To construct a 95% confidence interval for the true average nicotine content of the brand of cigarettes, we can use the formula for the confidence interval based on the t-distribution, since we have a small sample size (n < 30).
The formula for the confidence interval is:
\[ \text{CI} = \bar{x} \pm t^* \left( \frac{s}{\sqrt{n}} \right) \]
where:
- \(\bar{x}\) is the sample mean,
- \(t^*\) is the t-value from the t-distribution for \(n-1\) degrees of freedom at the desired confidence level,
- \(s\) is the sample standard deviation, and
- \(n\) is the sample size.
Given:
- Sample mean, \(\bar{x} = 2.6\) mg
- Sample standard deviation, \(s = 0.9\) mg
- Sample size, \(n = 8\)
Step 1: Calculate the degrees of freedom
\[ df = n - 1 = 8 - 1 = 7 \]
Step 2: Find the t-value for a 95% confidence level with 7 degrees of freedom
Using a t-table or calculator, the t-value \(t^*\) for 95% confidence and 7 degrees of freedom is approximately \(2.365\) (you can use a t-distribution calculator or look this up in a t-table).
Step 3: Calculate the standard error (SE)
\[ SE = \frac{s}{\sqrt{n}} = \frac{0.9}{\sqrt{8}} \approx \frac{0.9}{2.8284} \approx 0.3182 \]
Step 4: Calculate the margin of error (ME)
\[ ME = t^* \times SE = 2.365 \times 0.3182 \approx 0.7523 \]
Step 5: Construct the confidence interval
\[ \text{CI} = \bar{x} \pm ME = 2.6 \pm 0.7523 \]
Calculating the bounds:
- Lower bound: \(2.6 - 0.7523 \approx 1.8477\)
- Upper bound: \(2.6 + 0.7523 \approx 3.3523\)
Final Result
The 95% confidence interval for the true average nicotine content of this brand of cigarettes is approximately:
\[ (1.85, 3.35) \text{ milligrams} \]
(rounding to two decimal places).
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