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I see the table from your image, which includes two statistical problems, and you're likely looking for help filling out specific columns related to variance and standard error formulas.

### Here's how we can fill in the table:

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#### **1. Physics Quiz Completion Time**
- **Given Information (3 points):**
  - Average completion time: \( \mu = 25 \) minutes
  - Variance: \( \sigma^2 = 6.25 \) minutes²
  - Sample size: \( n = 30 \) students

- **Variance is Known/Unknown (1 point):**
  - **Known**, since the variance \( \sigma^2 = 6.25 \) is provided.

- **Standard Error Formula to be used (1 point):**
  - When variance is known, the standard error (SE) for the sample mean is calculated as:
    \[
    SE = \frac{\sigma}{\sqrt{n}} = \frac{\sqrt{6.25}}{\sqrt{30}} = \frac{2.5}{\sqrt{30}}
    \]

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#### **2. Random Sampling from a Population**
- **Given Information (3 points):**
  - Population size: \( N = 15 \)
  - Sample size: \( n = 6 \)
  - The population measurements are: 12, 11, 15, 8, 8, 9, 12, 15, 14, 13, 12, 11, 13, 12, 10

- **Variance is Known/Unknown (1 point):**
  - **Unknown**, because the sample is drawn from the population and the variance needs to be calculated from the sample.

- **Standard Error Formula to be used (1 point):**
  - Since it's sampling without replacement, and the population variance is unknown, we use:
    \[
    SE = \sqrt{\frac{(N - n)}{N - 1}} \times \frac{s}{\sqrt{n}}
    \]
  Where \( s \) is the sample standard deviation, \( N = 15 \), and \( n = 6 \).

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### Would you like me to go further into calculations or clarify anything? Let me know! Also, here are some questions that may expand your understanding:

1. How does the formula for standard error change when variance is unknown?
2. Why is the population size important when sampling without replacement?
3. How would you calculate the sample variance for the population in the second problem?
4. What assumptions are made when using the standard error formula for the first problem?
5. How would the results differ if the population variance was unknown in the first scenario?

**Tip:** Knowing whether the variance is known or unknown dictates whether you use a z-distribution or t-distribution for further hypothesis testing!

1. A college professor claims that the average time it takes a group of students to complete a physics quiz is 25 minutes, with a variance of 6.25 minutes squared. The professor selects 30 students to verify this claim. 2. A population composed of 15 items with different values is sampled randomly without replacement. Samples of 6 items are drawn.

Learn how to solve statistical problems involving the standard error and variance when analyzing physics quiz completion times and population sampling. Includes step-by-step formulas for known and unknown variances, sampling without replacement, and finite population correction.

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