Chapter 7, Hypothesis Testing I Video Solutions, The Essentials of Statistics: A Tool for Social Research

Problem 1

a. For each situation, find $Z$ (critical).
$\begin{array}{ccc}
\hline \text { Alpha } & \text { Form } & \text { Z(Critical) } \\
\hline .05 & \text { One-tailed } & \\
.10 & \text { Two-tailed } & \\
.06 & \text { Two-tailed } & \\
.01 & \text { One-tailed } & \\
.02 & \text { Two-tailed } & \\
\hline
\end{array}$
b. For each situation, find the critical $t$ score.
$\begin{array}{cccc}
\text { Alpha } & \text { Form } & N & t \text { (Critical) } \\
\hline .10 & \text { Two-tailed } & 31 & \\
.02 & \text { Two-tailed } & 24 & \\
.01 & \text { Two-tailed } & 121 & \\
.01 & \text { One-tailed } & 31 & \\
.05 & \text { One-tailed } & 61 & \\
\hline
\end{array}$
c. Compute the appropriate test statistic $(Z$ or $t)$ for each situation:
$\text { 1. } \begin{array}{ll}
\mu=2.40 & \bar{X}=2.20 \\
\sigma=0.75 & N=200
\end{array}$
$\text { 2. } \begin{aligned}
\mu=17.1 & \bar{X}=16.8 \\
s &=0.9
\end{aligned}$
$\text { 3. } \begin{aligned}
\mu=10.2 & \bar{X}=9.4 \\
s &=1.7
\end{aligned}$
$\begin{array}{ll}
\text { 4. } P_{u}=.57 & P_{s}=0.60 \\
& N=117
\end{array}$
$\text { 5. } \begin{aligned}
P_{u}=0.32 & P_{s}=0.30 \\
& N=322
\end{aligned}$

Prabhakar Kumar

Numerade Educator

03:48

Problem 2

The students at Littlewood Regional High School cut an average of $3.3$ classes per week. A random sample of 117 seniors averages $3.8$ cuts per week, with a standard deviation of $0.53 .$ Are seniors significantly different from the student body as a whole? (Hint: The wording of the research question suggests a two-tailed test. This means that the alternative, or research, hypothesis in step 2 will be stated as $H_{1}: \mu \neq 3.3$ and that the critical region will be split between the upper and lower tails of the sampling distribution. See Table $7.1$ for values of Z( critical) for various alpha levels.)

Harsh Gadhiya

Numerade Educator

01:12

Problem 3

What if the research question in problem $7.2$ were changed to "Do seniors cut a significantly greater number of classes"? How would the test change? (Hint: This wording implies a one-tailed test of significance. How would the research hypothesis change? For the alpha you used in problem $7.2$, what would the value of $Z($ critical $)$ be?)

James Kiss

Numerade Educator

02:59

Problem 4

Nationally, social workers average $10.2$ years of experience. In a random sample, 203 social workers in greater metropolitan Shinbone average only $8.7$ years, with a standard deviation of $0.52$. Are social workers in Shinbone significantly less experienced? (Hint: Note the wording of the research hypotheses. This situation may justify a one-tailed test of significance. If you chose a one-tailed test, what form would the research hypothesis take and where would the critical region begin?)

Sheryl Ezze

Numerade Educator

04:27

Problem 5

The same sample of social workers reports an average annual salary of $$\$ 25,782$$ with a standard deviation of $$\$ 622.$$ Is this figure significantly higher than the national average of $$\$ 24,509 ?$$ (Hint: The wording of the research hypotheses suggests a one-tailed test. What form would the research hypothesis take, and where would the critical region begin?)

Lucas Everson

Numerade Educator

00:59

Problem 6

Nat?onally, the average score on the college entrance exams (verbal test) is 453 , with a standard deviation of $95 .$ A random sample of 137 seniors at Littlewood Regional High School shows a mean score of 502 . Is there a significant difference?

Sheryl Ezze

Numerade Educator

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Problem 7

A random sample of 423 Chinese Americans has finished an average of $12.7$ years of formal education, with a standard deviation of $1.7$. Is this significantly different from the national average of $12.2$ years?

