Ali Is In Charge Of The Dinner Menu For His Senior Prom, And He Wants To Use A One-sample $z$ Interval To Estimate What Proportion Of Seniors Would Order A Vegetarian Option. He Randomly Selects 30 Of The 150 Total Seniors And Finds That 7 Of
Introduction
Ali, the senior prom organizer, is tasked with creating a dinner menu that caters to the diverse tastes of his fellow students. To ensure that the menu is inclusive and appealing to all, he wants to estimate the proportion of seniors who would order a vegetarian option. With a total of 150 seniors, Ali decides to conduct a random sample of 30 students to gather data on their dining preferences. In this article, we will explore how Ali can use a one-sample z interval to estimate the proportion of seniors who would order a vegetarian option.
Understanding the Problem
Ali's goal is to estimate the proportion of seniors who would order a vegetarian option. This can be represented as a probability, denoted by the symbol π (pi). The one-sample z interval is a statistical method used to estimate a population parameter, such as a proportion, based on a random sample of data. In this case, the population parameter is the true proportion of seniors who would order a vegetarian option, denoted by π.
The One-Sample z Interval
The one-sample z interval is a statistical method that provides a range of values within which the true population parameter is likely to lie. The interval is calculated using the following formula:
π̂ ± (Z * √(p̂ * (1-p̂) / n))
where:
- π̂ is the sample proportion
- Z is the z-score corresponding to the desired level of confidence
- p̂ is the sample proportion
- n is the sample size
Calculating the Sample Proportion
To calculate the sample proportion, Ali needs to divide the number of seniors who would order a vegetarian option by the total sample size. In this case, the sample size is 30, and the number of seniors who would order a vegetarian option is 7.
π̂ = 7/30 = 0.2333
Calculating the Standard Error
The standard error is a measure of the variability of the sample proportion. It is calculated using the following formula:
SE = √(p̂ * (1-p̂) / n)
where:
- SE is the standard error
- p̂ is the sample proportion
- n is the sample size
SE = √(0.2333 * (1-0.2333) / 30) = 0.0763
Calculating the z-Score
The z-score is a measure of the number of standard deviations that the sample proportion is away from the population mean. It is calculated using the following formula:
Z = (π̂ - π) / SE
where:
- Z is the z-score
- π̂ is the sample proportion
- π is the population mean
- SE is the standard error
Since we are estimating the population proportion, we will use the sample proportion as an estimate of the population mean.
Z = (0.2333 - 0.2333) / 0.0763 = 0
Calculating the One-Sample z Interval
Now that we have calculated the sample proportion, standard error, and z-score, we can calculate the one-sample z interval. The interval is calculated using the following formula:
π̂ ± (Z * √(p̂ * (1-p̂) / n))
where:
- π̂ is the sample proportion
- Z is the z-score
- p̂ is the sample proportion
- n is the sample size
π̂ ± (0 * √(0.2333 * (1-0.2333) / 30)) = 0.2333 ± 0
Interpretation of the Results
The one-sample z interval provides a range of values within which the true population parameter is likely to lie. In this case, the interval is 0.2333 ± 0, which means that the true population proportion is likely to be 0.2333.
Conclusion
In conclusion, Ali can use a one-sample z interval to estimate the proportion of seniors who would order a vegetarian option. By calculating the sample proportion, standard error, and z-score, Ali can determine the one-sample z interval, which provides a range of values within which the true population parameter is likely to lie. The interval is 0.2333 ± 0, which means that the true population proportion is likely to be 0.2333.
Limitations of the Method
While the one-sample z interval is a useful statistical method for estimating population parameters, it has some limitations. One of the main limitations is that it assumes that the sample is randomly selected from the population. If the sample is not randomly selected, the results may not be representative of the population.
Another limitation of the method is that it assumes that the population is normally distributed. If the population is not normally distributed, the results may not be accurate.
Future Research Directions
Future research directions for this study could include:
- Conducting a larger sample size to increase the accuracy of the results
- Using a more representative sample of the population
- Using a different statistical method, such as a confidence interval, to estimate the population parameter
References
- [1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
- [2] Larson, R. J., & Farber, B. A. (2017). Elementary statistics: Picturing the world. Cengage Learning.
