Question

Suppose the following data were collected relating the total number of crimes committed to the number of police officers and if the town is located in the Southwest. Use statistical software to find the regression equation. Is there enough evidence to support the claim that on average there are less crimes in the Southwest than in other regions of the country at the 0.050.05 level of significance? If yes, type the regression equation in the spaces provided with answers rounded to two decimal places. Else, select "There is not enough evidence."

Crimes   Officers   Southwest (1 if Southwest, 0 if otherwise)
277   34   1
388   27   0
303   36   0
424   20   0
293   31   1
350   30   0
363   29   0
427   23   0
387   24   0
295   31   1
383   26   0
321   26   1
263   35   1
317   27   1
366   21   1
372   21   1
339   31   0
286   36   0
307   29   1
279   34   1

Selecting a checkbox will replace the entered answer value(s) with the checkbox value. If the checkbox is not selected, the entered answer is used.

CRIMESi= _____+ ______ OFFICERSi + ______ b2 SOUTHWESTi+ei

or

there is not enough evidence

(Please explain as I truly want to understand this!)

Cheers

Answer 1

After running analysis in R software:

Code:

data = read.csv("..\\Desktop\\regression.csv")
model = lm(Crimes~., data = data)
summary(model)

Output:

Call:
lm(formula = Crimes ~ ., data = data)

Residuals:
Min 1Q Median 3Q Max
-17.6742 -5.2017 -0.0817 2.9555 21.1161

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 586.716 12.375 47.41 < 2e-16 ***
Officers -7.862 0.426 -18.46 1.11e-12 ***
Southwest -50.496 4.213 -11.99 1.02e-09 ***
---

Interpretation :

Crimesi = 586.716 - 7.862 * Officersi - 50.496 * Southwest, is the model

Meaning:

For a unit increase in Officers, Crimes decrease by 7.862 units and for a location Southwest, Crimes decrease by 50.496 units on an average

Also, p-values (bold in above table, last column) of all three coefficients (Intercept, Officers, Southwest) are approximately 0 and hence significant at 0.05 level. '*' indicate significance at 0.05 and here we have '***' for all three variables and is significant at 0.00 alpha.

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 9.396 on 17 degrees of freedom
Multiple R-squared: 0.9682,   Adjusted R-squared: 0.9645

Interpretation:

The model explains 96.82 % variability in Crimes

F-statistic: 259.1 on 2 and 17 DF, p-value: 1.845e-13

Interpretation:

Since p-value of entire model is 1.845e-13 (approx 0) , model is significant at alpha = 0 which means it is significant for all values of alpha above 0, be it 0.05, 0.1, etc.

is the output of the regression.

Note: I have added interpretation and meaning alongwith output statements for better understanding.

Please rate my answer and comment for doubt (if any)