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1 New York Survey Data Gage Korell Colorado State University-Global Stat-156 Mary Dereshiwsky 2/9/2024 \

2 Introduction In response to the need for comprehensive insights into the perspectives of New York City residents regarding their boroughs, a consulting firm was commissioned to conduct a thorough survey. The ensuing statistics presented in this report are derived from data collected over a one-month period within a broader six-month survey initiative. As New York City governmental personnel strive to gain a deeper understanding of the sentiments and opinions held by the city's diverse populace regarding its boroughs, this dataset serves as a valuable resource. By analyzing the collected data, policymakers and stakeholders can glean invaluable insights that inform strategic decision-making, urban planning initiatives, and community engagement efforts aimed at enhancing the overall quality of life for New Yorkers across all boroughs. Section 1: General Sentiment Toward Each Borough The initial segment of statistical analysis focuses on variables 1-5, comprising responses from 445 randomly selected residents of New York City who provided ratings concerning their perceptions of the five boroughs. Participants were instructed to assign a rating ranging from 1 (strongly dislike) to 5 (strongly like) to each borough based on their personal opinions. For Manhattan (variable 1), the point estimate stands at 3.23, accompanied by a standard deviation of 0.94. This suggests a moderately favorable attitude towards Manhattan among the sampled population. Furthermore, the 95% confidence interval spans from 3.15 to 3.32, indicating a range within which there is a 95% probability that the true sample mean lies (Mcleod, 2023). Moving to Brooklyn, the point estimate elevates significantly to 4.44, with a standard deviation of 1.12, indicating a notably positive sentiment towards this borough. The 95% confidence interval stretches from 4.34 to 4.55, indicating a higher positive attitude towards Brooklyn compared to Manhattan.

3 In contrast, Queens receives a lower point estimate of 2.28, with a standard deviation of 1.13, indicating a less favorable perception. The 95% confidence interval ranges from 2.17 to 2.38, indicating a more negative attitude towards Queens compared to Manhattan and Brooklyn. Similarly, the Bronx garners a point estimate of 2.41, with a standard deviation of 1.36. Despite not being the lowest-scoring borough, the Bronx exhibits a wider margin of error, reflected in its 95% confidence interval ranging from 2.28 to 2.54. This suggests a relatively lower level of certainty regarding the sample mean for the Bronx. Finally, Staten Island registers a point estimate of 4.29, with a standard deviation of 1.09, reflecting a predominantly positive attitude towards this borough. The 95% confidence interval extends from 4.19 to 4.39, indicating a high level of confidence in the favorable perception of Staten Island, ranking it as the second-highest-rated borough after Brooklyn within the sampled population. Section 2: Highest Level of Education Within the survey sample of 445 respondents, participants were queried about their highest attained level of education, with six distinct options provided. These options ranged from 1 (did not complete high school) to 6 (doctoral degree), encompassing a diverse spectrum of educational backgrounds. The point estimate for the highest level of education among respondents stands at 3.44, falling between the categories of an associate’s degree and a bachelor’s degree. This point estimate closely aligns with the median of the possible responses, situated midway between achieving an associate’s and a bachelor’s degree. Furthermore, the 95% confidence interval for this estimate spans from 3.33 to 3.56, indicating a high level of confidence in the accuracy of this assessment. Thus, it can be inferred that the average respondent possesses either an associate’s or a bachelor’s degree. In contrast, the proportion of respondents holding doctoral degrees is notably lower, with only 12 individuals reporting such qualifications out of the total sample size of 445. This

