SOLUTION: The percentages of police officers who are women are listed in the table for various city populations. City Population (in thousands) City Population Used to Represent City Popu

Question 1200507: The percentages of police officers who are women are listed in the table for various city populations.
City Population (in thousands) City Population Used to Represent City Population (in thousands) Percent
0-9.999 5 8.3
10-24.999 17.5 7.5
25-49.999 37.5 8.5
50-99.999 75 9.4
100-249.999 175 11.7
250 or more 300 17
Source: FBI Uniform Crime Report
Let f(n) be the percentage of police officers who are women in cities with populations of thousand. A model of the situation is f(n) = 0.00006n^2 + 0.012n + 7.884
.
Use a graphing calculator to draw the graph of the model and, in the same viewing window, the scattergram of the data. Does the model fit the data well?
The function is not a good model for the data ?
The function is a good model for the data. ?
Glen Ellyn, Illinois, has a population of 278.6 thousand. Estimate the percentage of police officers in Glen Ellyn who are women.
[ ] Round to 1 decimal place.
Find n when f(n)= 13
n= [ ] Round each to 1 decimal place.
What does it mean in this situation? (Answer this question on scratch paper prior to looking at the choices.)
Selection an answer.
wrong answer.
13% of police officers are women in cities with populations of about -408.7 thousand (model break down) and about 208.7 thousand

Answer by GingerAle(43) (Show Source): You can put this solution on YOUR website!

**Analysis:**
* **Visual Inspection:** By graphing the model and the scatterplot together, you can visually assess how well the model fits the data.
* If the curve of the function closely follows the trend of the scatterplot points, the model is considered a good fit.
* If the curve significantly deviates from the data points, the model is not a good fit.
* **Glen Ellyn:** The model predicts that approximately 13.5% of police officers in Glen Ellyn are women.
* **Finding n when f(n) = 13:**
* The calculator provides two solutions: 108.7 and -408.7.
* Since population cannot be negative, the relevant solution is 108.7 thousand.
* This means the model predicts that 13% of police officers are women in cities with populations of about 108.7 thousand.
**Based on visual inspection (which you would perform on your calculator), determine whether the model is a good fit for the data.**
**Key Points:**
* The model is an equation that attempts to describe the relationship between city population and the percentage of female police officers.
* The scatterplot helps visualize this relationship and assess how well the model captures the trend in the data.
* The model can be used to make predictions, such as estimating the percentage of female officers in a city with a given population.
I hope this helps! Let me know if you have any further questions.
**1. Graphing and Model Fit**
* **Using a graphing calculator:**
* Enter the data points (city population and percentage of female officers) into the lists.
* Graph the scatterplot of the data.
* Enter the model equation (f(n) = 0.00006n^2 + 0.012n + 7.884) into the function editor.
* Graph the model equation on the same graph as the scatterplot.
* **Visual Inspection:**
* Observe how closely the curve of the model fits the scatterplot points.
* If the curve closely follows the trend of the data points, the model is considered a good fit.
**2. Estimating Percentage of Female Officers in Glen Ellyn**
* **Substitute the population of Glen Ellyn (278.6 thousand) into the model equation:**
* f(278.6) = 0.00006 * (278.6)^2 + 0.012 * 278.6 + 7.884
* f(278.6) ≈ 13.5
* **The model estimates that approximately 13.5% of police officers in Glen Ellyn are women.**
**3. Finding n when f(n) = 13**
* **Set the model equation equal to 13:**
* 13 = 0.00006n^2 + 0.012n + 7.884
* **Solve for n:**
* This can be done using the quadratic formula or the "intersect" function on the graphing calculator.
* You should find two solutions for n:
* n ≈ 108.7
* n ≈ -408.7
* **Interpretation:**
* The model predicts that 13% of police officers are women in cities with populations of about 108.7 thousand.
* The negative value (-408.7 thousand) is not meaningful in this context as population cannot be negative.
**Conclusion:**
* Based on the visual inspection of the graph, you can determine whether the model fits the data well.
* The model can be used to estimate the percentage of female officers in cities with different populations.
* The interpretation of the results should consider the limitations of the model and the range of applicability of the data.
**Note:**
* This analysis assumes that the model is a reasonable representation of the relationship between city population and the percentage of female police officers.
* Factors other than city population may also influence the percentage of female officers.
I hope this helps! Let me know if you have any other questions.

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