Darsh’s simulated samples:

Sample 1 (size 50)

Sample 1 mean: 15.4

Sample 2 (size 50)

Sample 2 mean: 11.8

Sample 3 (size 100)

Sample 3 mean: 16.1

Sample 4 (size 100)

Sample 4 mean: 16.3

Darsh is trying to determine the average number of hours the students at his school study each week. He used a random number generator to simulate the possible results. His samples are shown.

If the school has 3,000 students, which sample size is more appropriate?




In order to determine which sample size is more appropriate, Darsh should consider **statistical reliability** and **accuracy** of his estimates. When the sample size is larger, the estimate of the population mean tends to be more reliable, because the **sampling variability** (or standard error) decreases.

Here’s a breakdown of the factors:

1. **Sample Size**:
   - **Larger sample sizes (e.g., Sample 3 and Sample 4 with 100 students)** provide more precise estimates of the population mean compared to smaller sample sizes (e.g., Sample 1 and Sample 2 with 50 students).
   - With a school population of 3,000 students, larger sample sizes would better represent the population.

2. **Mean Values**:
   - Even though Samples 1 and 2 have distinct means (15.4 and 11.8), the relatively small sample size (50) may lead to more variability and potential error in estimating the true population mean.
   - Samples 3 and 4, with a larger size (100), yield means that are much closer to each other (16.1 and 16.3). This suggests more stability and accuracy in capturing the true average.

### Conclusion:
For a population size of 3,000 students, a sample size of **100 students (like Samples 3 and 4)** would be more appropriate, as larger sample sizes tend to give a better estimate of the population mean and reduce sampling error. Darsh should opt for a larger sample size for more reliable results.

Would you like details on calculating the accuracy (e.g., margin of error) or other related aspects?

Here are some related questions:
1. How do you calculate the standard error for different sample sizes?
2. What is the impact of increasing sample size on the margin of error?
3. How do random number generators help in simulating sample data?
4. What is the significance of sample means in determining population characteristics?
5. How can Darsh ensure that his samples are random and unbiased?

**Tip**: Always check the **confidence intervals** when comparing sample means to understand how precise the estimates are!

Darsh’s simulated samples:

Sample 1 (size 50)
Sample 1 mean: 15.4
Sample 2 (size 50)
Sample 2 mean: 11.8
Sample 3 (size 100)
Sample 3 mean: 16.1
Sample 4 (size 100)
Sample 4 mean: 16.3

Darsh is trying to determine the average number of hours the students at his school study each week. He used a random number generator to simulate the possible results. His samples are shown.

If the school has 3,000 students, which sample size is more appropriate?

Explore how to determine the most appropriate sample size for estimating the average study hours of students in a school of 3,000. Learn about statistical concepts like mean, sampling, and standard error. This guide is perfect for high school students (Grades 9-12) looking to understand the impact of sample size on reliability and accuracy.

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