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4 yields a point estimate of 0.03, signifying that approximately 3% of the respondents hold doctoral degrees. The 95% confidence interval for this estimate ranges from 0.0225 to 0.0375, indicating a margin of error of 0.0075. Hence, with 95% confidence, it can be concluded that the proportion of respondents with doctoral degrees falls within this range, highlighting the comparatively smaller prevalence of individuals with advanced academic credentials within the surveyed population. Section 3: Marital Status In addition to capturing data on education levels, the survey also delved into the marital status of the respondents, who were prompted to select either 1 for married or 2 for single or other. The point estimate for marital status stands at 1.62, accompanied by a standard deviation of 0.49. Within a 95% confidence interval ranging from 1.58 to 1.67, it becomes evident that a higher proportion of respondents identified as single or other rather than married. While the point estimate falls slightly closer to the married category, its proximity to the midpoint of 1.5 suggests a balanced mix of marital statuses within the surveyed population. This balance is critical as it ensures that the survey results are representative of a diverse range of perspectives. For instance, had the majority of respondents been married, the outcomes might have skewed towards viewpoints more typical of married individuals, thereby potentially overlooking the experiences and opinions of single respondents. Digging deeper into the category of single or other, it's revealed that 276 respondents, representing 62% of the total sample size of 445, identified with this designation. This proportion yields a point estimate of 0.62, signifying that approximately 62% of the respondents fall into the single or other category. With a 95% confidence interval spanning from 0.60 to 0.64, and a margin of error of 0.02, it can be concluded with confidence that the percentage of respondents identifying as single or other lies within this range. This underscores the significance of acknowledging and representing diverse marital statuses within survey research, as it ensures a comprehensive understanding of the population's perspectives and experiences.

5 Section 4: Implications of a .05 Margin of Error Achieving a 95% confidence level with a margin of error as low as .05 necessitates considerably larger sample sizes compared to the original survey conducted with 445 respondents. To attain this level of precision, adjustments in sample sizes are imperative across all boroughs. Manhattan would require a substantial increase to a sample size of 1,358, while Brooklyn would need a sizable boost to 1,928 respondents. Similarly, Queens and Staten Island would need to expand their sample sizes to 1,963 and 1,825.68 respectively, to meet the desired confidence level. The Bronx, however, would necessitate the most substantial increase to a sample size of approximately 2,842.17. It is noteworthy that Brooklyn exhibited the smallest margin of error among the original 445 respondents compared to the other boroughs. This suggests that the data points for Brooklyn were less dispersed, resulting in a more compact range of values. Consequently, achieving a .05 margin of error requires a smaller increase in sample size for Brooklyn compared to other boroughs with more widely distributed data. This phenomenon underscores the importance of sample size in statistical analysis: larger sample sizes tend to yield more accurate estimates of the population mean, as they offer a more comprehensive representation of the population's variability. Therefore, increasing the sample size is essential for enhancing the reliability and precision of survey results, particularly when aiming for narrower margins of error and higher confidence levels. Conclusion In conclusion, the survey conducted by a consulting firm offers valuable insights into the opinions of New York City residents regarding their boroughs. Analyzing data collected over one month as part of a six-month initiative, policymakers gain valuable information for decision- making. The analysis reveals varying perceptions across boroughs, with Manhattan, Brooklyn, and Staten Island receiving positive ratings, while Queens and the Bronx score lower. Moreover, the survey provides insights into respondents' education levels and marital status.

6 Achieving a 95% confidence level with a narrow margin of error requires larger sample sizes, emphasizing the importance of sample size in statistical analysis. Overall, these findings aid governmental personnel in understanding residents' perspectives and making informed decisions to enhance the city's quality of life.

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7 Resources: Mcleod, S. (October, 2023). Confidence Intervals Explained: Examples, Formula & Interpretation. Retrieved from: Confidence Intervals in Statistics: Examples & Interpretation (simplypsychology.org)

8 Appendix Construct the 95% confidence interval for μ1= the average attitude toward Manhattan. 3.23 +/- .08 Construct the 95% confidence interval for π6= the population proportion of doctoral degrees. .03+/- .0075 Given the breakdown of responses for variable 7 (marital status of respondent), determine the point estimate 1.62 +/- .05 If the governmental agency personnel want to have 95% confidence that the sample mean will fall within this margin of error, how large should the sample sizes be for the Staten Island borough? 1,825.68