Question

The manager of a lightbulb factory wants to determine if there is any difference in the mean life expectancy of the bulbs manufactured on two different types of machines. A random sample of 4 lightbulbs from machine I indicates a sample mean of 352 hours and a sample standard deviation of 98 ​hours, and a similar sample of 4 lightbulbs from machine II indicates a sample mean of 364 hours and a sample standard deviation of 125 hours. Complete parts​ (a) and​ (b). a. Using the 0.05 level of​ significance, and assuming the population variances are​ equal, is there any evidence of a difference in the mean life of bulbs produced by the two types of​ machines?

Do or do not reject H0? There is evidence or there is insufficient evidence of a difference in the mean life of bulbs produced by the two types of​ machines?

Answer 1

Here for machine 1

sample mean = = 352 hours

Sample standard deviation = s₁ = 98 hours

sample size = n₁ = 4

Here for machine 2

sample mean = = 364 hours

Sample standard deviation = s₂ = 125 hours

sample size = n₂ = 4

Here ypothesis are

H₀ :

H_a :

Here population variances are equal so first we have to find the pooled standard deviation

s_p = sqrt [ {(n₁ -1)s₁² + (n₂ -1)s₂²}/(n₁ + n₂ -2)] = sqrt [ (3 * 98 * 98 + 3 * 125 * 125)/(4 + 4 - 2)] = 112.31

now standard error of difference in means = se_d = s_p * sqrt (1/n₁ + 1/n₂) = 112.31 * sqrt (1/4 + 1/4) = 79.42 hours

Here test statistic

t = ( - )/se_d

t = (352 - 364)/79.42 = -0.1511

here significance level = 0.05,

degree of freedom = 4 + 4 - 2 = 6

t_critical = TINV(0.05, 6) = 2.447

so here as we see that l t l < t_critical we would fail to reject the null hypothesis and can say that there is  insufficient evidence of a difference in the mean life of bulbs produced by the two types of​ machines