- Determine the critical t-value for \(df = 24\) at the 80% confidence level: \(t_{0.10} \approx 1.318\).
- Calculate the standard error of the mean: \(SE = \frac{9}{\sqrt{25}} = 1.8\).
- Compute the margin of error: \(E = 1.318 \times 1.8 \approx 2.4\).
- State the confidence interval in the requested format: \(\boxed{43 \pm 2.4}\).
Explanation
- Identify the parameters and critical value
To construct a confidence interval for the population mean \(\mu\) when the population standard deviation \(\sigma\) is unknown, we use the t-distribution.
Given data:
- Sample size \(n = 25\)
- Sample mean \(\bar{x} = \$43\)
- Sample standard deviation \(s = \$9\)
- Confidence level = \(80\%\)
First, we determine the significance level \(\alpha = 1 - 0.80 = 0.20\). We are looking for the critical t-value, \(t_{\alpha/2}\), with \(n-1 = 24\) degrees of freedom. Using the t-distribution table for \(\alpha/2 = 0.10\) and \(df = 24\), the critical t-value is approximately \(1.318\).
- Calculate the standard error and margin of error
The standard error of the mean (SE) estimates how far the sample mean is likely to be from the true population mean. It is calculated as:
\[SE = \frac{s}{\sqrt{n}} = \frac{9}{\sqrt{25}} = \frac{9}{5} = 1.8\]
Now, we calculate the margin of error (E) by multiplying the critical t-value by the standard error:
\[E = t_{0.10} \cdot SE \approx 1.318 \cdot 1.8 \approx 2.3721\]
Rounding this to one decimal place as requested, we get the margin of error \(E \approx 2.4\).
- Construct the confidence interval
The confidence interval is constructed using the formula \(\bar{x} \pm E\).
Substituting our values:
\[43 \pm 2.4\]
This gives an 80% confidence interval for the mean gift expenditure of roughly \(\$40.6\) to \(\$45.4\). Thus, we can conclude that the 80% confidence interval is \(43 \pm 2.4\).
Examples
Understanding confidence intervals is essential in real-world surveys and research. For example, if a store owner wants to know the average amount customers spend on gifts to stock their inventory appropriately, this interval provides a range that gives them 80% certainty about the true average population spending. It helps in making data-driven decisions while accounting for the uncertainty inherent in sampling.