Math Problem Statement

The lifetime of a certain type of battery is known to be normally distributed with standard deviation o = 17 hours. A sample of 50 batteries had a mean lifetime of 120.1 hours. It is desired to construct a 95% confidence interval for the mean lifetime for this type of battery. What is the standard error?

Solution

To calculate the standard error (SE) of the sample mean, use the formula:

SE=σnSE = \frac{\sigma}{\sqrt{n}}

Where:

  • σ=17\sigma = 17 hours (the population standard deviation),
  • n=50n = 50 (the sample size).

Step-by-Step Calculation:

SE=σn=1750SE = \frac{\sigma}{\sqrt{n}} = \frac{17}{\sqrt{50}}

507.071\sqrt{50} \approx 7.071

SE177.0712.404SE \approx \frac{17}{7.071} \approx 2.404

Final Answer:

The standard error (SE) is approximately 2.404 hours.


Would you like assistance calculating the confidence interval or more explanation? Here are 5 related questions to expand:

  1. How is the confidence interval constructed for a population mean?
  2. Why is the standard error important in statistical inference?
  3. What is the z-score for a 95% confidence interval, and how is it applied?
  4. How would the sample size affect the width of the confidence interval?
  5. How does the assumption of normality influence this calculation?

Tip: For normally distributed data, knowing the standard deviation helps directly compute the standard error and confidence intervals effectively!

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Math Problem Analysis

Mathematical Concepts

Statistics
Confidence Intervals
Standard Error

Formulas

SE = σ / √n

Theorems

Central Limit Theorem

Suitable Grade Level

Grades 10-12