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A simple random sample is a basic sampling technique where each member of a population has an equal chance of being chosen. It ensures that the sample is unbiased and accurately represents the population. This method is foundational in statistics because it enables researchers to draw general conclusions about a population based on a subset of collected data. To select a simple random sample:
- Define the population clearly.
- Assign a number to each member of the population.
- Use a random number generator to pick members from the population.
This technique is essential for maintaining randomness and reducing selection bias in studies.

A chi-square distribution is a statistical method that helps assess the variance in a data set. It's commonly used when constructing confidence intervals for variance because it handles sample variances derived from normally distributed populations. In our exercise, we use the chi-square distribution to calculate confidence intervals for the population variance. The steps involve:
- Determining degrees of freedom (df), which is typically the sample size minus one (n-1).
- Using chi-square tables or statistical software to find critical values for the desired confidence level (e.g., 90%, 98%).
- Applying these critical values in the formula \(\frac{(n-1)s^2}{\chi^2_{upper}}, \frac{(n-1)s^2}{\chi^2_{lower}}\) to get the confidence interval.
The chi-square distribution is pivotal for understanding how data variability might differ from the expected distribution in hypothesis testing and interval estimation.

The sample size directly influences the confidence interval's width. When sample size increases, the interval tends to get narrower, providing a more precise estimate.
Key effects of sample size include:
- **Larger Sample Sizes:** These yield more reliable and accurate estimates as they reduce the margin of error. Consequently, the confidence interval for the population variance becomes narrower.
- **Smaller Sample Sizes:** These can lead to wider confidence intervals, indicating less precision in the estimate of the population parameter.
In our exercise, increasing the sample size from 20 to 30 resulted in narrower confidence intervals for variance (7.94, 23.70 to 8.59, 22.77). Hence, it's crucial to consider sample size appropriately to ensure the statistical reliability of your results.