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Sample Statistics and Population Parameters Question 1: a) The formula for the confidence interval for a sample proportion is: The point estimate (or sample proportion) z-score standard error. This translates to: (Dies et al., 2019). b) To ensure that the confidence interval captures the population parameter, we must widen the confidence interval. By doing this, we include a greater range of plausible values, increasing the possibility of capturing the population parameter. To widen the confidence interval, we must increase the confidence level. Another way to ensure the confidence interval captures the population parameter is to increase the sample size. This would narrow the confidence interval and decrease the standard error. Therefore, there would be less variability, which increases the possibility that the population parameter is captured in the confidence interval (Dies et al., 2019). Question 2: For the Central Limit Theorem (CLT) to hold, the observations must be independent, and the success-failure condition must be met. If these conditions exist, the sample proportion can be modeled using a normal distribution. When we know the population distribution is normal, the conditions necessary for the CLT to hold include:  The observations in the sample must be independent, often requiring the sample to be a simple random sample.  We already know the sample size is sufficiently large (greater than 30) because it is modeled using the normal distribution. When we don’t know the population distribution, the conditions necessary for the CLT to hold include:  The observations in the sample must be independent, often requiring the sample to be a simple random sample.  The sample size is sufficiently large, i.e., and . In this case, n represents the sample size, and p represents the population proportion. This is the success-failure condition.

(Dies et al., 2019). When the sample size is barely over 30, given independence, the CLT still holds. A sample size above 30 can be modeled using a normal distribution (i.e., adhering to the CLT) (Ganti, 2023). The CLT will also hold when the sample size is large, given independence, allowing it to be modeled using a normal distribution. A sample size barely above 30 will result in smaller success-failure condition values, and the distribution graph may have a slight skew and appear more discrete. However, a large sample size will result in larger success-failure condition values, and the distribution graph will likely resemble a symmetric normal distribution graph that appears more continuous (Dies et al., 2019). Question 3: The margin of error and standard error have significant differences. The margin of error is a statistic that measures the extent of error in an observation in the sample. It predicts the acceptable difference between the research population and sample size. A lower margin of error increases the reliability of the investigation. However, standard error is a statistic that measures an investigation's dispersion. It measures how much the sample statistic differs from the population parameter. The lower the standard error, the more likely we can infer that the sample statistic is a reliable estimate of the population parameter (Longe, 2023). Words: 500 References: Dies, D. M., Barr, C. D., & Çetinkaya-R, M. (2019). Openintro statistics – Fourth edition: Chapter 5.1 & 5.2 . Open Textbook Library. [PDF]. https://www.biostat.jhsph.edu/~iruczins/teaching/books/2019.openintro.statistics.pdf . Ganti, A. (2023). Central Limit Theorem (CLT): Definition and Key Characteristics . Investopedia. https://www.investopedia.com/terms/c/central_limit_theorem.asp#:~:text=Sample %20sizes%20equal%20to%20or,population%20mean%20and%20standard%20deviation . Longe, B. (2023). Margin of error – Definition, Formula + Application . Formpl.us. https://www.formpl.us/blog/margin-of-error#:~:text=A%20margin%20of%20error %20is%20a%20statistical%20measure%20that%20accounts,deviation%20of%20the%20data %20set .

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