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Question

Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Identify the null and alternative​ hypotheses, test​ statistic, P-value, and state the final conclusion that addresses the original claim.

A simple random sample of 25 filtered 100 mm cigarettes is​ obtained, and the tar content of each cigarette is measured. The sample has a mean of 19.4 mg and a standard deviation of 3.77 mg. Use a

0.05 significance level to test the claim that the mean tar content of filtered 100 mm cigarettes is less than

21.1 ​mg, which is the mean for unfiltered king size cigarettes. What do the results​ suggest, if​ anything, about the effectiveness of the​ filters?

What are the​ hypotheses?

Identify the test statistic.

t=

Answer 1

The null hypothesis is the opposite of the 'claim' in the question. The 'claim' is the alternative hypothesis.

Null Hypothesis: mean >= 21.1

Alternative Hypothesis: mean < 21.1

Define the things you know:

n = 25

degrees of freedom = 25 - 1 = 24

mean = 19.4

SD = 3.77

SE = 3.77/sqrt(25) = 3.77/5 = 0.754

Compute your test statistic using the formula:

t = (sample mean - hypothesized mean)/SE = (19.4 - 21.1)/0.754 = -2.2546

Look up the critical t-statistic for 24 degrees of freedom for a 1-tailed test at a significance level of 0.05. This is -1.711.

Compare your obtained statistic with the critical statistic and get the p-value from a table or statistical program.

p-value for a left tailed test is: 0.01678

Since 0.01678 < 0.05, we can reject the null hypothesis in favor of the alternative that the mean is less than 21.1, suggesting that the filters are effective.