To test the manufacturer's claim regarding the average nicotine content, we need to set up the null and alternative hypotheses.
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Null Hypothesis (H0): The average nicotine content does not exceed 3.5 milligrams. Mathematically, this can be expressed as: \[ H_0: \mu \leq 3.5 \]
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Alternative Hypothesis (H1): The average nicotine content exceeds 3.5 milligrams. This can be expressed as: \[ H_1: \mu > 3.5 \]
We can now proceed to perform a statistical test to see if we have enough evidence to reject the null hypothesis in favor of the alternative hypothesis at the 0.01 significance level.
Steps for the Hypothesis Test:
- Sample Size (n): 8
- Sample Mean (\(\bar{x}\)): 4.2 mg
- Sample Standard Deviation (s): 1.4 mg
- Significance Level (\(\alpha\)): 0.01
We'll perform a one-sample t-test because the sample size is small (n < 30), and we assume the normality of the distribution.
Test Statistic Calculation:
The t statistic is calculated using the formula: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] Where:
- \(\bar{x}\) = sample mean
- \(\mu_0\) = population mean under the null hypothesis (3.5 mg)
- \(s\) = sample standard deviation
- \(n\) = sample size
Substituting in the values: \[ t = \frac{4.2 - 3.5}{1.4 / \sqrt{8}} = \frac{0.7}{1.4 / 2.8284} = \frac{0.7}{0.4949} \approx 1.415 \]
Critical Value:
Next, we determine the critical value from the t-distribution table for a one-tailed test with \(n - 1 = 7\) degrees of freedom at \(\alpha = 0.01\).
Using a t-table or calculator, the critical value for \(df = 7\) at \(\alpha = 0.01\) is approximately 2.998.
Decision Rule:
- If the calculated t value exceeds the critical t value, we reject the null hypothesis.
- If the calculated t value does not exceed the critical t value, we fail to reject the null hypothesis.
Conclusion:
- Calculated t value: approximately 1.415
- Critical t value: approximately 2.998
Since \(1.415 < 2.998\), we fail to reject the null hypothesis.
Final Statement:
There is not enough evidence at the 0.01 significance level to conclude that the average nicotine content of this brand exceeds 3.5 milligrams, supporting the manufacturer's claim.