Question

The recidivism rate for convicted sex offenders is 55%. A warden has implemented a new therapy with the hope that the rate will decrease. Of the 280 convicted sex offenders who had the new therapy before being released, 142 of them became repeat offenders. What can be concluded at the 0.05 level of significance?

H₀: p = 0.55

1. H_a: p__0.55: a. >

b. <

c. Not equal to

2. Test Statistic: a. 1.81

b. -1.44

c. -1.31

d. 2.62

3. Critical Value(s): a. -2.33, 2.33

b. 1.645

c. -1.96, 1.96

d. -1.645   

4. p-Value = a. 0.0163

b. 0.0747

c. 0.0129

d. 0.0081

5. Reject Ho or Fail to reject Ho?

6. Conclusion: a. Sufficient evidence

b. Insufficient evidence

Answer 1

Solution :

This is the left tailed test .

The null and alternative hypothesis is

H₀ : p = 0.55

H_a : p < 0.55

n = 280

x = 142

= x / n = 142 / 280 = 0.5071

P₀ = 0.55

1 - P₀ = 1 - 0.55 = 0.45

z = - P₀ / [P_{0 *} (1 - P₀ ) / n]

= 0.5071 - 0.55 / [(0.55 * 0.45) / 280]

= -1.44

Test statistic = -1.44

P(z < -1.44) = 0.0749

P-value = 0.0747

= 0.05

Z = Z0.05 = -1.645

P-value >

Fail to reject the H0 .

There is insufficient evidence .