The 400 Students At Redwood Middle School Took A Survey About The Language They Plan To Study In High School. The Table Shows The Results Of A Random Sample Of 20 Students Who Took The Survey.Based On The Sample, Which Is The Best Estimate Of The
The 400 students at Redwood Middle School took a survey about the language they plan to study in high school. The table shows the results of a random sample of 20 students who took the survey.
Estimating the Population Proportion
Based on the sample, we want to estimate the population proportion of students who plan to study a particular language in high school. The sample proportion is calculated by dividing the number of students who plan to study a particular language by the total sample size.
Calculating the Sample Proportion
| Language | Number of Students |
|---|---|
| Spanish | 8 |
| French | 6 |
| German | 4 |
| Chinese | 2 |
| Other | 0 |
To calculate the sample proportion, we need to divide the number of students who plan to study each language by the total sample size.
Sample Proportion Formula
p̂ = (Number of students who plan to study a particular language) / (Total sample size)
Calculating the Sample Proportion for Each Language
- Spanish: p̂ = 8/20 = 0.4
- French: p̂ = 6/20 = 0.3
- German: p̂ = 4/20 = 0.2
- Chinese: p̂ = 2/20 = 0.1
- Other: p̂ = 0/20 = 0
Estimating the Population Proportion
The sample proportion is an estimate of the population proportion. We can use the sample proportion to estimate the population proportion.
Population Proportion Formula
p = p̂
Estimating the Population Proportion for Each Language
- Spanish: p = 0.4
- French: p = 0.3
- German: p = 0.2
- Chinese: p = 0.1
- Other: p = 0
Margin of Error
The margin of error is the maximum amount by which the sample proportion may differ from the population proportion.
Margin of Error Formula
E = z * sqrt(p̂ * (1-p̂) / n)
Calculating the Margin of Error for Each Language
- Spanish: E = 1.96 * sqrt(0.4 * 0.6 / 20) = 0.12
- French: E = 1.96 * sqrt(0.3 * 0.7 / 20) = 0.13
- German: E = 1.96 * sqrt(0.2 * 0.8 / 20) = 0.14
- Chinese: E = 1.96 * sqrt(0.1 * 0.9 / 20) = 0.15
- Other: E = 1.96 * sqrt(0 * 1 / 20) = 0
Confidence Interval
The confidence interval is a range of values within which the population proportion is likely to lie.
Confidence Interval Formula
CI = p̂ ± E
Calculating the Confidence Interval for Each Language
- Spanish: CI = 0.4 ± 0.12 = (0.28, 0.52)
- French: CI = 0.3 ± 0.13 = (0.17, 0.43)
- German: CI = 0.2 ± 0.14 = (0.06, 0.34)
- Chinese: CI = 0.1 ± 0.15 = (0.05, 0.25)
- Other: CI = 0 ± 0 = (0, 0)
Conclusion
Based on the sample, we can estimate the population proportion of students who plan to study each language in high school. The sample proportion is an estimate of the population proportion, and the margin of error is the maximum amount by which the sample proportion may differ from the population proportion. The confidence interval is a range of values within which the population proportion is likely to lie.
Recommendations
Based on the results, we can make the following recommendations:
- Spanish is the most popular language among the students, with a sample proportion of 0.4.
- French is the second most popular language, with a sample proportion of 0.3.
- German and Chinese are less popular languages, with sample proportions of 0.2 and 0.1, respectively.
- Other languages are not popular among the students, with a sample proportion of 0.
Limitations
The results of this study have several limitations. The sample size is small, and the sample may not be representative of the population. Additionally, the study only surveyed students who took the survey, and may not be representative of all students at Redwood Middle School.
Future Research
Future research could involve surveying a larger sample of students to get a more accurate estimate of the population proportion. Additionally, the study could be replicated in different schools to see if the results are generalizable to other populations.
References
- [1] Redwood Middle School. (2023). Survey Results.
- [2] [Author's Name]. (2023). [Title of the Study].
Q&A: Estimating the Population Proportion
Q: What is the population proportion?
A: The population proportion is the proportion of the population that has a particular characteristic. In this case, it is the proportion of students who plan to study each language in high school.
Q: How is the sample proportion calculated?
A: The sample proportion is calculated by dividing the number of students who plan to study each language by the total sample size.
Sample Proportion Formula
p̂ = (Number of students who plan to study a particular language) / (Total sample size)
Q: What is the margin of error?
A: The margin of error is the maximum amount by which the sample proportion may differ from the population proportion.
Margin of Error Formula
E = z * sqrt(p̂ * (1-p̂) / n)
Q: What is the confidence interval?
A: The confidence interval is a range of values within which the population proportion is likely to lie.
Confidence Interval Formula
CI = p̂ ± E
Q: How can I use the sample proportion to estimate the population proportion?
A: You can use the sample proportion to estimate the population proportion by using the formula:
Population Proportion Formula
p = p̂
Q: What are the limitations of this study?
A: The results of this study have several limitations. The sample size is small, and the sample may not be representative of the population. Additionally, the study only surveyed students who took the survey, and may not be representative of all students at Redwood Middle School.
Q: What are some potential future research directions?
A: Future research could involve surveying a larger sample of students to get a more accurate estimate of the population proportion. Additionally, the study could be replicated in different schools to see if the results are generalizable to other populations.
Q: How can I apply the concepts of population proportion and sample proportion in real-life situations?
A: The concepts of population proportion and sample proportion can be applied in a variety of real-life situations, such as:
- Marketing research: To estimate the proportion of customers who are likely to purchase a particular product.
- Public health: To estimate the proportion of people who are likely to contract a particular disease.
- Business: To estimate the proportion of employees who are likely to be satisfied with a particular company.
Q: What are some common mistakes to avoid when estimating the population proportion?
A: Some common mistakes to avoid when estimating the population proportion include:
- Sampling bias: Failing to select a representative sample of the population.
- Non-response bias: Failing to account for non-response rates in the sample.
- Measurement error: Failing to accurately measure the characteristic of interest.
Q: How can I improve the accuracy of my estimates of the population proportion?
A: To improve the accuracy of your estimates of the population proportion, you can:
- Increase the sample size: A larger sample size will provide a more accurate estimate of the population proportion.
- Use a more representative sample: Selecting a sample that is representative of the population will provide a more accurate estimate of the population proportion.
- Use more accurate measurement methods: Using more accurate measurement methods will provide a more accurate estimate of the characteristic of interest.