The logistics company mentioned earlier has been collecting data on the delivery times for a specific route. A sample of 500 delivery times was taken, and the sample mean was found to be 90 minutes, with a sample standard deviation of 15 minutes. Assume that the delivery times are normally distributed. A-)Calculate a 95% Confidence Interval for the Mean Delivery Time. answer this with standard normal didtribution table

To calculate a 95% Confidence Interval (CI) for the mean delivery time, we will use the following steps, based on the formula for the CI of a normally distributed population mean when the population standard deviation is unknown (but the sample size is large enough):

### Confidence Interval Formula:
\[
CI = \bar{X} \pm Z_{\alpha/2} \cdot \frac{s}{\sqrt{n}}
\]
Where:
- \(\bar{X}\) = sample mean = 90 minutes
- \(s\) = sample standard deviation = 15 minutes
- \(n\) = sample size = 500
- \(Z_{\alpha/2}\) = Z-value corresponding to the desired confidence level (from the standard normal distribution table). For a 95% confidence level, \(Z_{\alpha/2} = 1.96\).

### Step 1: Calculate the Standard Error (SE)
The standard error of the sample mean is given by:
\[
SE = \frac{s}{\sqrt{n}} = \frac{15}{\sqrt{500}} \approx \frac{15}{22.36} \approx 0.671
\]

### Step 2: Calculate the Confidence Interval
Now, we apply the formula:
\[
CI = 90 \pm 1.96 \cdot 0.671
\]
\[
CI = 90 \pm 1.315
\]

### Step 3: Calculate the Lower and Upper Bounds
- Lower bound: \(90 - 1.315 = 88.685\)
- Upper bound: \(90 + 1.315 = 91.315\)

### Final Answer:
The 95% confidence interval for the mean delivery time is approximately:
\[
(88.685, 91.315) \, \text{minutes}
\]

This means we are 95% confident that the true mean delivery time lies between 88.685 minutes and 91.315 minutes.

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Would you like more details or have any further questions? Here are some additional questions you might find useful:

1. How does changing the confidence level (e.g., from 95% to 99%) affect the confidence interval?
2. What is the effect of increasing the sample size on the width of the confidence interval?
3. How would the confidence interval change if the sample standard deviation increased to 20 minutes?
4. Why do we use the standard normal distribution (Z-distribution) in this case instead of the t-distribution?
5. What if the delivery times were not normally distributed—would the confidence interval calculation still be valid?

**Tip**: Confidence intervals become narrower when the sample size increases because larger samples provide more accurate estimates of the population mean.

The logistics company mentioned earlier has been collecting data on the delivery times for a specific route. A sample of 500 delivery times was taken, and the sample mean was found to be 90 minutes, with a sample standard deviation of 15 minutes. Assume that the delivery times are normally distributed. Calculate a 95% Confidence Interval for the Mean Delivery Time.

Learn how to calculate a 95% Confidence Interval for the mean delivery time of a logistics company's sample of 500 delivery times. The sample mean is 90 minutes with a standard deviation of 15 minutes. Use statistical concepts like confidence intervals and normal distribution to determine the accuracy of the mean delivery time.

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