Homework Assignment 9 Due In Week 10 And Worth 30 Points

Homework Assignment 9due In Week 10 And Worth 30 Pointssuppose That Th

Homework Assignment 9 due in week 10 and worth 30 points. Suppose that there are two candidates (Jones and Johns) in the upcoming presidential election. Sara has discussed the election with 15 friends, and 10 of them said they are voting for Jones. Sara believes that Jones will win the election because he has more than 50% of the votes based on this discussion.

Answer the following questions:

1. Is Sara’s conclusion a logical inference based on what you know about statistical inference? Why or why not?

2. How many friends should Sara sample to confidently conclude with a 95% confidence interval that Jones will win? Why?

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

Paper For Above instruction

In approaching Sara’s conclusion about the presidential election, it is crucial to understand the principles of statistical inference and how small sample sizes impact the reliability of such conclusions. While Sara observes that 10 out of 15 friends support Jones, this percentage suggests a 66.7% support rate within her sample. However, drawing a definitive conclusion about the entire population of voters—who are much larger and more diverse—requires careful statistical analysis, which involves sampling methods, confidence intervals, and understanding margin of error.

Firstly, Sara’s conclusion that Jones will win based on her small sample is not entirely logical in the context of statistical inference. Her sample size of 15 friends is quite limited and may not accurately reflect the broader voter population. The sample might be biased or unrepresentative because friends tend to share similar perspectives, demographics, or social environments, and these factors can skew results. According to statistical theory, in order to make a reliable generalization about an entire population, a sufficiently large and randomly selected sample is necessary. Small samples tend to have higher variability, meaning that the results are less stable and more likely to fluctuate from the true population value.

Secondly, to estimate how many people Sara should sample to be 95% confident that her results accurately reflect the entire voter population, we must consider the margin of error and the size of the population. Using the standard sample size formula for proportions, the sample size (n) can be calculated considering the estimated proportion (p), the desired confidence level (usually 95%), and a margin of error (e). Assuming Sara expects support for Jones to be around 50% (which provides the maximum sample size), and aiming for a 5% margin of error, the calculation is as follows:

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

Where Z is the z-score corresponding to 95% confidence (approximately 1.96). Plugging in the numbers: n = (1.96^2 0.5 0.5) / 0.05^2 ≈ 384.16. Therefore, Sara should survey approximately 385 friends to be 95% confident that her estimate of support for Jones in the entire population is within ±5%. This larger sample size reduces the likelihood of sampling error and offers a more reliable basis for predicting election outcomes.

Finally, to explain these concepts to Sara without statistical jargon, I would say: “Imagine you want to know whether most people like a new product. If you ask only 15 friends, their opinions might not match everyone’s because they might all be from similar areas or share certain views. To be more sure about what most people think, you need to ask more friends—closer to 385 people—so that your guess is more likely to be accurate for everyone. The more people you ask, the more confident you can be that your opinion reflects the true situation.”

In conclusion, Sara’s initial conclusion based on her small sample isn’t very reliable. To make a confident prediction about the election, she needs to survey a much larger group of friends using proper sampling methods. Explaining this process in simple terms helps clarify why larger, more representative samples lead to more trustworthy conclusions.

References

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