4. A buyer of electric bulbs purchases 400 bulbs; 200 bulbs of each brand. Upon testing these bulbs be found that brand A has an average of 1225 hrs with a S.D of 42 hrs.
Where as brand B had a mean life of 1265 hrs with a S.D of 60hrs. Can the buyer be certain that brands B is superior than brand A in equality
The following null and alternative hypotheses need to be tested:
"H_0:\\mu_1\\leq\\mu_2"
"H_1:\\mu_1>\\mu_2"
This corresponds to a right-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.
Since the brand A and brand B are the different populations, we assume that the population variances are unequal, so then the number of degrees of freedom is computed as follows:
It is found that the critical value for this right-tailed test is "t_c=1.6491," for "\\alpha=0.05"
and "df=359.842."
The rejection region for this right-tailed test is "R=\\{t:t>1.6491\\}."
Since it is assumed that the population variances are unequal, the t-statistic is computed as follows:
Since it is observed that "t=7.7238>1.6491=t_c," it is then concluded that the null hypothesis is rejected.
Therefore, there is enough evidence to claim that population mean "\\mu_1" is greater than "\\mu_2," at the 0.05 significance level.
Using the P-value approach: The p-value for the right-tailed test with "df=359.842" is "p\\approx0," and since "p=0<0.05=\\alpha," it is concluded that the null hypothesis is rejected.
Therefore, there is enough evidence to claim that population mean "\\mu_1" is greater than "\\mu_2," at the 0.05 significance level.
Therefore, there is enough evidence to claim that brands B is superior than brand A in equality at the 0.05 significance level.
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