Mat 510 Homework Assignment 9 Due In Week 055462

Mat 510 Homework Assignmenthomework 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 presented involves Sara's informal poll of her friends to predict the outcome of the upcoming presidential election. She observes that 10 out of 15 friends support candidate Jones, leading her to believe Jones will win, reasoning from her limited sample that more than 50% of all voters support Jones. This situation provides a foundation to assess concepts of statistical inference, sample size calculation, and effective communication of statistical findings without jargon.

Analysis of Sara’s Conclusion Based on Statistical Inference

Sara's conclusion that Jones will win the election because she observed that 10 out of her 15 friends support Jones is an intuitive but statistically flawed inference. Statistical inference involves making generalizations about a population based on a sample. However, her sample of 15 friends is small and may not accurately represent the wider electorate's preferences, which is essential for reliable predictions.

In statistical terms, her sample size is insufficient for reliable inference at a broader population level. Small samples are susceptible to sampling variability, which can lead to misleading conclusions. For example, in her limited sample, a 66.7% support rate for Jones (10 out of 15) might not be reflective of the true proportion of support among all voters.

Therefore, while her observed support in her small sample suggests a possibility that Jones might win, it is not a definitive or logically sound conclusion without considering larger, more representative samples and the inherent variability in small samples. Statistically, her reasoning ignores potential sampling bias and the need for a confidence interval to gauge the estimate's reliability.

Determining the Appropriate Sample Size for 95% Confidence

To estimate the population support with a high degree of confidence, Sara would need a larger sample size. Sample size calculation for proportions involves factors such as the expected proportion (support for Jones), desired confidence level (95%), and acceptable margin of error.

Assuming Sara expects support around 50% (a conservative estimate that maximizes required sample size), and desiring a margin of error of ±5%, she can use the formula for sample size estimation in proportion studies:

n = (Z^2 p (1 - p)) / E^2

Where:

• n = required sample size
• Z = Z-score for 95% confidence (approximately 1.96)
• p = estimated proportion support (assumed 0.5)
• E = margin of error (0.05)

Calculating:

n = (1.96^2 0.5 0.5) / 0.05^2 ≈ (3.8416 * 0.25) / 0.0025 ≈ 0.9604 / 0.0025 ≈ 384.16

Thus, Sara would need to survey approximately 385 people to estimate the support for Jones within a ±5% margin of error at 95% confidence. If she tolerates a larger margin of error, the required sample size could be smaller, but 385 represents a balanced, reliable estimate.

Explaining the Findings to Sara in Non-Technical Terms

To help Sara understand why she needs to look beyond her small group of friends, I would explain that asking only 15 people is like trying to judge a large crowd’s opinion by talking to just a few individuals—sometimes they might not represent everyone accurately. To be more confident that her guess about who will win is correct, she needs to ask more people, enough to give a clear picture of the overall opinion.

I would say that if she talks to around 385 people, she can be more certain that her estimate of how many support Jones is close to what most people think. This bigger number helps reduce the chances of being misled by a few friends who may have different opinions from the general population.

In short, the more people she asks, the more trustworthy her prediction becomes. Just like polling a larger group gives a better idea of the overall opinion, asking a bigger sample allows her to be more confident about her prediction for the election.

Conclusion

Sara's initial conclusion based on her small sample is not statistically sound because small samples are prone to variability and may not accurately reflect the larger population. To make a reliable prediction, she should survey a much larger group—around 385 people—at a 95% confidence level, to ensure her estimate is both precise and dependable. Communicating this to Sara in simple language underscores the importance of sample size in making accurate predictions, emphasizing that broader input leads to more trustworthy conclusions in statistical reasoning.

References

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• Groves, R. M., et al. (2009). Survey Methodology. Wiley Series in Survey Methodology.
• Measuring Support in Elections (2020). Pew Research Center. https://www.pewresearch.org
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