Question

The recidivism rate for convicted sex offenders is 11%. A warden suspects that this percent is lower if the sex offender is also a drug addict. Of the 352 convicted sex offenders who were also drug addicts, 32 of them became repeat offenders. What can be concluded at the αα = 0.01 level of significance?

1. For this study, we should use Select an answer t-test for a population mean z-test for a population proportion
2. The null and alternative hypotheses would be:
   Ho: ? μ p  ? < = > ≠   (please enter a decimal)   
   H₁: ? p μ  ? > = < ≠   (Please enter a decimal)

3. The test statistic ? z t  =  (please show your answer to 3 decimal places.)
4. The p-value =  (Please show your answer to 4 decimal places.)
5. The p-value is ? ≤ >  αα
6. Based on this, we should Select an answer accept fail to reject reject  the null hypothesis.
7. Thus, the final conclusion is that ...
   • The data suggest the population proportion is not significantly lower than 11% at αα = 0.01, so there is statistically significant evidence to conclude that the population proportion of convicted sex offender drug addicts who become repeat offenders is equal to 11%.
   • The data suggest the population proportion is not significantly lower than 11% at αα = 0.01, so there is statistically insignificant evidence to conclude that the population proportion of convicted sex offender drug addicts who become repeat offenders is lower than 11%.
   • The data suggest the populaton proportion is significantly lower than 11% at αα = 0.01, so there is statistically significant evidence to conclude that the population proportion of convicted sex offender drug addicts who become repeat offenders is lower than 11%.
8. Interpret the p-value in the context of the study.
   • If the sample proportion of convicted sex offender drug addicts who become repeat offenders is 9% and if another 352 convicted sex offender drug addicts are surveyed then there would be a 12.62% chance of concluding that fewer than 11% of convicted sex offender drug addicts become repeat offenders.
   • There is a 12.62% chance that fewer than 11% of all convicted sex offender drug addicts become repeat offenders.
   • There is a 11% chance of a Type I error.
   • If the population proportion of convicted sex offender drug addicts who become repeat offenders is 11% and if another 352 inner city residents are surveyed then there would be a 12.62% chance that fewer than 9% of the 352 convicted sex offender drug addicts in the study become repeat offenders.
9. Interpret the level of significance in the context of the study.
   • If the population proportion of convicted sex offender drug addicts who become repeat offenders is lower than 11% and if another 352 convicted sex offender drug addicts are observed then there would be a 1% chance that we would end up falsely concluding that the proportion of all convicted sex offender drug addicts who become repeat offenders is equal to 11%.
   • There is a 1% chance that Lizard People aka "Reptilians" are running the world.
   • There is a 1% chance that the proportion of all convicted sex offender drug addicts who become repeat offenders is lower than 11%.
   • If the population proportion of convicted sex offender drug addicts who become repeat offenders is 11% and if another 352 convicted sex offender drug addicts are observed, then there would be a 1% chance that we would end up falsely concluding that the proportion of all convicted sex offender drug addicts who become repeat offenders is lower than 11%.

Answer 1

a. We should use z test for a population proportion.

b. H₀ : p = 0.11

H_a : p < 0.11

c. here sample proportion = p^ = x/n= 32/352 = 0.0910

standard error of sample proportion = se_p =  sqrt [0.11 * (1 - 0.11)/352] = 0.0167

z = (p^ - p)/se_p​​​​​​​ = (0.0910 - 0.11)/0.0167 = -1.145

(d) p - value = P(Z < -1.145) = 0.1262

(e) the p - value is greater than alpha.

(f) we fail to reject the null hypothesis.

(g) The data suggest the population proportion is not significantly lower than 11% at αα = 0.01, so there is statistically insignificant evidence to conclude that the population proportion of convicted sex offender drug addicts who become repeat offenders is lower than 11%. Option b is correct here.

(h) If the population proportion of convicted sex offender drug addicts who become repeat offenders is 11% and if another 352 inner city residents are surveyed then there would be a 12.62% chance that fewer than 9% of the 352 convicted sex offender drug addicts in the study become repeat offenders.

(i) If the population proportion of convicted sex offender drug addicts who become repeat offenders is 11% and if another 352 convicted sex offender drug addicts are observed, then there would be a 1% chance that we would end up falsely concluding that the proportion of all convicted sex offender drug addicts who become repeat offenders is lower than 11%.

Last option is correct here.