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The mean tar content of a simple random sample of 35 unfiltered cigarettes is 21.1 mg, with a standard deviation of 3.2 mg. The mean tar content of a simple random sample of 30 filtered cigarettes is 13.2 mg with a standard deviation of 3.7 mg. At a significance level of 0.01, do the results suggest that, on average, filtered cigarettes have less tar than unfiltered cigarettes?

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Answer:

There is significant evidence at 0.01 significance level that filtered cigarettes have less tar than unfiltered cigarettes

Step-by-step explanation:

Let M(f) be the true mean tar content of unfiltered cigarettes

And M(u) be the true mean tar content of filtered cigarettes

Then

[tex]H_{0}[/tex]: M(f) = M(u)

[tex]H_{a}[/tex]: M(f) < M(u)

test statistic can be calculated using the formula:

[tex]z=\frac{X-Y}{\sqrt{\frac{s(x)^2}{N(x)}+\frac{s(y)^2}{N(y)}}}[/tex] where

  • X is the sample mean tar content of unfiltered cigarettes (21.1 mg)
  • Y is the sample mean tar content of filtered cigarettes (13.2 mg)
  • s(x) is the sample standard deviation of unfiltered cigarettes (3.2 mg)
  • s(y) is the sample standard deviation of filtered cigarettes (3.7 mg)
  • N(x) is the sample size  of unfiltered cigarettes (35)
  • N(y) is the sample size of filtered cigarettes (30)

Then [tex]z=\frac{21.1-13.2}{\sqrt{\frac{3.2^2}{35}+\frac{3.7^2}{30}}}[/tex]

≈9.13

p-value of the statistic ≈0 <0.01 (significance level) Thus we can reject the null hypothesis.

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