Appendix
The following is a list of the calculations used to determine the one-sample z interval:
| Calculation | Formula | |
|---|---|---|
| Sample proportion | π̂ = 7/30 = 0.2333 | |
| Standard error | SE = √(p̂ * (1-p̂) / n) = 0.0763 | |
| z-score | Z = (π̂ - π) / SE = 0 | |
| One-sample z interval | π̂ ± (Z * √(p̂ * (1-p̂) / n)) = 0.2333 ± 0 |
Introduction
In our previous article, we explored how to use one-sample z intervals to estimate proportions. We discussed the formula for calculating the one-sample z interval, and how to interpret the results. In this article, we will answer some frequently asked questions about one-sample z intervals.
Q: What is the purpose of a one-sample z interval?
A: The purpose of a one-sample z interval is to estimate a population parameter, such as a proportion, based on a random sample of data. It provides a range of values within which the true population parameter is likely to lie.
Q: How do I calculate the sample proportion?
A: To calculate the sample proportion, you need to divide the number of observations in the sample that have a particular characteristic by the total sample size.
Q: What is the standard error, and how do I calculate it?
A: The standard error is a measure of the variability of the sample proportion. It is calculated using the formula: SE = √(p̂ * (1-p̂) / n), where p̂ is the sample proportion and n is the sample size.
Q: What is the z-score, and how do I calculate it?
A: The z-score is a measure of the number of standard deviations that the sample proportion is away from the population mean. It is calculated using the formula: Z = (π̂ - π) / SE, where π̂ is the sample proportion and SE is the standard error.
Q: How do I interpret the results of a one-sample z interval?
A: The one-sample z interval provides a range of values within which the true population parameter is likely to lie. You can interpret the results by looking at the lower and upper bounds of the interval.
Q: What are some common mistakes to avoid when using one-sample z intervals?
A: Some common mistakes to avoid when using one-sample z intervals include:
- Not checking the assumptions of the method, such as normality of the population and random sampling
- Not calculating the standard error correctly
- Not interpreting the results correctly
Q: Can I use one-sample z intervals for other types of data?
A: No, one-sample z intervals are specifically designed for estimating proportions. If you have other types of data, such as means or medians, you will need to use a different statistical method.
Q: How do I choose the level of confidence for my one-sample z interval?
A: The level of confidence is the probability that the true population parameter lies within the interval. Common levels of confidence include 95% and 99%.
Q: Can I use one-sample z intervals for small sample sizes?
A: While one-sample z intervals can be used for small sample sizes, the results may not be as accurate as those obtained with larger sample sizes.
Q: How do I report the results of a one-sample z interval?
A: When reporting the results of a one-sample z interval, you should include the following information:
- The sample proportion
- The standard error
- The z-score
- The one-sample z interval
- The level of confidence
Conclusion
In conclusion, one-sample z intervals are a useful statistical method for estimating proportions. By understanding how to calculate and interpret the results of a one-sample z interval, you can make informed decisions about your data. Remember to check the assumptions of the method, calculate the standard error correctly, and interpret the results correctly.
References
- [1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
- [2] Larson, R. J., & Farber, B. A. (2017). Elementary statistics: Picturing the world. Cengage Learning.
Appendix
The following is a list of the calculations used to determine the one-sample z interval:
| Calculation | Formula |
|---|---|
| Sample proportion | π̂ = 7/30 = 0.2333 |
| Standard error | SE = √(p̂ * (1-p̂) / n) = 0.0763 |
| z-score | Z = (π̂ - π) / SE = 0 |
| One-sample z interval | π̂ ± (Z * √(p̂ * (1-p̂) / n)) = 0.2333 ± 0 |
Glossary
- Sample proportion: The proportion of observations in the sample that have a particular characteristic.
- Standard error: A measure of the variability of the sample proportion.
- z-score: A measure of the number of standard deviations that the sample proportion is away from the population mean.
- One-sample z interval: A range of values within which the true population parameter is likely to lie.
- Level of confidence: The probability that the true population parameter lies within the interval.