Question

In a study of the role of young drivers in automobile accidents, data on percentage of licensed drivers under the age of 21 and the number of fatal accidents per 1000 licenses were determined for 32 cities. The data are stored in Table B. The first column contains a number as the city code, the second column contains the percentage of drivers who are under 21, and the third column contains the number of fatal accidents per 1000 drivers. The primary interest is whether or not the number of fatal accidents is dependent upon the proportion of licensed drivers that are under 21.

City Number	% of drivers under 21	# of fatal accidents per 1000 drivers
1	5	0
2	5	0
3	5	0
4	13	2.029
5	17	5.12
6	7	0.468
7	13	1.463
8	14	3.412
9	8	2.104
10	15	3.146
11	11	2.081
12	14	3.612
13	11	2.117
14	12	2.758
15	9	1.819
16	8	1.483
17	14	3.211
18	10	1.157
19	10	0.871
20	9	1.34
21	15	2.751
22	6	0
23	9	0.712
24	12	1.93
25	17	3.899
26	4	0
27	14	2.992
28	9	0.577
29	8	1.819
30	11	2.218
31	9	1.075
32	15	2.105

Regression analysis, where one variable depends on another, can be used to predict levels of a dependent variable for specified levels of an independent variable. Use the EXCEL REGRESSION command to calculate the intercept and slope of the least‑squares line, as well as the analysis of variance associated with that line. Fill in the following table and use the results to answer the next few questions. Carefully choose your independent and dependent variables and input them correctly using EXCEL’s regression command. In this example, the percentage of drivers under the age of 21 affects the number of Fatals/1000 licenses.

The regression equation (least‑squares line) is

            Fatals/1000 licenses =                   +                   % under 21

                                                 (intercept)       (slope)

Analysis of variance

Source                         DF          SS                 MS          F               P

Regression                   1         ________   _______  ________   _______                      

Residual (Error)           30        ________   _______

10. What is the estimated increase in number of fatal accidents per 1000 licenses due to a one percent increase in the percentage of drivers under 21 (i.e. the slope)?

11. What is the standard deviation of the estimated slope?

12. What is the estimated number of fatal accidents per 1000 licenses if there were no drivers under the age of 21 (i.e. the y intercept)?

13. What percentage of the variation in accident fatalities can be explained by the linear relationship with drivers under 21 (i.e. 100 ´ the unadjusted coefficient of determination)?

Answer 1

x	y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
5	0	31.29	3.32	10.19
5	0	31.29	3.32	10.19
5	0	31.29	3.32	10.19
13	2.029	5.79	0.04	0.50
17	5.12	41.04	10.88	21.13
7	0.468	12.92	1.83	4.86
13	1.463	5.79	0.12810	-0.8612
14	3.412	11.60	2.53158	5.420
8	2.104	6.73	0.08	-0.73
15	3.146	19.42	1.76	5.84
11	2.081	0.17	0.07	0.11
14	3.612	11.60	3.21	6.10
11	2.117	0.17	0.09	0.12
12	2.758	1.98	0.88	1.32
9	1.819	2.54	0.00	0.00
8	1.483	6.73	0.11	0.88
14	3.211	11.60	1.93	4.74
10	1.157	0.35	0.44	0.39
10	0.871	0.35	0.90	0.56
9	1.34	2.54	0.23	0.77
15	2.751	19.42	0.87	4.10
6	0	21.10	3.32	8.36
9	0.712	2.54	1.23	1.77
12	1.93	1.978	0.012	0.153
17	3.899	41.040	4.318	13.313
4	0	43.478	3.316	12.007
14	2.992	11.603	1.371	3.989
9	0.577	2.540	1.547	1.982
8	1.819	6.728	0.000	0.005
11	2.218	0.165	0.158	0.161
9	1.075	2.540	0.556	1.189
15	2.105	19.415	0.081	1.252

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	339.00	58.27	407.72	51.83	129.98
mean	10.59	1.82	SSxx	SSyy	SSxy

sample size ,   n =   32          
here, x̅ = Σx / n=   10.594   ,     ȳ = Σy/n =   1.821  
                  
SSxx =    Σ(x-x̅)² =    407.7188          
SSxy=   Σ(x-x̅)(y-ȳ) =   130.0          
                  
estimated slope , ß1 = SSxy/SSxx =   130.0   /   407.719   =   0.31881
                  
intercept,   ß0 = y̅-ß1* x̄ =   -1.55643          
                  
so, regression line is   Ŷ =   -1.5564   +   0.3188   *x
==================

Fatals/1000 licenses =           -1.5564 +           0.3188 % under 21

Anova table
variation	SS	df	MS	F-stat	p-value
regression	41.439	1	41.4392	119.6079	0.0000
error,	10.394	30	0.3465
total	51.833	31

10)

answer: 0.3188

11) estimated std error of slope =Se(ß1) = Se/√Sxx =    0.589   /√   407.72   =   0.0292

12) -1.5564

13) R² =    (Sxy)²/(Sx.Sy) =    0.799 or 79.9%