The table shows the rank attained by male and female...

The table shows the rank attained by male and female officers in the New York City Police Department (NYPD). Do these data indicate that men and women are equitably represented at all levels of the department? $$ \begin{array}{|c|c|c|c|} \hline & & \text { Male } & \text { Female } \\ \hline & \text { Officer } & 21,900 & 4281 \\ \hline & \text { Detective } & 4058 & 806 \\ \hline & \text { Sergeant } & 3898 & 415 \\ \hline & \text { Lieutenant } & 1333 & 89 \\ \hline & \text { Captain } & 359 & 12 \\ \hline & \text { Higher ranks } & 218 & 10 \\ \hline \end{array} $$ a) What's the probability that a person selected at random from the NYPD is a female? b) What's the probability that a person selected at random from the NYPD is a detective? c) Assuming no bias in promotions, how many female detectives would you expect the NYPD to have? d) To see if there is evidence of differences in ranks attained by males and females, will you test goodness-of-fit, homogeneity, or independence? e) State the hypotheses. f) Test the conditions. g) How many degrees of freedom are there? h) Find the chi-square value and the associated $\mathrm{P}$-value. i) State your conclusion. j) If you concluded that the distributions are not the same, analyze the differences using the standardized residuals of your calculations.

The table shows the rank attained by male and female officers in the New York City Police Department (NYPD). Do these data indicate that men and women are equitably represented at all levels of the department? $$
\begin{array}{|c|c|c|c|}
\hline & & \text { Male } & \text { Female } \\
\hline & \text { Officer } & 21,900 & 4281 \\
\hline & \text { Detective } & 4058 & 806 \\
\hline & \text { Sergeant } & 3898 & 415 \\
\hline & \text { Lieutenant } & 1333 & 89 \\
\hline & \text { Captain } & 359 & 12 \\
\hline & \text { Higher ranks } & 218 & 10 \\
\hline
\end{array}
$$
a) What's the probability that a person selected at random from the NYPD is a female?
b) What's the probability that a person selected at random from the NYPD is a detective?
c) Assuming no bias in promotions, how many female detectives would you expect the NYPD to have?
d) To see if there is evidence of differences in ranks attained by males and females, will you test goodness-of-fit, homogeneity, or independence?
e) State the hypotheses.
f) Test the conditions.
g) How many degrees of freedom are there?
h) Find the chi-square value and the associated $\mathrm{P}$-value.
i) State your conclusion.
j) If you concluded that the distributions are not the same, analyze the differences using the standardized residuals of your calculations.

Key Concepts

Probability Computation

This involves determining the likelihood of an event by dividing the number of favorable outcomes by the total number of outcomes. In categorical data problems, probabilities are computed by summing relevant counts from a contingency table and then dividing by the overall total to understand proportions in the population.

Contingency Tables and Expected Frequencies

A contingency table organizes categorical data and allows us to compare counts across different groups. Expected frequencies are computed under the assumption of no association between the categorical variables (such as gender and rank) by using the product of the marginal totals divided by the grand total. This is essential to determine whether observed deviations are statistically significant.

Chi-Square Test of Independence

The chi-square test of independence is used to determine whether there is a significant association between two categorical variables. It compares the observed frequencies in a contingency table to the expected frequencies obtained under the null hypothesis of independence, helping to decide if any deviation from expected counts is due to sampling variability or indicates a real association.

Degrees of Freedom in Contingency Tables

Degrees of freedom in a contingency table are calculated based on the number of categories in each variable. Specifically, for a table with r rows and c columns, the degrees of freedom are (r - 1) × (c - 1). This value is crucial for determining the appropriate chi-square distribution to reference when testing hypotheses.

Hypothesis Testing in Categorical Data

Hypothesis testing in this context involves formulating a null hypothesis that states there is no association between the categorical variables (i.e., the distributions are the same across groups), and an alternative hypothesis that suggests there is an association. These hypotheses provide the framework for statistical tests to assess whether observed differences are significant.

Standardized Residuals in Chi-Square Tests

Standardized residuals are computed for each cell of the contingency table to identify which cells contribute the most to the overall chi-square statistic. They provide insight into the nature of the association by indicating where observed counts differ significantly from expected counts, allowing for a detailed post-hoc analysis after a significant chi-square result.

Testing Conditions for the Chi-Square Test

Before performing a chi-square test, certain conditions must be satisfied: the sample should be randomly selected; the expected frequency in each cell should be at least 5 to justify the approximation to the chi-square distribution; and the observations should be independent. These conditions ensure the validity and reliability of the test results.

Transcript

00:01 So the first question we want to find, what is the probability that a randomly chosen person from the new york police department is female? and so that would be that total is 5 ,613 divided by that table total which is 37 ,379.

00:16 So 5613 divided by that value, i happen to store that as x, and that's approximately 15%, .510.

00:25 Now part b asks, what's the probability that a person selected from random is a detective? probability of being a detective.

00:35 And so we have two ways to be a detective.

00:37 A detective can take place with 4 ,058 and with, let me get my total for the females, 806.

00:45 And so that total, 4 ,058 plus 806 is a total of 4 ,864.

00:55 So we would be taking that 4 ,864 divided by that total.

01:01 And i'm, well i'll write it down one more time and then i'll just call it t for total.

01:07 And so we have to divide that by that total which i actually store as x and that's about 13%, .130.

01:15 Now on part c, and why don't we change some color, just, it says assuming no biases and promotions, how many female detectives would you expect? well if they're independent, the probability of female and detect, we should be able to multiply those two probabilities together if they're independent.

01:34 And that should be the probability.

01:36 So we would take that .15 times .13 and so we should expect it to happen .019, .0195.

01:47 That's how often it should happen.

01:49 And i'll write that .0195.

01:52 So about 2 % of the time it should happen...

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