- Calculate the sample mean: \(\bar{x} = 9.07\).
- Find the sample standard deviation: \(s \approx 2.66\).
- Determine the margin of error using the t-distribution: \(ME = t_{0.025, 9} \times \frac{s}{\sqrt{n}} \approx 1.90\).
- The 95% confidence interval is \(\bar{x} \pm ME\), resulting in \(\boxed{7.17 \text{ to } 10.97 \text{ minutes}}\).
Explanation
- Problem Analysis and Methodology Selection
To find the 95% confidence interval for the population mean time, we first analyze our sample data. The dataset consists of \(n = 10\) measurements: \(7.0, 12.8, 11.1, 7.7, 8.0, 13.0, 7.2, 7.9, 10.8, 5.2\). Because the population standard deviation is unknown and the sample size is small, we use the t-distribution with \(df = n - 1 = 9\) degrees of freedom.
- Calculating Sample Statistics
First, we calculate the sample mean \(\bar{x}\):
\[\bar{x} = \frac{7.0 + 12.8 + 11.1 + 7.7 + 8.0 + 13.0 + 7.2 + 7.9 + 10.8 + 5.2}{10} = \frac{90.7}{10} = 9.07\]
Next, we calculate the sample standard deviation \(s\):
\[s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \approx 2.6588\]
These values describe the center and spread of our sample.
- Determining the Margin of Error
For a 95% confidence level with \(df = 9\), we look up or calculate the critical t-value, \(t_{\alpha/2}\), where \(\alpha = 0.05\). The t-value \(t_{0.025, 9}\) is approximately \(2.262\).
Now, we calculate the standard error of the mean (SE):
\[SE = \frac{s}{\sqrt{n}} = \frac{2.6588}{\sqrt{10}} \approx 0.8408\]
Finally, the margin of error (ME) is:
\[ME = t_{0.025, 9} \times SE = 2.262 \times 0.8408 \approx 1.902\]
- Constructing the Confidence Interval
The confidence interval is constructed as \([\bar{x} - ME, \bar{x} + ME]\):
\[\text{Lower Bound} = 9.07 - 1.902 = 7.168\]
\[\text{Upper Bound} = 9.07 + 1.902 = 10.972\]
Rounding these to two decimal places, we get the interval from \(7.17\) to \(10.97\) minutes. This matches the second provided option. Therefore, the 95% confidence interval is \(\boxed{7.17 \text{ to } 10.97 \text{ minutes}}\).
Examples
Confidence intervals are essential for coaches or analysts who want to estimate the average performance time of a group while acknowledging the uncertainty that comes from a small sample size. By calculating this interval, a coach can state with 95% confidence that the true average drill time for all players likely falls within the range of \([7.17, 10.97]\) minutes. This helps in setting realistic expectations for training benchmarks.