Question

A criminologist developed a test to measure recidivism, where low scores indicated a lower probability of repeating the undesirable behavior. The test is normed so that it has a mean of 140 and a standard deviation of 40.
a. What is the percentile rank of a score of 172?

b. What is the Z score for a test score of 200?

c. What percentage of scores falls between 100 and 160?

d. What proportion of respondents should score above 190?

e. Suppose an individual is in the 67th percentile in this test, what is his or her corresponding recidivism score?

Answer 1

Answer:

Step-by-step explanation:

We know that when X is normal

X-mu/sigma is std normal

Here the scores X is normal with mean =140 and sd =40

a) X=172

gives Z = 32/40 = 0.80

P(Z<0.8) = 0.5+0.2881

=0.7881

Hence 79th percentile

b) X =200 then x-mu = 60

So z=60/40 = 1.5

c) 100<x<160 means

1.5<z<0.5

d) X>190 means Z>1.25 hence proportion = P(Z>1.25) =0.5-0.3944

=0.1056

e) For 67th percentile z value = 0.0675

Hence x score = 140+0.0675(40) = 142.7

Answer 2

a). The percentile of a score of $172$ is $\boxed{79{\text{th}}\,{\text{percentile}}}$.

b). The Z score for a test of 200 is $\boxed{1.5}$.

c). The percentage of score between $100$ to $160$ is $\boxed{53.3\% }$.

d). The proportion of respondents should score above $190$ is $\boxed{0.10565}$.

e). The corresponding score for $67{\text{th}}$ percentile is $\boxed{142.7}$.

Further Explanation:

The Z score of the standard normal distribution can be obtained as,

$\boxed{{\text{Z}} = \dfrac{{X - \mu }}{\sigma }}$

Given:

The mean of test is $\boxed{140}$.

The standard deviation of the score is $\boxed{40}$.

Explanation:

Part (a).

The Z score of $172$ can be obtained as,

$\begin{aligned}{\text{Z}}&= \frac{{172 - 140}}{{40}} \\ &= \frac{{32}}{{40}} \\ &= 0.8 \\ \end{aligned}$

The percentile of a score of $172$ can be obtained as,

${\text{Percentile}} = {\text{P}}\left( {{\text{Z}$

From Z-table the percentile is $78.81\%$.

Approximately $\boxed{79{\text{th}}\,{\text{percentile}}}$.

Part (b).

The Z score for a test score $200$ can be obtained as,

$\begin{aligned} {\text{Z}}&= \frac{{200 - 140}}{{40}} \\ &= \frac{{60}}{{40}} \\ &= 1.5 \\\end{aligned}$

The Z-score is $\boxed {1.5}$.

Part (c).

The percentage of scores fall between $100$ and $160$ can be obtained as,

$\begin{aligned} {\text{Percentage}} &= {\text{P}}\left( {\frac{{100 - 140}}{{40}} < {\text{Z}} < \frac{{160 - 140}}{{40}}} \right) \\ &= {\text{P}}\left( {\frac{{ - 40}}{{40}} < {\text{Z}} < \frac{{20}}{{40}}} \right) \\ &= {\text{P}}\left( { - 1 < {\text{Z}} < 0.5} \right) \\ &= {\text{P}}\left( {{\text{Z} < 0}}{\text{.5}}} \right) - {\text{P}}\left( {{\text{Z} > }} - {\text{1}}} \right) \\ \end{gathered}$

Further solve above equation.

$\begin{aligned}{\text{Percentage}} &= {\text{P}}\left( {{\text{Z} < 0.5}}\right) -\left[ 1 - P({Z$

Approximately the percentage is $\boxed{53.3\% }$.

Part (d).

The proportion of respondent score above $190$ can be obtained as,

$\begin{aligned}{\text{Proportion}} &= {\text{P}}\left( {{\text{Z}} > \frac{{190 - 140}}{{40}}} \right) \\ &= {\text{P}}\left( {{\text{Z}} > \frac{{50}}{{40}}} \right) \\ &= {\text{P}}\left( {{\text{Z}} > 1.25} \right) \\ &= 1 - {\text{P}}\left( {\text{Z}< 1.25} \right) \\ &= 1 - 0.89435 \\ &= 0.10565 \\\end{gathered}$

The proportion of respondents should score above $190$ is $\boxed{0.10565}$.

Part (e).

The Z-value of $67{\text{th}}$ is $0.0675$.

The corresponding score for $67{\text{th}}$ percentile can be obtained as,

$\begin{aligned}0.0675 &= \frac{{X - 140}}{{40}} \\ 40\left( {0.0675} \right) &= X - 140 \\ 2.7 + 140 &= X \\ 142.7 &= X \\ \end{aligned}$

The corresponding score for $67{\text{th}}$ percentile is $\boxed{142.7}$.

Learn more:

1. Learn more about normal distribution brainly.com/question/12698949

2. Learn more about standard normal distribution brainly.com/question/13006989

Answer details:

Grade: College

Subject: Statistics

Chapter: Normal distribution

Keywords: Z-score, standard normal distribution, standard deviation, criminologist, test, measure, probability, low score, mean, repeating, indicated, normal distribution, percentile, percentage, undesirable behavior, proportion.