Math Problem Statement

A delivery company provides expected time of arrival. The company wants to provide an accurate suggestion with a 90% confidence level. If the sample size is 50, the sample average is 35 minutes, the standard deviation is 10 minutes, and the z-score is 1.65, what is the standard error of the mean?

Solution

To calculate the standard error of the mean (SEM), you can use the following formula:

SEM=sn\text{SEM} = \frac{s}{\sqrt{n}}

where:

  • ss is the sample standard deviation,
  • nn is the sample size.

Given:

  • s=10s = 10 minutes,
  • n=50n = 50.

Substitute these values into the formula:

SEM=1050=107.0711.414minutes\text{SEM} = \frac{10}{\sqrt{50}} = \frac{10}{7.071} \approx 1.414 \, \text{minutes}

Answer:

The standard error of the mean (SEM) is approximately 1.414 minutes.

Would you like a detailed explanation of the steps or any clarification?


Here are 5 related questions to explore this topic further:

  1. How would the confidence interval change if the confidence level were 95% instead of 90%?
  2. What would happen to the SEM if the sample size increased to 100?
  3. How is the confidence interval calculated using the SEM and z-score?
  4. Why is the z-score chosen for the 90% confidence level in this problem?
  5. How does increasing the sample size affect the accuracy of the expected time of arrival?

Tip: The standard error decreases as sample size increases, making estimates more precise.

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Math Problem Analysis

Mathematical Concepts

Statistics
Confidence Intervals
Standard Error

Formulas

Standard Error of the Mean (SEM) = s / √n

Theorems

Central Limit Theorem

Suitable Grade Level

Grades 10-12