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Chapter 8: Problem 16

What would be the correlation between the annual salaries of males and females at a company if for a certain type of position men always made (a) $$\$ 5,000$$ more than women? (b) $$25 \%$$ more than women? (c) $$15 \%$$ less than women?

Short Answer

Expert verified
Correlation is 1 in all cases.

Step by step solution

01

Define Correlation

The correlation measures the strength and direction of the linear relationship between two variables. In this context, it assesses how changes in men's salaries correspond to changes in women's salaries within a company.
02

Solve Case (a): Fixed Increase

If every man's salary is consistently $5,000 more than a woman's salary for the same position, then we can represent men's salary as \( M = W + 5000 \), where \( M \) is the men's salary and \( W \) is the women's salary. The relationship is perfectly linear, and the correlation between them is 1.
03

Solve Case (b): Percentage Increase

For a 25% increase, men's salary can be written as \( M = 1.25W \). This is a perfectly proportional (linear) relationship where any change in women's salary results in a proportional change in men's salary by 25%. Hence, the correlation is 1.
04

Solve Case (c): Percentage Decrease

When men earn 15% less, their salary is \( M = 0.85W \). This is still a linear relationship, as any change in women's salaries results in a proportional change in men's salaries by 85%. The correlation is again 1.
05

Conclusion of Correlation

In all cases (a, b, c), the men's salary is linearly dependent on the women's salary. Since they maintain a perfect linear relationship, the correlation coefficient is 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Relationships in Salaries
A linear relationship describes a direct and consistent connection between two variables. This means if one variable changes, the other changes as well in a predictable manner. In the context of salaries, a linear relationship implies that any change in women's salaries reflects a specific, consistent change in men's salaries.
For example, if we know the formula connecting men's and women's salaries, such as \( M = W + 5000 \) or \( M = 1.25W \), we establish a linear relationship. These equations state that men's salaries can be predicted based on women's salaries.
Understanding this relationship helps ensure transparency and fairness in compensation, allowing for gender-neutral salary adjustments.
Salary Comparison Techniques
Salary comparison is an essential exercise in evaluating the fairness and equity of pay practices within an organization. Comparing salaries involves assessing the pay level of different employees, particularly across variables like gender, position, and experience.
When using formulas to compare salaries, we can directly see how one salary interacts with another. For instance, if men's salaries are \( \$5000 \) more than women's, we use the formula \( M = W + 5000 \). This provides a clear basis for comparison.
  • Direct comparisons like the above help highlight discrepancies and ensure equitable pay structures.
  • Numerical comparisons using fixed amounts or percentages make discrepancies tangible and facilitate necessary adjustments.
Understanding and Addressing the Gender Pay Gap
The gender pay gap refers to the difference in earnings between men and women. It is often expressed as a percentage difference and can stem from factors such as discrimination, occupational segregation, and work experience.
By analyzing pay using percentages, such as \( M = 0.85W \), where men's salaries are 15% less than women's, one can gauge the alignment or disparity within an organization.
Tackling the gender pay gap requires:
  • Awareness and transparency in salary data between genders.
  • Implementing policies that promote equal pay for equivalent work.
  • Regular monitoring and adjustments to compensation practices.
Addressing these gaps contributes to a fairer and more inclusive workplace.
Proportional Change and Its Impact
Proportional change refers to how a variable changes relative to another in a constant ratio. In salary comparisons, this means evaluating how changes in one salary affect changes in another based on a set percentage.
For example, a proportional change of 25% in women's salaries implies men's salaries would be 1.25 times women's salaries, represented by the formula \( M = 1.25W \). This connection helps in understanding pay dynamics.
Proportional change ensures consistency in compensation adjustments, facilitating:
  • Fair and predictable pay changes across employee categories.
  • Easy to implement policies for salary updates, sustaining equity.
Such insights into proportional change enable organizations to maintain balance and fairness in salary structures.

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Most popular questions from this chapter

The following regression output is for predicting the heart weight (in g) of cats from their body weight (in \(\mathrm{kg}\) ). The coefficients are estimated using a dataset of 144 domestic cats. $$\begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -0.357 & 0.692 & -0.515 & 0.607 \\ \text { body wt } & 4.034 & 0.250 & 16.119 & 0.000 \\ \hline s=1.452 & R^{2}=64.66 \% & \quad R_{a d j}^{2}=64.41 \% \end{array}$$ (a) Write out the linear model. (b) Interpret the intercept. (c) Interpret the slope. (d) Interpret \(R^{2}\). (e) Calculate the correlation coefficient.

Exercise 8.25 presents regression output from a model for predicting annual murders per million from percentage living in poverty based on a random sample of 20 metropolitan areas. The model output is also provided below. $$\begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -29.901 & 7.789 & -3.839 & 0.001 \\ \text { poverty\% } & 2.559 & 0.390 & 6.562 & 0.000 \\ \hline\end{array}$$ \(s=5.512 \quad R^{2}=70.52 \% \quad R_{a d j}^{2}=68.89 \%\) (a) What are the hypotheses for evaluating whether poverty percentage is a significant predictor of murder rate? (b) State the conclusion of the hypothesis test from part (a) in context of the data. (c) Calculate a \(95 \%\) confidence interval for the slope of poverty percentage, and interpret it in context of the data. (d) Do your results from the hypothesis test and the confidence interval agree? Explain.

Exercise 8.12 introduces data on the average monthly temperature during the month babies first try to crawl (about 6 months after birth) and the average first crawling age for babies born in a given month. A scatterplot of these two variables reveals a potential outlying month when the average temperature is about \(53^{\circ} \mathrm{F}\) and average crawling age is about 28.5 weeks. Does this point have high leverage? Is it an influential point?

The scatterplot shows the relationship between socioeconomic status measured as the percentage of children in a neighborhood receiving reduced-fee lunches at school (lunch) and the percentage of bike riders in the neighborhood wearing helmets (helmet). The average percentage of children receiving reduced-fee lunches is \(30.8 \%\) with a standard deviation of \(26.7 \%\) and the average percentage of bike riders wearing helmets is \(38.8 \%\) with a standard deviation of \(16.9 \%\). (a) If the \(R^{2}\) for the least-squares regression line for these data is \(72 \%,\) what is the correlation between lunch and helmet? (b) Calculate the slope and intercept for the least-squares regression line for these data. (c) Interpret the intercept of the least-squares regression line in the context of the application. (d) Interpret the slope of the least-squares regression line in the context of the application. (e) What would the value of the residual be for a neighborhood where \(40 \%\) of the children receive reduced-fee lunches and \(40 \%\) of the bike riders wear helmets? Interpret the meaning of this residual in the context of the application.

The following regression output is for predicting annual murders per million from percentage living in poverty in a random sample of 20 metropolitan areas. $$\begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -29.901 & 7.789 & -3.839 & 0.001 \\ \text { poverty\% } & 2.559 & 0.390 & 6.562 & 0.000 \\ \hline s=5.512 & R^{2}=70.52 \% & R_{a d j}^{2}=68.89 \%\end{array}$$ (a) Write out the linear model. (b) Interpret the intercept. (c) Interpret the slope. (d) Interpret \(R^{2}\). (e) Calculate the correlation coefficient.
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