Wei Zhang

Numerade Educator

02:55

Problem 8

A sample of 105 sanitation workers for the city of Euonymus, Texas, earns an average of $$\$ 24,375$$ per year. The average salary for all Euonymus city workers is $$\$ 24,230,$$ with a standard deviation of $$\$ 523$$ Is the sanitation workers' salary significantly different from that of city workers as a whole? Conduct both one- and two-tailed tests.

Sheryl Ezze

Numerade Educator

03:33

Problem 9

a. Nationally, the population as a whole watches $6.2$ hours of TV per day. A random sample of 1,017 senior citizens report watching an average of $5.9$ hours per day, with a standard deviation of $0.7$. Is the difference significant?
b. The same sample of senior citizens reports that they belong to an average of $2.1$ voluntary organizations and clubs, with a standard deviation of $0.5$. Nationally, the average is $1.7$. Is the difference significant?

Gus Steppen

Numerade Educator

08:15

Problem 10

A school system has assigned several hundred "chronic and severe" underachievers to an alternative educational experience. To assess the program, a random sample of 35 students has been selected for comparison with all students in the system.
a. In terms of GPA, did the program work?
$\begin{aligned}
&\text { a. In terms of GPA, did the program work? }\\
&\begin{array}{lc}
\hline \hline \text { Systemwide GPA } & \text { Program GPA } \\
\hline \mu=2.47 & \bar{X}=2.55 \\
& s=0.70 \\
& N=35
\end{array}
\end{aligned}$
b. In terms of absenteeism (number of days missed per year), what can be said about the success of the program?
$\begin{array}{ll}
\hline \hline \text { Systemwide } & \text { Program } \\
\hline \mu=6.137 & \bar{X}=4.78 \\
& s=1.11 \\
& N=35 \\
\hline
\end{array}$
c. In terms of standardized test scores in math and
reading, was the program a success?
$\begin{aligned}
&\begin{array}{lc}
\hline \hline \begin{array}{l}
\text { Math Test- } \\
\text { Systemwide }
\end{array} & \begin{array}{c}
\text { Math Test- } \\
\text { Program }
\end{array} \\
\hline \mu=103 & \bar{X}=106 \\
& s=2.0 \\
& N=35 \\
\hline
\end{array}\\
&\begin{array}{lc}
\hline \hline \begin{array}{l}
\text { Reading Test- } \\
\text { Systemwide }
\end{array} & \begin{array}{c}
\text { Reading Test - } \\
\text { Program }
\end{array} \\
\hline \mu=110 & \bar{X}=113 \\
& s=2.0 \\
& N=35 \\
\hline
\end{array}
\end{aligned}$

Sid Wan

University of Louisville

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Problem 11

A random sample of 26 local sociology graduates scored an average of 458 on the GRE advanced sociology test, with a standard deviation of
20. Is this significantly different from the national average $(\mu=440)$ ?

Wei Zhang

Numerade Educator

01:45

Problem 12

Nationally, the per capita monthly fuel oil bill is $$\$ 110.$$ A random sample of 36 southeastern cities averages $$\$ 78,$$ with a standard deviation of $$\$ 4.$$ Is the difference significant? Summarize your conclusions in a sentence or two.

James Kiss

Numerade Educator

02:21

Problem 13

A survey shows that $10 \%$ of the population is victimized by property crime each year. A random sample of 527 older citizens ( 65 years or more of age) shows a victimization rate of $14 \%$. Are older people more likely to be victimized? Conduct both one- and two-tailed tests of significance.

Sheryl Ezze

Numerade Educator

04:04

Problem 14

A random sample of 113 felons convicted of nonviolent crimes in a state prison system completed a program designed to improve their education and work skills before being released on parole. Fifty-eight eventually became repeat offenders. Is this recidivism rate significantly different from the rate for all offenders in that state (57\%)? Summarize your conclusions in a sentence or two. (Hint: You must use the information given in the problem to compute a sample proportion. Remember to convert the population percentage to a proportion.)

