Question

A delivery company is logging the number of miles each employee drives each day. The company found an average of about 85.6 miles. Assume that the standard deviation is 9.85 miles and that a random sample of 50 drivers was used to determine the average. Which statements are true about this situation?

Answer 1

To evaluate the statements about the situation involving the delivery company's average miles driven by employees, we can consider some statistical properties, particularly those related to the central limit theorem and the characteristics of the sampling distribution.

1. Sample Size and Central Limit Theorem: Since the sample size is 50 (which is greater than 30), the sampling distribution of the sample mean will be approximately normally distributed due to the Central Limit Theorem, regardless of the distribution of the population.

2. Mean and Standard Error: The mean (μ) of the sampling distribution will be equal to the population mean (85.6 miles). The standard deviation of the sample (σ) is given as 9.85 miles. To find the standard error (SE) of the mean, we can use the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{9.85}{\sqrt{50}} \approx 1.39 \text{ miles} \] This SE will represent the average amount of variability in the sample means.

3. Confidence Intervals and z-scores: We can construct confidence intervals around the sample mean if needed, using the standard error. For instance, for a 95% confidence interval, we would use a z-score of approximately 1.96: \[ CI = \mu \pm z \times SE = 85.6 \pm 1.96 \times 1.39 \] This gives a range of values around the mean to estimate where the true population mean would lie.

4. Normality in Sample Distribution: While the individual miles driven by each driver may or may not be normally distributed, the average miles (mean) from the sample will follow a normal distribution because of the sufficiently large sample size.

5. Interpretation of Standard Deviation: The standard deviation of 9.85 miles describes how much the individual mileages for the drivers vary from the mean on average. It does not directly affect the mean but is essential for understanding the variability.

In summary, the key points that can be derived from this situation, assuming no statements are explicitly provided, include:

• The sampling distribution of the mean of miles driven by drivers is approximately normal.
• The average number of miles driven by the employees is 85.6 miles.
• The standard error of the mean is approximately 1.39 miles.
• Individual driving mileage may vary significantly around the average (mean) driving distance due to the standard deviation.