The recidivism rate for convicted sex offenders is 55%. A warden has implemented

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

Ha: p                            [ Select ]                       ["<", "Not equal to", ">"]         0.55

p-Value =                            [ Select ]                       ["0.0163", "0.0747", "0.0081"]          

                           [ Select ]                       ["Fail to reject Ho", "Reject Ho"]      

Conclusion: There is                            [ Select ]                       ["sufficient", "insufficient"]         evidence to make the conclusion that the population proportion of convicted sex offenders who receive the therapy and then become repeat offenders is less than 0.55.

p-Value Interpretation: If the proportion of convicted sex offenders who receive the therapy and then become repeat offenders is equal to                            [ Select ]                       ["0.55", "0.62", "0.81"]       and if another study was done with a new randomly selected collection of 280 who received the therapy, then there is a                            [ Select ]                       ["1.63", "7.47", "0.81"]         percent chance that the proportion who become repeat offenders for this new sample would be less than                            [ Select ]                       ["0.48", "0.51", "0.62"]         .

Level of significance interpretation: If the proportion of convicted sex offenders who receive the therapy and then become repeat offenders is equal to                           [ Select ]                       ["0.48", "0.55", "0.62"]         and if many studies are done each with a new randomly selected collection of 280 who receive the therapy then                           [ Select ]                       ["5", "10", "1"]         percent of these studies would result in the false conclusion that the proportion of convicted sex offenders who receive the therapy and then become repeat offenders is less than 0.55.

Explanation / Answer

Given that,
possibile chances (x)=142
sample size(n)=280
success rate ( p )= x/n = 0.5071
success probability,( po )=0.55
failure probability,( qo) = 0.45
null, Ho:p=0.55
alternate, H1: p<0.55
level of significance, = 0.05
from standard normal table,left tailed z /2 =1.64
since our test is left-tailed
reject Ho, if zo < -1.64
we use test statistic z proportion = p-po/sqrt(poqo/n)
zo=0.50714-0.55/(sqrt(0.2475)/280)
zo =-1.4415
| zo | =1.4415
critical value
the value of |z | at los 0.05% is 1.64
we got |zo| =1.441 & | z | =1.64
make decision
hence value of |zo | < | z | and here we do not reject Ho
p-value: left tail - Ha : ( p < -1.4415 ) = 0.07472
hence value of p0.05 < 0.07472,here we do not reject Ho
ANSWERS
---------------
H0: p = 0.55
Ha: p  <  0.55
p-Value =   0.07472          
"Fail to reject Ho
There is insufficient evidence to make the conclusion that the population proportion of convicted sex offenders who receive the therapy and then become repeat offenders is less than 0.55.

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