Understanding the Random Condition

Oct 13, 2025 by windailys.com 13 views

Introduction

Leon, a high school student, wants to estimate the proportion of seniors at his school who like to watch football. To achieve this, he conducts a simple random sample of 50 seniors out of the total 200 seniors on campus. After interviewing them, he finds that 32 seniors like to watch football. In this article, we will discuss whether the random condition for finding the sample is met and how Leon can proceed with his analysis.

Understanding the Random Condition

The random condition, also known as the random sampling condition, is a fundamental requirement for making inferences about a population based on a sample. It states that every member of the population has an equal chance of being selected for the sample. In other words, the sample should be representative of the population, and the selection of the sample should be free from bias.

Checking the Random Condition

To check if the random condition is met, we need to examine the method used by Leon to select the sample. Since Leon conducted a simple random sample, we can assume that he used a random number generator or a random selection method to choose the 50 seniors from the total 200 seniors on campus.

Calculating the Probability of Selection

To further verify the random condition, we can calculate the probability of selection for each senior. Since there are 200 seniors in total, and 50 were selected for the sample, the probability of selection for each senior is:

P(selection) = (Number of selected seniors) / (Total number of seniors) = 50 / 200 = 0.25

This means that each senior has a 25% chance of being selected for the sample.

Testing for Randomness

To test for randomness, we can use statistical tests such as the chi-squared test or the Kolmogorov-Smirnov test. However, since we are dealing with a simple random sample, we can assume that the sample is representative of the population.

Conclusion

Based on the information provided, it appears that Leon has met the random condition for finding the sample. The simple random sample of 50 seniors was selected from the total 200 seniors on campus, and each senior had an equal chance of being selected. Therefore, we can proceed with the analysis to estimate the proportion of seniors who like to watch football.

Estimating the Proportion

To estimate the proportion of seniors who like to watch football, we can use the sample proportion formula:

p̂ = (Number of seniors who like to watch football) / (Total number of seniors in the sample) = 32 / 50 = 0.64

This means that approximately 64% of the seniors in the sample like to watch football.

Margin of Error

To calculate the margin of error, we can use the formula:

Margin of Error = (Z-score) * (Standard Error) = (1.96) * (sqrt((p̂ * (1-p̂)) / n)) = (1.96) * (sqrt((0.64 * (1-0.64)) / 50)) = 0.12

This means that the margin of error is approximately 12%.

Confidence Interval

To construct a confidence interval, we can use the formula:

CI = p̂ ± (Margin of Error) = 0.64 ± 0.12 = (0.52, 0.76)

This means that we can be 95% confident that the true proportion of seniors who like to watch football is between 52% and 76%.

Conclusion

Q: What is the purpose of estimating the proportion of seniors who like to watch football?

A: The purpose of estimating the proportion of seniors who like to watch football is to understand the preferences and interests of the senior population at Leon's high school. This information can be useful for planning events, activities, and resources that cater to the interests of the seniors.

Q: What is a simple random sample, and how is it different from other types of samples?

A: A simple random sample is a type of sample where every member of the population has an equal chance of being selected. This is different from other types of samples, such as stratified samples or cluster samples, where the selection process is more complex.

Q: How do you calculate the probability of selection for each senior in a simple random sample?

A: To calculate the probability of selection for each senior in a simple random sample, you divide the number of selected seniors by the total number of seniors in the population. For example, if 50 seniors are selected from a total of 200 seniors, the probability of selection for each senior is 50/200 = 0.25.

Q: What is the margin of error, and how is it used in estimating the proportion of seniors who like to watch football?

A: The margin of error is a measure of the maximum amount by which the sample proportion may differ from the true population proportion. It is used to construct a confidence interval, which provides a range of values within which the true population proportion is likely to lie.

Q: How do you calculate the margin of error for a simple random sample?

A: To calculate the margin of error for a simple random sample, you use the formula: Margin of Error = (Z-score) * (Standard Error), where the Z-score is a value that depends on the desired level of confidence, and the Standard Error is a measure of the variability of the sample proportion.

Q: What is the confidence interval, and how is it used in estimating the proportion of seniors who like to watch football?

A: The confidence interval is a range of values within which the true population proportion is likely to lie. It is constructed by adding and subtracting the margin of error from the sample proportion. For example, if the sample proportion is 0.64 and the margin of error is 0.12, the 95% confidence interval is (0.52, 0.76).

Q: How do you determine the sample size for a simple random sample?

A: The sample size for a simple random sample is determined by the desired level of precision and the expected variability of the population proportion. A larger sample size provides more precise estimates, but it also requires more resources and time.

Q: What are some common pitfalls to avoid when estimating the proportion of seniors who like to watch football?

A: Some common pitfalls to avoid when estimating the proportion of seniors who like to watch football include:

Not meeting the random condition, which can lead to biased estimates
Not accounting for non-response or missing data, which can lead to underestimation or overestimation of the true population proportion
Not using a sufficient sample size, which can lead to imprecise estimates
Not considering the potential for sampling error, which can lead to incorrect conclusions

Q: How do you ensure that the sample is representative of the population?

A: To ensure that the sample is representative of the population, you should:

Use a random sampling method to select the sample
Ensure that the sample is large enough to provide reliable estimates
Account for non-response or missing data
Consider the potential for sampling error
Use statistical methods to analyze the data and draw conclusions.