Confidence Intervals About A Proportion Each Week ✓ Solved

Confidence Intervals about a Proportion Each week, you will

Each week, you will be asked to respond to the prompt or prompts in the discussion forum. Your initial post should be 75-150 words in length, and is due on Sunday. By Tuesday, you should respond to two additional posts from your peers. For the last several weeks, you have been working with this scenario: ABC Enterprises is a growing company in which employees must frequently travel by car for business a few times each week. Currently, each employee is allotted 30 dollars each travel day to cover all meals. Sales employees often travel in a 1.5-hour radius of the office and frequently have multiple appointments each week. They have complained that that dollar amount for food is not enough, especially on days with multiple meetings. A survey, given by the CEO, was taken to get a sense of what the employees recommended for a new allotment. Although not all responded, the response rate was pretty good – 67%. Note: The 30 dollars is for food. Gas is a different amount given to the employees - please do not assume the 30 dollars includes fuel.

Introduction

This paper aims to analyze the proportion of male and female employees at ABC Enterprises using the provided sample data, along with creating and interpreting confidence intervals for the male population proportion. The analysis will also address the concern raised by Annie regarding whether a higher male population impacts expenditure on food during work-related travel.

Proportion of Male Employees in the Sample

From the given dataset, we can quantify the number of male and female employees in the sample. For analysis purposes, let's denote the number of males as M and females as F.

Given that the sample of 52 employees reveals that a certain number are male while the rest are female, the proportion of males in the sample can be computed as:

Proportion of males (p̂) = Number of males in the sample / Total number of employees in the sample

If we obtain, for example, 25 males in the sample, the proportion would be:

p̂ = 25 / 52 = 0.4808 (approximately 48.08%).

Validity of Extrapolating to the Population

It is crucial to acknowledge that the proportion obtained from our sample does not necessarily reflect the entire population. This line of reasoning is grounded in the fact that the sampling method may introduce bias, and the characteristics of the sample may not be representative of the population at large. Factors such as the non-random selection of respondents or an uneven response rate across gender could hinder the assumption that the sample's proportion mirrors that of the whole company.

Confidence Intervals for the Male Proportion

95% Confidence Interval

To derive the confidence interval for the population proportion of male employees, we apply the relevant statistical formula. First, we determine the z sub alpha over two value for a 95% confidence interval, which is approximately 1.96.

The next step requires knowing n, the total sample size of 52. Using the formula for the confidence interval:

CI = p̂ ± z * √(p̂(1-p̂)/n)

Substituting the values, we get:

CI = 0.4808 ± 1.96 * √(0.4808(1-0.4808)/52)

This calculation will yield a range for the confidence interval, indicating that we can be 95% confident that the proportion of male employees in the entire company falls within that range.

99% Confidence Interval

The process to calculate the 99% confidence interval is similar, except we substitute the z value to approximately 2.576:

CI = p̂ ± z * √(p̂(1-p̂)/n)

Utilizing the same values:

CI = 0.4808 ± 2.576 * √(0.4808(1-0.4808)/52)

The resulting interval will be wider than the 95% confidence interval, indicating a greater degree of certainty but a lack of precision in the estimate.

Comparison Between Confidence Intervals

When comparing the ranges from the two confidence intervals, it becomes evident that the 99% interval is broader than the 95% interval. The rationale for this observation lies in the increased level of confidence associated with the 99% interval—increasing certainty regarding our estimate necessitates a more expansive range of values.

Addressing Annie's Concern

In response to Annie's assertion that the male population influences the dollar amount allocated for meals, it is imperative to consider several aspects. Firstly, the allocation for meals, set at $30 per travel day, might not be reflecting the actual expenses encountered by employees, irrespective of their gender. Annie's inference may overlook the variability in spending habits among individuals, regardless of gender. A potential approach to alleviate these concerns could involve conducting further studies to understand actual meal costs better or adjusting the daily allowances based on statistical findings emanating from the sample.

Conclusion

In conclusion, analyzing the proportions of male employees and developing confidence intervals allow us to gather insights regarding ABC Enterprises' workforce. It is essential to use this statistical data judiciously, being cautious about extrapolating sample findings to the broader employee population.

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