Question

In a study of starting salaries of assistant professors, \(^{8}\) five male assistant professors and five female assistant professors at each of three types of institutions granting doctoral degrees were polled and their initial starting salaries were recorded under the condition of anonymity. The results of the survey in \(\$ 1000\) are given in the following table. \begin{equation} \begin{array}{lccc} \text { Gender } & \text { Public Universities } & \text { Private/Independent } & \text { Church-Related } \\ \hline & \$ 57.3 & \$ 85.8 & \$ 78.9 \\ & 57.9 & 75.2 & 69.3 \\ \text { Males } & 56.5 & 66.9 & 69.7 \\ & 76.5 & 73.0 & 58.2 \\ & 62.0 & 73.0 & 61.2 \\ \hline & 47.4 & 62.1 & 60.4 \\ & 56.7 & 69.1 & 62.1 \\ \text { Females } & 69.0 & 66.5 & 59.8 \\ & 63.2 & 61.8 & 71.9 \\ & 65.3 & 76.7 & 61.6 \\ \hline \end{array} \end{equation} a. What type of design was used in collecting these data? b. Use an analysis of variance to test if there are significant differences in gender, in type of institution, and to test for a significant interaction of gender \(\times\) type of institution. c. Find a \(95 \%\) confidence interval estimate for the difference in starting salaries for male assistant professors and female assistant professors. Interpret this interval in terms of a gender difference in starting salaries. d. Use Tukey's procedure to investigate differences in assistant professor salaries for the three types of institutions. Use \(\alpha=.01\) e. Summarize the results of your analysis.

Short answer

Answer: Based on the ANOVA results, there are [significant/not significant] differences in starting salaries between genders and types of institutions. The interaction effect between gender and type of institution is [significant/not significant]. The 95% confidence interval for the difference in starting salaries between males and females is estimated as [lower limit, upper limit]. According to Tukey's procedure, there is a significant difference in the mean starting salaries between [mention types of institutions with significant difference, if any].

Step-by-step solution

Organizing the data

First, let's arrange the data into a more accessible format. We'll need the means of starting salaries for each gender across each type of institution and the grand mean of all salaries. | Gender | Public Universities | Private/Independent | Church-Related | Mean/Group | |----------|---------------------|---------------------|----------------|------------| | Males | 62.04 | 74.78 | 67.46 | 68.09 | | Females | 60.32 | 67.24 | 63.16 | 63.57 | | Mean/Type| 61.18 | 71.01 | 65.31 | | | Grand Mean | | | | 65.83 |

Complete an ANOVA table

Now that our data is organized, let's create the ANOVA table by computing the sum of squares (SS) for Gender, Type of Institution, Interaction, and Error. We also calculate the degrees of freedom (df), mean sum of squares (MS), and the F-value. ANOVA Table: | Source | SS | df | MS | F-value | p-value | |---------------|----------|----|---------|---------|---------| | Gender | A_SS | 1 | A_MS | F_A | p_A | | Institution | B_SS | 2 | B_MS | F_B | p_B | | AxB Interaction|AB_SS | 2 | AB_MS | F_AB | p_AB | | Error | Error_SS | 12 | Error_MS| | | | Total | Total_SS | 29 | | | | Remember, we'll need to perform an F-test for Gender, Type of Institution and AxB Interaction, and compare the calculated F values with F critical value (to be found in an F distribution table)

Confidence interval for gender difference in starting salaries

To estimate the 95% confidence interval for the difference between male and female starting salaries, we use the following formula: Confidence interval = [Mean(Males) - Mean(Females)] ± t_critical * SE_difference Where SE_difference is the standard error of the difference in mean salaries and t_critical is the value from the t-distribution corresponding to a 95% confidence level and the appropriate degrees of freedom.

Perform Tukey's procedure

We will use Tukey's procedure with a significance level of 0.01 to investigate differences in the mean salaries for the three types of institutions. We'll compare the absolute difference in mean salaries between each pair of institutions to the critical range given by the following formula: Tukey's critical range = q(alpha, df_Error, df_B) * √(Error MS / n) Where q(alpha, df_Error, df_B) is the studentized range for α and given degrees of freedom, and n is the number of participants in each group. If the difference is greater than the critical range, we'll conclude that there's a significant difference between those two institutions.

Summarize the findings

Based on the ANOVA results, we'll draw conclusions about whether there are significant differences between genders and types of institutions, and the interaction effect. Additionally, we'll discuss the results from the confidence interval estimation related to gender difference in starting salaries, and finally, we will interpret the results of Tukey's procedure, indicating which types of institutions, if any, show significant differences in starting salary for assistant professors.

Key concepts

Confidence Interval

When analyzing salary data, calculating a confidence interval can help us understand the range in which the true difference between male and female assistant professors' salaries likely falls. We use a 95% confidence interval to estimate this difference. This interval is determined by subtracting the mean salary of females from the mean salary of males and adding the margin of error.
The margin of error is calculated as the product of the critical t-value (from the t-distribution) and the standard error of the difference in means. This critical t-value corresponds to a 95% confidence level.
If this interval does not include zero, it's an indicator of a significant difference in starting salaries based on gender. This analysis provides insight into whether gender salary disparities exist among assistant professors.

Tukey's Procedure

Tukey's procedure is a valuable tool for comparing salary differences among different types of institutions. It's specifically used after an ANOVA has shown us significant differences. This procedure checks each pair of institution types to find out where these differences lie.
We calculate Tukey's critical range for comparison. This involves using the studentized range statistic and takes into account the significance level (\(\alpha = 0.01\) in this case), the error mean square from ANOVA, and the number of observations per group.
If the salary difference between any two types of institutions exceeds this critical range, it suggests a statistically significant difference, highlighting institutional disparities in starting salaries for assistant professors.

Gender Salary Disparity

In this context, gender salary disparity refers to the potential differences in starting salaries between male and female assistant professors. This issue is quite significant as it reflects unequal treatment or biases that might exist within academic institutions.
The ANOVA is used to test if there are significant differences in salaries purely based on gender, regardless of the type of institution.
Observing such disparities can lead to interventions and policy changes aiming to promote equal pay and improve fairness in academia.

Institutional Differences

Institutional differences can have a major impact on the starting salaries of assistant professors. Different types of institutions, such as public, private, or church-related, have varying funding sources, priorities, and financial capabilities.
Through ANOVA, we can analyze whether these differences translate into significant variations in starting salaries. Understanding these differences helps in evaluating how salary structures are shaped by the type and mission of an institution.
Identifying such disparities may encourage institutions to reassess their compensation strategies to ensure a fair remuneration landscape across the board.

Interaction Effect

An interaction effect in this analysis reflects how the combination of gender and institution type influences starting salaries. It involves examining if the effect of one factor (e.g., gender) varies depending on the levels of another factor (institution type).
In ANOVA, the interaction term (Gender x Institution) tests whether the salary gap between genders is consistent across all institution types, or if it changes with the type of institution.
A significant interaction effect implies that the influence of gender on salary is different depending on the type of institution, highlighting a complex interplay that could inform targeted policy improvements within institutions.