Final answer:
The hypotheses for the mean tar content in cigarettes are set as H0: μ = 21.1 mg and H1: μ < 21.1 mg. The test statistic is -1.29, which is not smaller than the critical value for a one-tailed test at the 0.05 significance level (-1.645). Therefore, we do not reject the null hypothesis, indicating insufficient evidence to support the claim that the mean tar content is less than 21.1 mg.
Step-by-step explanation:
Hypothesis Testing for Mean Tar Content in Cigarettes
When conducting a hypothesis test for the mean tar content of a brand of cigarettes, the null hypothesis (H0) and the alternative hypothesis (H1) serve as foundational elements. Given the sample data and the information provided, let's set up the hypotheses and perform the calculations needed.
The null hypothesis (H0): μ = 21.1 mg, represents the claim that the mean tar content for the population (all cigarettes of this brand) is equal to the given value of 21.1 mg. The alternative hypothesis (H1): μ < 21.1 mg, represents the claim we're testing for: that the mean tar content for the population is less than the given value.
To find the test statistic, we use the following formula for a z-test, since the population standard deviation is known:
Z = (X - μ) / (σ / √ n)
Plugging in the numbers:
Z = (20.2 - 21.1) / (3.48 / √ 25)
Z ≈ -1.29
For a one-tailed test at the 0.05 significance level, the critical z-value is approximately -1.645. We compare the test statistic to this critical value to make a decision about the null hypothesis.
Since the test statistic of -1.29 is greater than the critical value of -1.645, we do not have enough evidence to reject the null hypothesis.
Conclusion: There is insufficient evidence at the 0.05 significance level to conclude that the mean tar content of this brand of cigarettes is less than the mean of all cigarettes at 21.1 mg. Therefore, the claim is not supported by the sample data provided.