Question

The recidivism rate for convicted sex offenders is 12%. A warden suspects that this percent is different if the sex offender is also a drug addict. Of the 343 convicted sex offenders who were also drug addicts, 58 of them became repeat offenders. What can be concluded at the α = 0.01 level of significance?

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

3. The test statistic? t z = ( ) (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 ? ≤ > α
6. Based on this, we should Select an answer reject, fail to reject, accept the null hypothesis.
7. Thus, the final conclusion is that ...
   • The data suggest the population proportion is not significantly different from 12% 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 different from 12%.
   • The data suggest the population proportion is not significantly different from 12% 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 12%.
   • The data suggest the population proportion is significantly different from 12% 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 different from 12%.
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 17% and if another 343 convicted sex offender drug addicts are observed then there would be a 0.52000000000001% chance that we would conclude either fewer than 12% of all convicted sex offender drug addicts become repeat offenders or more than 12% of all convicted sex offender drug addicts become repeat offenders.
   • There is a 0.52000000000001% chance of a Type I error.
   • There is a 0.52000000000001% chance that the percent of all convicted sex offender drug addicts who become repeat offenders differs from 12%.
   • If the population proportion of convicted sex offender drug addicts who become repeat offenders is 12% and if another 343 convicted sex offender drug addicts are observed, then there would be a 0.52000000000001% chance that either more than 17% of the 343 convicted sex offender drug addicts in the study become repeat offenders or fewer than 7% of the 343 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 12% and if another 343 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 different from 12%.
   • If the population proportion of convicted sex offender drug addicts who become repeat offenders is different from 12% and if another 343 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 12%.
   • There is a 1% chance that the proportion of all convicted sex offender drug addicts who become repeat offenders is different from 12%.
   • There is a 1% chance that Lizard People aka "Reptilians" are running the world.

Answer 1

(a) z-test for a population proportion. Ho: p=0.12 H:p=0.12 Excel add-in (PH Stat) procedure: Step 1: In EXCEL, Select Add-Ins > PH Stat > One-Sample Tests > Z Test for the proportion. Step 2: Enter 0.12 under Null hypothesis. Step 3: Enter 0.01 under Level of significance. Step 4: Enter 58 under Number of Items Interest. Step 5: Enter 343 under Sample size. Step 6: In Test Options, choose Two Tail Test. Step 7: Click OK Excel add-in (PH Stat) output:

Z Test of Hypothesis for the Proportion 0.12 Data Null Hypothesis n = Level of significance Number of Items of Interest Sample Size 0.01 58 343 Intermediate Calculations Sample Proportion 0.16909621 Standard Error 0.0175 Z Test Statistic 2.7981 Two-Tail Test Lower Critical Value -2.5758 Upper Critical Value 2.5758 p-Value 0.0051 Reject the null hypothesis The value of the test statistic is z = 2.798. The p-value = 0.0051. The p-value is sa

(d) We should reject the null hypothesis. The data suggest the population proportion is significantly different from 12% at aa=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 different from 12%. If the sample proportion of convicted sex offender drug addicts who become repeat offenders is 17% and if another 343 convicted sex offender drug addicts are observed then there would be a 0.52000000000001% chance that we would conclude either fewer than 12% of all convicted sex

offender drug addicts become repeat offenders or more than 12% of all convicted sex offender drug addicts become repeat offenders. If the population proportion of convicted sex offender drug addicts who become repeat offenders is different from 12% and if another 343 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 12%.