To calculate the 95% confidence interval for the population mean, we can use the formula for the confidence interval when the population standard deviation is known.
The formula for the confidence interval at a given confidence level is:
\[ \text{CI} = \bar{x} \pm z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \(\bar{x}\) is the sample mean,
- \(z\) is the z-score corresponding to the desired confidence level,
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.
Step 1: Calculate the Sample Mean \(\bar{x}\)
First, let's calculate the sample mean (\(\bar{x}\)) from the given data:
- We have:
- 10 people said 15 minutes,
- 18 people said 30 minutes,
- 12 people said 25 minutes.
The total number of responses \(n\) is: \[ n = 10 + 18 + 12 = 40 \]
Now we calculate the weighted total time: \[ \text{Total time} = (10 \times 15) + (18 \times 30) + (12 \times 25) \] \[ = 150 + 540 + 300 = 990 \text{ minutes} \]
Now, calculate the sample mean \(\bar{x}\): \[ \bar{x} = \frac{\text{Total time}}{n} = \frac{990}{40} = 24.75 \text{ minutes} \]
Step 2: Identify the z-score for 95% Confidence Level
For a 95% confidence level, the z-score is approximately \(z = 1.96\).
Step 3: Use the Known Population Standard Deviation and Calculate the Confidence Interval
Given that the population standard deviation \(\sigma\) is 5.64 minutes:
Now calculate the standard error (SE): \[ \text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{5.64}{\sqrt{40}} \approx \frac{5.64}{6.32} \approx 0.89 \text{ minutes} \]
Step 4: Calculate the Margin of Error
Now, calculate the margin of error (ME): \[ \text{ME} = z \cdot \text{SE} = 1.96 \cdot 0.89 \approx 1.74 \text{ minutes} \]
Step 5: Construct the Confidence Interval
Finally, we can construct the 95% confidence interval: \[ \text{CI} = \bar{x} \pm \text{ME} = 24.75 \pm 1.74 \] Calculating the bounds: \[ \text{Lower bound} = 24.75 - 1.74 \approx 23.01 \text{ minutes} \] \[ \text{Upper bound} = 24.75 + 1.74 \approx 26.49 \text{ minutes} \]
Thus, the 95% confidence interval for the population mean is approximately:
\[ \boxed{(23.01, 26.49)} \text{ minutes} \]