0.01

where mu1=mean of liftetime of the cheap lamps

mu2=mean of lifetime of the expensive lamps

t=(x1-x2)-(mu1-mu2)/sqrt(s1^2/n1+s2^2/n2)

x1,s1,n1 are mean,standard devaition ,sample size of cheap lamps

x2,s2,n2 are mean,standard devaition ,sample size of expensive lamps

t=(400-460)-(50)/sqrt(15^2/55+23^2/55))

t=-9.449537

df=n1+n2-2=55+55-2=108

p value in excel

==T.DIST.2T( 9.449537,108)

p=8.32682E-16

p=0.0000

p<0.01

reject null hypothesis

Accept alternative hypothesis.
Conclusion:
There is sufficient statistical evidence at 1% level of significance to conclude that true mean of of the lifetime of the cheap lamps is less than 50 days than that of the expensive lamps.