In a study, the heights of individuals in a large population are normally 
distributed with a population variance of σ
2 = 36. A researcher takes a random 
sample of 49 individuals from this population and calculates the sample mean and 
sample variance. Assuming that the sample provides an unbiased estimate of the 
population, what is the expected value of the sample variance s
2
?
 a. 49
 b. 6
 c. 36
 d. 7

Given the problem, we know the following:

- The population variance \( \sigma^2 = 36 \),
- The sample size \( n = 49 \),
- The sample provides an unbiased estimate of the population.

### Key Concepts:
- The sample variance \( s^2 \) is an unbiased estimator of the population variance \( \sigma^2 \). Therefore, the expected value of the sample variance is equal to the population variance.

That is:

\[
\mathbb{E}[s^2] = \sigma^2
\]

### Answer:
Since the population variance \( \sigma^2 \) is 36, the expected value of the sample variance \( s^2 \) is also:

\[
\mathbb{E}[s^2] = 36
\]

Thus, the correct answer is:

**c. 36**

---

Would you like more details or have any questions about this? 

Here are 5 related questions to consider:
1. How does sample size affect the variance of a sample?
2. What is the difference between population variance and sample variance?
3. How do you calculate sample variance from raw data?
4. Why is the sample variance an unbiased estimator of the population variance?
5. How does increasing the sample size impact the accuracy of the estimate of population variance?

**Tip:** When calculating sample variance, remember to divide by \(n-1\), not \(n\), to get an unbiased estimate.

In a study, the heights of individuals in a large population are normally distributed with a population variance of σ² = 36. A researcher takes a random sample of 49 individuals from this population and calculates the sample mean and sample variance. Assuming that the sample provides an unbiased estimate of the population, what is the expected value of the sample variance s²? a. 49 b. 6 c. 36 d. 7

Explore the expected value of the sample variance for a normally distributed population with a variance of σ² = 36. This problem explains key statistical concepts like sample variance, population variance, and unbiased estimators. Suitable for college-level statistics students.

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