- Use the t-distribution with \(df = 16\) and a critical value \(t_{0.01, 16} \approx 2.583\) for a 98% confidence interval.
- Calculate the standard error: \(SE = \frac{9.2}{\sqrt{17}} \approx 2.2313\).
- Calculate the margin of error: \(E = 2.583 \times 2.2313 \approx 5.7646\).
- The 98% confidence interval is \(\bar{x} \pm E = 56.6 \pm 5.7646\), which rounds to \(\boxed{50.8 < \mu < 62.4}\).
Explanation
- Analyze the given data
To estimate the population mean \(\mu\), we start by identifying the key parameters from the provided sample data:
- Sample size \(n = 17\)
- Sample mean \(\bar{x} = 56.6\)
- Sample standard deviation \(s = 9.2\)
- Confidence level \(CL = 0.98\), which corresponds to a significance level \(\alpha = 1 - 0.98 = 0.02\).
Since the population standard deviation is unknown and the sample size is small, we use the Student's t-distribution with degrees of freedom \(df = n - 1 = 16\).
- Calculate the margin of error
To find the margin of error, we first need the critical value \(t_{\alpha/2}\) for a 98% confidence level with 16 degrees of freedom. Using the t-distribution, we find the critical value \(t_{0.01, 16} \approx 2.583\).
Next, we calculate the standard error of the mean (SE):
\[SE = \frac{s}{\sqrt{n}} = \frac{9.2}{\sqrt{17}} \approx \frac{9.2}{4.1231} \approx 2.2313\]
Finally, the margin of error \(E\) is given by:
\[E = t_{\alpha/2} \times SE = 2.583 \times 2.2313 \approx 5.7646\]
- Determine the confidence interval
With the margin of error determined, we calculate the lower and upper bounds of the confidence interval:
- Lower Bound = \(\bar{x} - E = 56.6 - 5.7646 = 50.8354\)
- Upper Bound = \(\bar{x} + E = 56.6 + 5.7646 = 62.3646\)
Rounding these values to one decimal place, we obtain 50.8 and 62.4, respectively. Thus, the 98% confidence interval for the population mean \(\mu\) is \(50.8 < \mu < 62.4\).
Examples
Confidence intervals are widely used in quality control and market research. For instance, if a company tests the battery life of 17 randomly selected smartphones and finds a mean life of 56.6 hours with a standard deviation of 9.2, they can use this method to tell customers with 98% confidence that the true average battery life of all their phones falls within a specific range, helping to set realistic expectations for consumers.