Question

Suppose there is a linear association between crime rate and percentage of high school graduates.

a) State the full and reduced model

b)Obtain SSE(F), SSE(R), df(F), fd(R), test statistics F for the general linear test and decision rule.

crime rate, high school grad %

  8487    74
   8179    82
   8362    81
   8220    81
   6246    87
   9100    66
   6561    68
   5873    81
   7993    74
   7932    82
   6491    75
   6816    82
   9639    78
   4595    84
   5037    82
   4427    79
   6226    78
  10768    73
   8335    77
  12311    65
  10104    77
  10503    76
   7562    79
   8593    79
   7133    78
  10205    84
  14016    78
   5959    81
   3764    89
   4297    85
   7562    77
   4844    74
   5777    80
   3599    84
   3219    88
  11187    75
   2105    77
   6650    78
  11371    61
   4517    91
   7348    83
   5696    77
   4995    85
   9248    70
   6860    88
   9776    80
   4280    82
  11154    82
   3442    82
   9674    70
   7309    64
   4530    79
   4017    83
   7122    77
   5689    76
   6109    80
   3343    84
   5029    82
   4330    81
   5425    74
   8769    81
   6880    76
   6538    78
   6521    78
   9423    79
   9697    83
   3805    79
   3134    83
   3433    81
   2979    84
   6836    64
   5804    67
   7986    75
  10994    73
  11322    77
   8937    64
   8807    75
  11087    80
  10355    83
   7858    85
   3632    91
   8040    88
   6981    83
   7582    76

Answer 1

Using Excel

data -> data analysis -> regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.4127
R Square	0.1703
Adjusted R Square	0.1602
Standard Error	2356.2920
Observations	84
ANOVA
	df	SS	MS	F	Significance F
Regression	1	93462942.2670	93462942.2670	16.8338	0.0001
Residual	82	455273165.2925	5552111.7719
Total	83	548736107.5595
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	20517.5999	3277.6427	6.2599	0.0000	13997.3245	27037.8754
x	-170.5752	41.5743	-4.1029	0.0001	-253.2798	-87.8706

a)
Full model
y^ = a+ b*x
reduced model
y^ =a
b)
SSE (F) = 93462942.2670

SSE (R) = 455273165.2925

df F = 1
df R = 82
F = 16.8338
we reject the null hypothesis
as p-value = 0.0001 < alpha