Mat 510 Homework Assignment 9 Due In Week

Mat 510 Homework Assignment homework Assignment 9due In Week 10 And W

Suppose that there are two (2) candidates (i.e., Jones and Johns) in the upcoming presidential election. Sara notes that she has discussed the presidential election candidates with 15 friends, and 10 said that they are voting for candidate Jones. Sara is therefore convinced that candidate Jones will win the election because Jones gets more than 50% of votes. Answer the following questions in the space provided below:

1. Based on what you now know about statistical inference, is Sara’s conclusion a logical conclusion? Why or why not?

2. How many friend samples Sara should have in order to draw the conclusion with 95% confidence interval? Why?

3. How would you explain your conclusion to Sara without using any statistical jargon? Why?

Paper For Above instruction

The scenario provided involves Sara's interpretation of a small sample survey of her friends concerning their voting intentions for the upcoming presidential election. While her intuition and the data she gathered suggest a majority preference for candidate Jones, a thorough understanding of statistical inference is necessary to assess the validity of her conclusion. This paper explores whether Sara’s conclusion is justified based on her sample size, confidence levels, and how to communicate statistical findings in layman’s terms.

Introduction

Public opinion polling and individual surveys often serve as preliminary indicators of election outcomes. However, drawing reliable conclusions from small samples poses significant challenges. In Sara's case, she surveyed only 15 friends, and 10 of them expressed support for candidate Jones. The question arises whether such a sample can reliably predict the election outcome. Understanding the principles of statistical inference, including sampling variability, confidence intervals, and the limitations of small samples, is vital to evaluating Sara’s conclusion.

Is Sara’s Conclusion a Logical Inference?

Sara’s conclusion that candidate Jones will win based solely on her informal sample is not entirely justified. Statistical inference involves estimating how well a sample represents the entire population, which requires larger and more representative samples. Her sample of 15 friends, with a 66.7% support rate for Jones, provides an initial indication but is subject to considerable sampling error. The small sample size results in a wide margin of error, meaning the true proportion of support among all voters could be significantly different from her observed proportion.

For example, with such a small sample, it is possible that her result is an anomaly or unrepresentative of the larger voter population. Statistical theory demonstrates that the variability of a sample proportion increases as sample size decreases. Therefore, her conclusion that Jones will definitely win is premature. Instead, she should regard her result as suggestive but not definitive evidence of the overall voting trend.

In conclusion, Sara's reasoning lacks the statistical robustness needed to confidently predict an election outcome based solely on her small sample. Proper inference requires considering sampling variability and utilizing larger, more representative samples to minimize error.

Determining the Necessary Sample Size for 95% Confidence

The second question concerns how large a sample Sara would need to draw a conclusion with 95% confidence. To calculate this, we utilize the formula for sample size in estimating proportions:

n = (Z² p (1 - p)) / E²

where:

• Z is the Z-score corresponding to the desired confidence level (for 95%, Z ≈ 1.96)
• p is the estimated proportion supporting Jones (initially 0.667 from her sample)
• E is the margin of error she is willing to accept (e.g., 0.05 for 5%)

Using p = 0.667, Z = 1.96, and an E of 0.05, her required sample size is estimated as:

n = (1.96² 0.667 0.333) / 0.05² ≈ (3.8416 * 0.222) / 0.0025 ≈ 0.852 / 0.0025 ≈ 341

Therefore, Sara would need to survey approximately 341 friends to achieve a 95% confidence level with a margin of error of 5%. This larger sample size reduces sampling variability and provides a more reliable estimate of the true support for Jones.

In essence, increasing sample size improves the precision of the estimate and enhances the confidence she can have in her prediction.

Explaining Results Without Statistical Jargon

To explain the findings to Sara without using technical language, I would say:

"Your small survey gives us a first look at what some people think, but it's like trying to guess the final score of a big game by watching only a few minutes. Just because most of your friends support Jones doesn't mean everyone will. To be more sure about who will win, we need to ask more people, kind of like talking to a bigger crowd. The more people we ask, the closer we'll get to knowing what most voters will do. So, while your current numbers are interesting, they are just a starting point, and we need to check with a bigger group to be more certain."

This way, the idea of sampling, uncertainty, and confidence is conveyed in relatable terms, emphasizing the importance of larger and more representative data for making accurate predictions.

Conclusion

In summary, Sara's conclusion based on her small sample is not entirely reliable due to the inherent variability and potential bias of small samples. To draw a more confident conclusion about the election outcome, she would need a significantly larger sample size—around 341 respondents for a 95% confidence level with acceptable precision. Communicating these concepts in everyday language highlights that predictions become more accurate as more data is collected. Ultimately, while her initial findings suggest a trend, they should be interpreted cautiously until supported by broader, more comprehensive surveys.

References

• DerSimonian, R., & Laird, N. (1986). Meta-analysis in clinical trials. Controlled Clinical Trials, 7(3), 177-188.
• Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver & Boyd.
• Lohr, S. L. (2009). Sampling: Design and Analysis. Cengage Learning.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics. W.H. Freeman.
• Salant, P., & Dillman, D. A. (1994). How to Conduct Your Own Survey: Leading Do-It-Yourself Guides. Wiley.
• Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6(2), 461-464.
• Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics. Pearson.
• Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
• Zar, J. H. (1999). Biostatistical Analysis. Prentice Hall.
• Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences. Pearson.