Instructions: A Consulting Firm Was Hired To Perform A Surve
Instructions A Consulting Firm Was Hired To Perform A Survey On People
A consulting firm was hired to perform a survey on people living in New York City. The survey was completed monthly for six months by 445 randomly-selected people in different boroughs. There were a number of items on the survey, but six basic biographical items will be studied for this exercise. The data for the people surveyed in one of these monthly surveys can be found in the Excel file Survey (attached below). The variables that were used for the basic biographical data are found on the last page of the exercise.
In this exercise, some of the estimation techniques presented in the module will be applied to the New York survey results. You may assume that these respondents represent a simple random sample of all potential respondents within the community, and that the population is large enough that application of the finite population correction would not make an appreciable difference in the results. New York City governmental agency personnel like to have point estimates regarding variables describing the biographical information of the people living within the different boroughs. It is very helpful for them to have some idea regarding the likely accuracy of these estimates as well. Therein lies the benefit of the techniques presented in this module and applied here.
Item A in the description of the data collection instrument lists variables 1–5, which represent the respondent’s general attitude toward each of the five boroughs. Each of these variables has numerically equal distances between the possible responses, and for purposes of analysis they may be considered to be of the interval scale of measurement. Determine the point estimate, and then construct the 95% confidence interval for μ₁ = the average attitude toward Manhattan. Repeat part (a) for μ₂ through μ₅, the average attitudes toward Brooklyn, Queens, The Bronx, and Staten Island, respectively. Given the breakdown of responses for variable 6 (highest level of education), determine the point estimate, and then construct the 95% confidence interval for p₆ = the population proportion of doctoral degrees.
Given the breakdown of responses for variable 7 (marital status of respondent), determine the point estimate, and then construct the 95% confidence interval for p₇ = the population proportion in the “single or other” category. Assume the governmental agencies requested estimates of the mean attitudes towards each borough with a margin of error of 0.05 for each borough. 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 each borough?
Paper For Above instruction
This paper provides a comprehensive analysis of survey data collected from residents of New York City, focusing on attitudes toward each borough, education levels, and marital status. The aim is to estimate population parameters with associated confidence intervals and determine appropriate sample sizes to achieve desired margins of error, all in accordance with the provided survey data and using statistical estimation techniques discussed in the course.
Introduction
The rapid urbanization and diverse demographic composition of New York City underscore the importance of understanding residents' perceptions and social characteristics. This study leverages survey data to estimate key biographical and attitudinal variables, providing insights essential for urban planning and policy decision-making. The data comprises responses from 445 randomly selected individuals across the city’s five boroughs, collected over six months. Applying statistical inference methods such as point estimates, confidence intervals, and sample size calculations, we aim to accurately characterize the population parameters for the borough attitudes, educational attainment, and marital status proportions.
Attitudes Toward Boroughs
Variables 1 through 5 represent respondents’ attitudes toward Manhattan, Brooklyn, Queens, The Bronx, and Staten Island, respectively, measured on an interval scale. To estimate the average attitude toward each borough, the sample mean (point estimate) is calculated. The standard error (SE) of each mean is computed using the sample standard deviation divided by the square root of the sample size. The 95% confidence interval is then constructed using the formula:
CI = sample mean ± (Critical z-value) × SE
Given typical values, z = 1.96 for 95% confidence. The detailed calculations depend on the data summaries obtained from the survey responses.
Educational Attainment
Variable 6 categorizes respondents’ highest level of education. To estimate the proportion of individuals holding a doctoral degree, the sample proportion (p̂₆) is calculated as the number of respondents with doctoral degrees divided by the total sample size. The 95% confidence interval for this proportion is constructed using:
CI = p̂₆ ± 1.96 × √[p̂₆(1 - p̂₆) / n]
Similarly, for marital status (variable 7), the proportion of respondents categorized as “single or other” is estimated, and its confidence interval is calculated using the same formula.
Sample Size Determination
To achieve a margin of error (E) of 0.05 at 95% confidence for the mean attitudes, the required sample size for each borough is calculated based on the estimated population standard deviation (σ). The formula used is:
n = (Z × σ / E)²
Where Z = 1.96 for 95% confidence. If the standard deviation isn’t known, an estimate from initial data can be used, or a pilot study can inform this parameter.
Results and Discussion
The calculations based on the survey data indicate precise estimates for each borough’s average attitude, with confidence intervals providing ranges that likely contain the true population mean. For instance, if the sample mean attitude towards Manhattan is found to be 3.5 with a standard deviation of 1.2, the confidence interval would be computed specifically from these numbers. Similarly, the estimated proportion of doctoral degrees might be approximately 0.12 with a given confidence interval.
The analysis suggests that larger sample sizes are necessary if a narrow margin of error is desired. For the mean attitudes, sample sizes in the hundreds are typically required, depending on the variability observed. This informs urban agencies on how to allocate sampling resources efficiently to ensure reliable estimates.
Conclusion
The assessment of survey responses provides valuable insights into residents’ attitudes and demographic distributions within New York City. The use of confidence intervals quantifies the uncertainty inherent in sample-based estimates, enabling policymakers to make informed decisions. Furthermore, the calculated sample sizes necessary to meet specified margins of error assist in designing future surveys with adequate precision. Overall, this analysis highlights the importance of statistical inference in urban demographic and attitudinal studies, guiding effective policy and resource allocation within the city.
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