To construct a 95% confidence interval for the population mean vehicle speed, we will use the following formula for the confidence interval of the mean when the population standard deviation ($\sigma$) is known:

$\text{Confidence Interval} = \left( \bar{x} - Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}, \bar{x} + Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \right)$

Where:

• $\bar{x}$ is the sample mean
• $\sigma$ is the population standard deviation
• $n$ is the sample size
• $Z_{\alpha/2}$ is the Z-score corresponding to the confidence level (for a 95% confidence interval, $Z_{\alpha/2} = 1.96$)

Given:

• Sample mean $\bar{x} = 68.1$ kmph
• Population standard deviation $\sigma = 5.9$ kmph
• Sample size $n = 40$
• Confidence level = 95%, so $Z_{\alpha/2} = 1.96$

Step-by-step Calculation:

1. Standard error of the mean (SE): $SE = \frac{\sigma}{\sqrt{n}} = \frac{5.9}{\sqrt{40}} = \frac{5.9}{6.32} \approx 0.9332$

2. Margin of error (ME): $ME = Z_{\alpha/2} \cdot SE = 1.96 \cdot 0.9332 \approx 1.828$

3. Confidence interval: $\text{Lower limit} = \bar{x} - ME = 68.1 - 1.828 \approx 66.27$ $\text{Upper limit} = \bar{x} + ME = 68.1 + 1.828 \approx 69.93$

Final Confidence Interval:

$(66.27 \, \text{kmph}, 69.93 \, \text{kmph})$

So, the 95% confidence interval for the population mean vehicle speed is approximately (66.27 kmph, 69.93 kmph).

Would you like more details or explanations?

Here are some related questions:

1. How does changing the confidence level affect the confidence interval?
2. What is the importance of the population standard deviation in this calculation?
3. How would the confidence interval change if the sample size were larger?
4. Can you explain how to interpret the 95% confidence interval in a practical sense?
5. What would happen if the vehicle speeds were not normally distributed?

Tip: A wider confidence interval means more uncertainty in estimating the population mean.