Question

The mean tar content of a simple random sample of 25 unfiltered king-size cigarettes is 20.8 mg, with a standard deviation of 4 mg. The mean tar content of a simple random sample of 25 filtered 100 mm cigarettes is 13.4 mg with a standard deviation of 3.9 mg.

Assume that the two samples are independent, simple, random samples, selected from normally distributed populations. Do not assume that the population standard deviations are equal.

o Use a 0.05 significance level to test the claim that unfiltered king-size cigarettes have mean tar content greater than that of filtered 100 mm cigarettes.

o What does the result suggest about the effectiveness of cigarette filters?

Answer 1

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ =μ2​

Ha: μ1​ >μ2​

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.

(2) Rejection Region: Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=47.969. In fact, the degrees of freedom are computed as follows, assuming that the population variances are unequal:

Hence, it is found that the critical value for this right-tailed test is tc​=1.677, for α=0.05 and df=47.969.

The rejection region for this right-tailed test is R={t:t>1.677}.

(3) Test Statistics

Since it is assumed that the population variances are unequal, the t-statistic is computed as follows:

(4) Decision about the null hypothesis: Since it is observed that t=6.623>tc​=1.677, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.000, and since p=0.000<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion: It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μ1​ is greater than μ2​, at the 0.05 significance level.

Cigarette filters are effective. Less amount of tar consumed by using filtered 100mm cigarettes