- The sample mean is \(\bar{x} = 42\), population standard deviation is \(\sigma = 14.5\), sample size \(n = 28\), and the critical Z-value for 95% is \(1.96\).
- Calculate the standard error: \(SE = \frac{14.5}{\sqrt{28}} \approx 2.74\).
- Calculate the margin of error: \(ME = 1.96 \times 2.7402 \approx 5.37\).
- The 95% confidence interval is \([42 - 5.37, 42 + 5.37] = \boxed{[36.63, 47.37]}\).
Explanation
- Analyze the problem and identify data
To estimate the population mean weight \(\mu\) of all textbooks based on our sample, we use the provided data: a sample mean \(\bar{x} = 42\) ounces, a sample size \(n = 28\), and a known population standard deviation \(\sigma = 14.5\) ounces. Since the population standard deviation is known, we apply the Z-distribution to construct our confidence interval. The goal is to determine the range where we can be 95% confident that the true population mean lies.
- Determine the critical Z-value
The confidence level of 95% implies an alpha level \(\alpha = 1 - 0.95 = 0.05\). For a two-tailed test, we look for the critical Z-value \(Z_{\alpha/2}\), which corresponds to \(Z_{0.025}\). Consulting standard Z-tables, the critical value that captures the middle 95% of the distribution is approximately \(1.96\).
- Calculate the standard error
The standard error (SE) of the mean represents how much the sample mean is expected to vary from the population mean. It is calculated using the formula \(SE = \frac{\sigma}{\sqrt{n}}\). Substituting our values, we get:
\[SE = \frac{14.5}{\sqrt{28}} \approx 2.7402\]
- Calculate the margin of error
The margin of error (ME) is the product of the critical Z-value and the standard error:
\[ME = Z_{\alpha/2} \times SE = 1.96 \times 2.74024 \approx 5.3709\]
This value represents the maximum likely difference between our sample mean and the true population mean at a 95% confidence level.
- Calculate and conclude the confidence interval
Finally, we construct the confidence interval by subtracting and adding the margin of error to the sample mean:
\[\text{Lower Limit} = \bar{x} - ME = 42 - 5.3709 = 36.6291\]
\[\text{Upper Limit} = \bar{x} + ME = 42 + 5.3709 = 47.3709\]
Rounding these results to two decimal places, we get the interval \([36.63, 47.37]\). Thus, we are 95% confident that the true population mean textbook weight falls between 36.63 ounces and 47.37 ounces.
Examples
Confidence intervals are widely used in quality control and manufacturing. For instance, if a company wants to ensure that a new batch of cereal boxes meets specific weight requirements, they would weigh a random sample, calculate the confidence interval for the population mean weight, and check if that interval falls within the desired target range. This helps the manufacturer determine if the production process is consistent and meets quality standards without having to weigh every single box produced. Calculation: If the mean is \(\bar{x}\) and the margin of error is \(ME\), the interval is \([\bar{x} - ME, \bar{x} + ME]\).