Jameson Kuper

Numerade Educator

01:48

Problem 15

In a recent statewide election, $55 \%$ of the voters rejected a proposal to institute a state lottery. In a random sample of 150 urban precincts, $49 \%$ of the voters rejected the proposal. Is the difference significant? Summarize your conclusions in a sentence or two.

James Kiss

Numerade Educator

03:20

Problem 16

Statewide, the police clear by arrest $35 \%$ of the robberies and $42 \%$ of the aggravated assaults reported to them. A researcher takes a random sample of all the robberies $(N=207)$ and aggravated assaults $(N=178)$ reported to a metropolitan police department in one year and finds that 83 of the robberies and 80 of the assaults were cleared by arrest. Are the local arrest rates significantly different from the statewide rate? Write a sentence or two in-
terpreting your decision.

Sheryl Ezze

Numerade Educator

02:00

Problem 17

A researcher has compiled a file of information on a random sample of 317 families that have chronic, long-term patterns of child abuse. Reported here are some of the characteristics of the sample, along with values for the city as a whole. For each trait, test the null hypothesis of “no difference” and summarize your findings.

Joshua Argo

Numerade Educator

03:01

Problem 18

You are the head of an agency seeking funding for a program to reduce unemployment among teenage males. Nationally, the unemployment rate for this group is 18%. A random sample of 323 teenage males in your area reveals an unemployment rate of 21.7%. Is the difference significant? Can you demonstrate a need for the program? Should you use a one-tailed test in this situation? Why? Explain the result of your test of significance as you would to a funding agency.

Sheryl Ezze

Numerade Educator

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Problem 19

PA The city manager of Shinbone has received a complaint from the local union of firefighters to the effect that they are underpaid. Not having much time, the city manager gathers the records of a random sample of 27 firefighters and finds that their average salary is $$\$ 38,073,$$ with a standard deviation of $$\$ 575.$$ If she knows that the average salary nationally is $$\$ 38,202,$$ how can she respond to the complaint? Should she use a one-tailed test in this situation?
Why? What would she say in a memo to the union that would respond to the complaint?

Rashmi Sinha

Numerade Educator

13:55

Problem 20

The following essay questions review the basic principles and concepts of inferential statistics.
The order of the questions roughly follows the fivestep model.
a. Hypothesis testing or significance testing can be conducted only with a random sample. Why?
b. Under what specific conditions can it be assumed that the sampling distribution is normal in shape?
c. Explain the role of the sampling distribution in a test of hypothesis.
d. The null hypothesis is an assumption about reality that makes it possible to test sample outcomes for their significance. Explain.
e. What is the critical region? How is the size of the critical region determined?
f. Describe a research situation in which a onetailed test of hypothesis would be appropriate.
g. Thinking about the shape of the sampling distribution, why does use of the $t$ distribution (as opposed to the $\mathrm{Z}$ distribution) make it more difficult to reject the null hypothesis?
h. What exactly can be concluded in the onesample case when the test statistic falls into the critical region?

Harsh Gadhiya

Numerade Educator

02:10

Problem 21

A researcher is studying changes in the student body at her university and has selected a random sample of 163 from the freshman class.
The table below compares their characteristics to those of the student body as a whole. Which differences are significant?
$\begin{array}{llcl}
& \text { Variable } & \text { Freshmen } & \text { All Students } \\
\hline \text { a. } & \text { Proportion Republican } & P_{s}=0.44 & P_{u}=0.49 \\
\text { b. } & \text { Proportion majoring in math and science } & P_{s}=0.17 & P_{u}=0.09 \\
\text { c. } & \text { Average family income } & \bar{X}=63,542 & \mu=59,235 \\
& & s=1,568 & \\
\text { d. } & \text { Average number of siblings } & \bar{X}=0.9 & \mu=1.2 \\
& & s=0.2 & \\
\text { e. } & \text { Average number of telephone calls } & \bar{X}=1.5 & \mu=0.4 \\
& \text { to parents per week } & s=0.4 & \\
\text { f. } & \text { Average number of close friends living } & \bar{X}=4.5 & \mu=7.8 \\
& \text { on campus } & s=0.3 & \\
\hline
\end{array}$

Bon Zapata

Numerade Educator