To calculate the 98% confidence interval for the mean commute time using the provided data, we can follow these steps:

Step 1: Identify the parameters.

• Standard deviation (σ): 4.5 minutes
• Sample size (n): 104
• Confidence level: 98%
• Sample mean ($\bar{x}$): Calculated from the provided data.

Step 2: Calculate the sample mean ($\bar{x}$).

To calculate the sample mean, sum all the provided commute times and divide by the total number of observations (104). I'll calculate it first.

Step 3: Calculate the standard error of the mean (SE).

The formula for the standard error (SE) is:

$SE = \frac{\sigma}{\sqrt{n}}$

Where:

• $\sigma = 4.5$ (standard deviation)
• $n = 104$ (sample size)

Step 4: Find the z-value for a 98% confidence level.

For a 98% confidence interval, the z-value (z*) corresponds to the critical value for a two-tailed distribution. This can be found using statistical tables or Excel, and it is approximately 2.33 for 98% confidence.

Step 5: Calculate the margin of error (ME).

The margin of error is:

$ME = z* \times SE$

Step 6: Calculate the confidence interval.

The confidence interval (CI) is given by:

$\text{CI} = \bar{x} \pm ME$

Now, I’ll calculate the sample mean, standard error, margin of error, and the final confidence interval.

Let's proceed with the calculations. numpy as np

Provided data (commute times)

commute_times = [ 21, 15, 35, 40, 40, 34, 32, 11, 18, 35, 9, 31, 15, 29, 41, 37, 40, 10, 31, 14, 39, 14, 11, 33, 38, 21, 35, 33, 34, 27, 31, 26, 35, 27, 32, 18, 40, 17, 32, 38, 28, 11, 21, 30, 33, 12, 24, 23, 24, 37, 30, 28, 10, 13, 33, 10, 10, 14, 41, 15, 36, 11, 13, 38, 39, 15, 37, 39, 41, 11, 12, 35, 16, 41, 34, 29, 36, 32, 20, 27, 34, 32, 25, 16, 34, 13, 29, 27, 33, 20, 17, 40, 28, 23, 34, 12, 33, 37, 39, 36, 11, 42, 33, 20 ]

Known values

sigma = 4.5 # Standard deviation n = len(commute_times) # Sample size

Step 1: Calculate the sample mean

sample_mean = np.mean(commute_times)

Step 2: Calculate the standard error of the mean (SE)

SE = sigma / np.sqrt(n)

Step 3: Find the z-value for a 98% confidence level

z_value = 2.33 # For a 98% confidence interval

Step 4: Calculate the margin of error (ME)

ME = z_value * SE

Step 5: Calculate the confidence interval

CI_lower = sample_mean - ME CI_upper = sample_mean + ME

sample_mean, SE, ME, CI_lower, CI_upperThe calculations for the 98% confidence interval for the mean commute time are as follows:

• Sample Mean ($\bar{x}$): 26.9 minutes (rounded to one decimal place)
• Standard Error (SE): 0.44 minutes
• Margin of Error (ME): 1.03 minutes
• 98% Confidence Interval: [25.9, 27.9] minutes

This means that we can be 98% confident that the true mean commute time lies between 25.9 and 27.9 minutes.

Would you like more details on any part of this calculation, or do you have any questions? Here are some related topics you might find interesting:

1. How to calculate the margin of error for different confidence levels.
2. The difference between confidence intervals for population and sample means.
3. How to perform a similar calculation with a t-distribution when the population standard deviation is unknown.
4. Why we use the z-distribution in this case despite having a sample.
5. How to interpret the results of a hypothesis test in the context of confidence intervals.

Tip: Always check if the sample size is large enough (typically $n \geq 30$) to apply the z-distribution, as it helps simplify calculations.