Question

Do long prison sentences result in a lower crime rate? You divide 24 states where you have violent crime rate data into how severe their punishments are for violent felonies into three groups:

(1) Low Severity (sentences on average for violent offenses are less than 5 years)

(2) Medium Severity (sentences on average for violent offenses are between 5 and 10 years)

(3) High Severity (sentences on average for violent offenses are more than 10 years)

Then, using the FBI’s Uniform Crime Reports you get the violent crime rate for these 24 states.

Here are your data where each entry is the rate of violent crime in that state in 2014:

Low Severity	Medium Severity	High Severity
8.3	9.2	10.5
7.6	11.1	12.7
8.0	10.7	10.4
6.9	9.7	8.9
8.0	12.5	11.9
7.3	10.8	11.1
7.9	8.7	9.2
8.6	13.7	14.2
Xbar= 7.825	Xbar = 10.800	Xbar = 11.112
n1 = 8	n2  = 8	n3 = 8

xbar = 9.91

1. Step 1 (1pt each; 2pts total)

- H0 =  

- H1 =

2. Step 2 (1pt each; 2pts total)

• Type of test:
• Distribution:

3. Step 3 (1pt total)

• α = .01 (given to you)
• critical value:

4. Step 4 The observed value is 12.633 (given to you)
5. Step 5: Make a decision about the null hypothesis; interpret the results (1pt each; 2pts total)

Answer 1

a)

Ho: µ1=µ2=µ3  
H1: not all means are equal  

b)

One way ANOVA

c)

count, ni =	8	8	8
mean , x̅ i =	7.825	10.80	11.11
std. dev., si =	0.544	1.672	1.777
sample variances, si^2 =	0.296	2.797	3.158
total sum	62.6	86.4	88.9		237.9
grand mean , x̅̅ =	Σni*x̅i/Σni =	9.91

square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	4.358	0.788	1.440
					TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	34.861	6.301	11.520		52.6825
SS(within ) = SSW = Σ(n-1)s² =	2.075	19.580	22.109		43.7637
no. of treatment , k =	3
df between = k-1 =	2
N = Σn =	24
df within = N-k =	21
mean square between groups , MSB = SSB/k-1 =	26.3413
mean square within groups , MSW = SSW/N-k =	2.0840
F-stat = MSB/MSW =	12.6398

	SS	df	MS	F	p-value	F-critical
Between:	52.68	2	26.34	12.633	0.0002	5.78
Within:	43.76	21	2.08
Total:	96.45	23
					α =	0.01
			conclusion :	p-value<α , reject null hypothesis

e)

F-stat> critical value, reject Ho  

Thanks in advance!

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