Problem #20: To compare two brands of cigarettes, Brand 1 and Brand 2, for their

Question

Problem #20: To compare two brands of cigarettes, Brand 1 and Brand 2, for their tar content, a sample of 60 was inspected from Brand 1 and a sample of 40 from Brand 2. The results of the tests are summarized as follows. Let mu_1 be the mean tar content of all cigarettes of Brand 1 and mu_2 be the mean tar content of all cigarettes of Brand 2, and perform a hypothesis test at the 0.05 significance level to determine whether or not Brand 1 has a lower tar content than Brand 2. Make sure to state the null and alternative hypotheses, state whether it is a left-, right-, or two-tailed test, show all necessary steps to arrive at your conclusion, and state your final conclusion in the context of this problem.

Explanation / Answer

Here we have to test the hypothesis that,

H0 : Brand1 has a equal tar content than brand2.

H1 : Brand1 has a lower tar content than brand2.

The test is one sided with < (left tailed test) sign.

Assume alpha = level of significance = 5% = 0.05

Given values are :

For brand1 :

n1 = 60

X1bar = 15.4

s1 = 1.732

For brand2 :

n2 = 40

X2bar = 16.8

s2 = 2.000

First we have to check whether variances are equal or not.

For that the hypothesis is,

H0 : Variances are equal.

H1 : Variances are not equal.

The test statistic is,

F = larger variance / smaller variance

This we can done by using TI-83 calculator.

steps :

STAT --> TESTS --> D:2-SampFTest --> ENTER --> Highlight on Stats --> ENTER --> Input all the values --> select alternative not= --> Calculate --> ENTER

Test statistic F = 1.333

P-value = 0.3131

P-value > alpha

Accept H0 at 5% level of significance.

COnclusion : The variances are equal.

So we use pooled variances.

Now we test here two means using two sample t-test with equal variances.

Ti-83 steps :

STAT --> TESTS --> 4:2-SampTTest --> ENTER --> Highlight on Stats --> ENTER --> Input all the values --> select alternative " < mu2 " --> ENTER --> Pooled : Yes --> ENTER --> Calculate --> ENTER

The test statistic is,

t = -3.7207

df = 98

P-value = 1.6567E-4 = 0.00016567

P-value < alpha

Reject H0 at 5% level of significance.

Conclusion : There is sufficient evidence to say that brand1 has a lower tar content than brand2.

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