To test the manufacturer's claim that the average nicotine content exceeds 3.5 milligrams, we can set up our hypothesis tests as follows:
- Null hypothesis (\(H_0\)): \(\mu \leq 3.5\) (The average nicotine content is less than or equal to 3.5 mg.)
- Alternative hypothesis (\(H_a\)): \(\mu > 3.5\) (The average nicotine content is greater than 3.5 mg.)
Given:
- Sample size (\(n\)) = 8
- Sample mean (\(\bar{x}\)) = 4.2 mg
- Sample standard deviation (\(s\)) = 1.4 mg
- Significance level (\(\alpha\)) = 0.1
We use the t-test for the sample mean since the sample size is small and we are assuming the distribution is normal.
The test statistic for the one-sample t-test is calculated using the formula:
\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]
Where:
- \(\bar{x}\) is the sample mean,
- \(\mu_0\) is the population mean under the null hypothesis (3.5 mg),
- \(s\) is the sample standard deviation,
- \(n\) is the sample size.
Substituting in the values:
\[ t = \frac{4.2 - 3.5}{1.4 / \sqrt{8}} \]
Calculating the denominator:
\[ 1.4 / \sqrt{8} \approx 1.4 / 2.828 \approx 0.495 \]
Now we compute the t-statistic:
\[ t = \frac{4.2 - 3.5}{0.495} \approx \frac{0.7}{0.495} \approx 1.414 \]
Thus, the value of the test statistic is approximately 1.414.