The data for this graph was taken from the 50 cities in the United States. a. Lo

Question

The data for this graph was taken from the 50 cities in the United States. a. Looking at the regression, what is the slope and what does it mean in this example? b. What is the y-intercept? What does it represent in this example? Is it meaningful? c Interpret what the R - Sq means in this example AND find the linear correlation coefficient r? d. Given a city's population is 2, 500,000 determine the predicted number of annual crimes. If you can't answer this question, please give reason why. e. Given a city's population is 4, 500,000 determine the predicted number of annual crimes. If you can't answer this question, please give a reason why. f. Someone in Berlin Germany asked you to predict the number of annual crimes it the population was 1, 500,000 what would the prediction be? It you can't answer this question please give a reason why.

Explanation / Answer

Solution

The given regression equation is read as: y = 12951 + 0.05150 x, where y = annual crime and x = city’s population. 12951 and 0.05150 are usually represented by a and b and are referred as sample estimates of and respectively of the population regression equation, y = + x.   

Part (a)

In the estimated regression equation ycap = a + bx, b represents the slope. So, in our case, the slope is 0.05150 ANSWER 1

The slope represents the change in y per unit change in y. So, in our case, it would mean that the annual crime is expected to increase [increase because b is positive], by 0.0515 whenever the population in up by 1 member. Or, for every increase of 100 in the population, annual crime is expected to go up by 5.15. ANSWER 2

Part (b)

In the estimated regression equation ycap = a + bx, a represents the y-intercept. So, in our case, the y-intercept is 12951 ANSWER 1

Interpretation of y-intercept is that it represents the expected value of y when x = 0. So, it should mean, in our case, that when there is no population, there would be 12951 crimes annually. This does not make any sense. So, not meaningful in our case.ANSWER 2

Part (c)

R2 represents the proportion of variation in dependent variable y that is explained by independent variable x.

Not able to compute R since standard deviation of x and y are not given in the question. ANSWER

[Given standard deviation of x and y, as sx and sy, R = b x (sx/sy)]

Part (d)

Given population of a city is 2,500,000, the predicted annual crime

= 12951 + (0.05150 x 2,500,000) = 12951 + 128750 = 141701 ANSWER

[Cautionary note: The above prediction is valid only if the city is in the US. If the city is not in the US, no prediction is possible]

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