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Chapter 9: Q54 E (page 398)

Teen Court is a juvenile diversion program designed to circumvent the formal processing of first-time juvenile offenders within the juvenile justice system. The article "An Experimental Evaluation of Teen Courts" (J. of Experimental Criminology, 2008: 137-163) reported on a study in which offenders were randomly assigned either to Teen Court or to the traditional Department of Juvenile Services method of processing. Of the \(56TC\) individuals, 18 subsequently recidivated (look it up!) during the 18 -month follow-up period, whereas 12 of the 51 DJS individuals did so. Does the data suggest that the true proportion of TC individuals who recidivate during the specified follow-up period differs from the proportion of DJS individuals who do so? State and test the relevant hypotheses using a significance level of 0.10.

Short Answer

Expert verified

There is not sufficient evidence to support the claim that the true proportion of TC individuals who recidivate during the specified follow-up period differs from the proportion of DJS individuals who do so.

Step by step solution

01

Step 1: Given information

\(\begin{array}{l}{\rm{Sample size of first sample: }}{n_1} = 56{\rm{ and }}{x_1} = 18\\{\rm{Sample size of second sample:}}{n_2} = 51{\rm{ and }}{x_2} = 12\\{\rm{Level of significance }}\alpha = 0.10\end{array}\)

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis and the alternative hypothesis state the opposite of each other. The null hypothesis needs to contain an equality.

\(\begin{array}{l}{H_0}:{p_1} = {p_2}\\{H_a}:{p_1} \ne {p_2}\end{array}\)

02

Test statistic

\({\rm{ The sample proportion is the number of successes divided by the sample size: }}\)

\(\begin{array}{c}{{\hat p}_1} = \frac{{{x_1}}}{{{n_1}}} = \frac{{18}}{{56}} \approx 0.3214\\{{\hat p}_2} = \frac{{{x_2}}}{{{n_2}}} = \frac{{12}}{{51}} \approx 0.2353\\{{\hat p}_p} = \frac{{{x_1} + {x_2}}}{{{n_1} + {n_2}}}\\ = \frac{{18 + 12}}{{56 + 51}}\\ = \frac{{30}}{{107}}\\ \approx 0.2804\end{array}\)

Determine the value of the test statistic:

\(\begin{array}{c}z = \frac{{{{\hat p}_1} - {{\hat p}_2}}}{{\sqrt {{{\hat p}_p}\left( {1 - {{\hat p}_p}} \right)} \sqrt {\frac{1}{{{n_1}}} + \frac{1}{{{n_2}}}} }}\\ = \frac{{0.3214 - 0.2353}}{{\sqrt {0.2804(1 - 0.2804)} \sqrt {\frac{1}{{56}} + \frac{1}{{51}}} }}\\ \approx 0.99\end{array}\)

03

Finding P-value

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, assuming that the null hypothesis is true. Determine the P-value using table the normal probability table in the appendix:

\(\begin{array}{c}P = P(Z < - 0.99{\rm{ or }}Z > 0.99)\\ = 2P(Z < - 0.99)\\ = 2(0.1611)\\ = 0.3222\end{array}\)

If the P-value is smaller than the significance level, then reject the null hypothesis:

\(P > 0.10 \Rightarrow {\rm{ Fail to reject }}{H_0}\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, assuming that the null hypothesis is true. Determine the P-value using table the normal probability table in the appendix:

\(\begin{array}{c}P = P(Z < - 0.99{\rm{ or }}Z > 0.99)\\ = 2P(Z < - 0.99)\\ = 2(0.1611)\\ = 0.3222\end{array}\)

If the P-value is smaller than the significance level, then reject the null hypothesis:

\(P > 0.10 \Rightarrow {\rm{ Fail to reject }}{H_0}\)

04

Final conclusion

There is not sufficient evidence to support the claim that the true proportion of TC individuals who recidivate during the specified follow-un period differs from the proportion of DJS individuals who do so.

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Most popular questions from this chapter

The article "Flexure of Concrete Beams Reinforced with Advanced Composite Orthogrids"\((J\). of Aerospace Engr., 1997: 7-15) gave the accompanying data on ultimate load\((kN)\)for two different types of beams.

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a. Assuming that the underlying distributions are normal, calculate and interpret a\(99\% \)CI for the difference between true average load for the fiberglass beams and that for the carbon beams.

b. Does the upper limit of the interval you calculated in part (a) give a\(99\% \)upper confidence bound for the difference between the two\(\mu \)'s? If not, calculate such a bound. Does it strongly suggest that true average load for the carbon beams is more than that for the fiberglass beams? Explain.

It is well known that a placebo, a fake medication or treatment, can sometimes have a positive effect just because patients often expect the medication or treatment to be helpful. The article "Beware the Nocebo Effect" (New York Times, Aug. 12, 2012) gave examples of a less familiar phenomenon, the tendency for patients informed of possible side effects to actually experience those side effects. The article cited a study reported in The Journal of Sexual Medicine in which a group of patients diagnosed with benign prostatic hyperplasia was randomly divided into two subgroups. One subgroup of size 55 received a compound of proven efficacy along with counseling that a potential side effect of the treatment was erectile dysfunction. The other subgroup of size 52 was given the same treatment without counseling. The percentage of the no-counseling subgroup that reported one or more sexual side effects was 15.3 %, whereas 43.6 % of the counseling subgroup reported at least one sexual side effect. State and test the appropriate hypotheses at significance level .05 to decide whether the nocebo effect is operating here. (Note: The estimated expected number of "successes" in the no-counseling sample is a bit shy of 10, but not by enough to be of great concern (some sources use a less conservative cutoff of 5 rather than 10).)

An experimenter wishes to obtain a CI for the difference between true average breaking strength for cables manufactured by company I and by company II. Suppose breaking strength is normally distributed for both types of cable with s1 5 30 psi and s2 5 20 psi.

  1. If costs dictate that the sample size for the type I cable should be three times the sample size for the type II cable, how many observations are required if the 99% CI is to be no wider than 20 psi?
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\(\begin{array}{*{20}{c}}{ \ge 93:}&{100}&{100}&{60}&{135}&{195}&{195}&{}\\{}&{125}&{135}&{95}&{42}&{75}&{72}&{}\\{ \le 89:}&{80}&{75}&{75}&{85}&{75}&{35}&{85}\\{}&{65}&{45}&{100}&{28}&{38}&{50}&{28}\end{array}\)

Assume that these are both random samples of prices from the population of all wines recently reviewed that received ratings of at least 93 and at most 89 , respectively.

a. Investigate the plausibility of assuming that both sampled populations are normal.

b. Construct a comparative boxplot. What does it suggest about the difference in true average prices?

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The article "Pine Needles as Sensors of Atmospheric Pollution" (Environ. Monitoring, 1982: 273-286) reported on the use of neutron-activity analysis to determine pollutant concentration in pine needles. According to the article's authors, "These observations strongly indicated that for those elements which are determined well by the analytical procedures, the distribution of concentration is lognormal. Accordingly, in tests of significance the logarithms of concentrations will be used." The given data refers to bromine concentration in needles taken from a site near an oil-fired steam plant and from a relatively clean site. The summary values are means and standard deviations of the log-transformed observations.

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Steamplant 8 18.0 4.9

Clean 9 11.0 4.6

Let \(\mu _1^*\) be the true average log concentration at the first site, and define \(\mu _2^*\) analogously for the second site.

a. Use the pooled t test (based on assuming normality and equal standard deviations) to decide at significance level .05 whether the two concentration distribution means are equal.

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