Essentials of Statistics for the Behavioral Science

Frederick J Gravetter, Larry B. Wallnau, Jon-David Hague

Chapter 8 Introduction to Hypothesis Testing - all with Video Answers

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Chapter Questions

Problem 1

In the $z$-score formula as it is used in a hypothesis test,
a.Explain what is measured by $M-\mu$ in the numerator.
b.Explain what is measured by the standard error in the denominator.

Karen Song

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The value of the $z$-score in a hypothesis test is influenced by a variety of factors. Assuming that all other variables are held constant, explain how the value of $z$ is influenced by each of the following:
a.Increasing the difference between the sample mean and the original population mean.
b.Increasing the population standard deviation.
c.Increasing the number of scores in the sample.

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Problem 3

In words, define the alpha level and the critical region for a hypothesis test.

Rashmi Sinha

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02:38

Problem 4

If the alpha level is changed from $\alpha=.05$ to $\alpha=.01$,
a. What happens to the boundaries for the critical region?
b. What happens to the probability of a Type I error?

Melissa A

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03:45

Problem 5

Although there is a popular belief that herbal remedies such as Ginkgo biloba and Ginseng may improve learning and memory in healthy adults, these effects are usually not supported by wellcontrolled research (Persson, Bringlov, Nilsson, \& Nyberg, 2004). In a typical study, a researcher obtains a sample of $n=36$ participants and has each person take the herbal supplements every day for 90 days. At the end of the 90 days, each person takes a standardized memory test. For the general population, scores from the test are normally distributed with a mean of $\mu=$ 80 and a standard deviation of $\mu=18$. The sample of research participants had an average of $M=84$.
a.In a sentence, state the null hypothesis being tested.
b.Using symbols, state the null hypothesis and the alternative
hypothesis.
c.Conduct the hypothesis test using a two-tailed test with $\alpha=$ 05 .

Sheryl Ezze

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01:38

Problem 6

A researcher is investigating the effectiveness of a new studyskills training program for elementary school children. A sample of $n=25$ third-grade children is selected to participate in the program and each child is given a standardized achievement test at the end of the year. For the regular population of third-grade children, scores on the test form a normal distribution with a mean of $\mu=150$ and a standard deviation of $\sigma=25$. The mean for the sample is $M=158$.
a.Identify the independent and the dependent variables for this study.
b.Assuming a two-tailed test, state the null hypothesis in a sentence that includes the independent variable and the dependent variable.
c.Using symbols, state the hypotheses $\left(H_0\right.$ and $\left.H_1\right)$ for the twotailed test.
d.Sketch the appropriate distribution, and locate the critical region for $\alpha=.05$.
e. calculate the test statistic ( $z$-score) for the sample.
f. What decision should be made about the null hypothesis, and what decision should be made about the effect of the program?

Dominador Tan

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04:04

Problem 7

Childhood participation in sports, cultural groups, and youth groups appears to be related to improved self-esteem for adolescents (McGee, Williams, Howden-Chapman, Martin, \& Kawachi, 2006). In a representative study, a sample of $n=100$ adolescents with a history of group participation is given a standardized self-esteem questionnaire. For the general population of adolescents, scores on this questionnaire form a normal distribution with a mean of $\mu=40$ and a standard deviation of $\mu=12$. The sample of group-participation adolescents had an average of $M=43.84$.
a. Does this sample provide enough evidence to conclude that self-esteem scores for these adolescents are significantly different from those of the general population? Use a two-tailed test with $\alpha$ $=.01$.
b. Compute Cohens's $d$ to measure the size of the difference.
c. Write a sentence describing the outcome of the hypothesis test and the measure of effect size as it would appear in a research report.

Sheryl Ezze

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06:25

Problem 8

State College is evaluating a new English composition course for freshmen. A random sample of $n=25$ freshmen is obtained and the students are placed in the course during their first semester. One year later, a writing sample is obtained for each student and the writing samples are graded using a standardized evaluation technique. The average score for the sample is $M=76$. For the general population of college students, writing scores form a normal distribution with a mean of $\mu=70$.
a.If the writing scores for the population have a standard deviation of $\sigma=20$, does the sample provide enough evidence to conclude that the new composition course has a significant effect? Assume a two-tailed test with $\alpha=.05$.
d.If the population standard deviation is $\sigma=10$, is the sample sufficient to demonstrate a significant effect? Again, assume a two-tailed test with $\alpha=.05$.
c.Comparing your answers for parts a and $b$, explain how the magnitude of the standard deviation influences the outcome of a hypothesis test.

Sheryl Ezze

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03:31

Problem 9

A random sample is selected from a normal population with a mean of $\mu=50$ and a standard deviation of $\sigma=12$. After a treatment is administered to the individuals in the sample, the sample mean is found to be $M=55$.
a.If the sample consists of $n=16$ scores, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with $\alpha=.05$.
b.If the sample consists of $n=36$ scores, is the sample mean sufficient to conclude that the treatment has a significant effect. Use a two-tailed test with $\alpha=.05$.
c.Comparing your answers for parts a and $b$, explain how the size of the sample influences the outcome of a hypothesis test.

Sheryl Ezze

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Problem 10

A random sample of $n=36$ scores is selected from a normal population with a mean of $\mu=60$. After a treatment is administered to the individuals in the sample, the sample mean is found to be $M=52$.
a.If the population standard deviation is $\sigma=18$, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with $\alpha=.05$.
b.If the population standard deviation is $\sigma=30$, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with $\alpha=.05$.
c.Comparing your answers for parts a and b, explain how the magnitude of the standard deviation influences the outcome of a hypothesis test.

Rashmi Sinha

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Problem 11

Miller (2008) examined the energy drink consumption of college undergraduates and found that males use energy drinks significantly more often than females. To further investigate this phenomenon, suppose a researcher selects a random sample of $n$ $=36$ male undergraduates and a sample of $n=25$ females. On average, the males reported consuming $M=2.45$ drinks per month and females had an average of $M=1.28$. Assume that the overall level of consumption for college undergraduates averages $\mu=1.85$ energy drinks per month, and that the distribution of monthly consumption scores is approximately normal with a standard deviation of $\sigma=1.2$.
a.Did this sample of males consume significantly more energy drinks than the overall population average? Use a one-tailed test with $\alpha=.01$.
b.Did this sample of females consume significantly fewer energy drinks than the overall population average? Use a onetailed test with $\alpha=.01$

Rashmi Sinha

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Problem 12

A random sample is selected from a normal population with a mean of $\mu=40$ and a standard deviation of $\sigma=10$. After a treatment is administered to the individuals in the sample, the sample mean is found to be $M=42$.
a.How large a sample is necessary for this sample mean to be statistically significant? Assume a two-tailed test with $\alpha=.05$
b.If the sample mean were $M=41$, what sample size is needed to be significant for a two-tailed test with $\alpha=.05$.

Rashmi Sinha

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03:09

Problem 13

There is some evidence that REM sleep, associated with dreaming, may also play a role in learning and memory processing. For example, Smith and Lapp (1991) found increased REM activity for college students during exam periods. Suppose that REM activity for a sample of $n=16$ students during the final exam period produced an average score of $M=143$. Regular REM activity for the college population averages $\mu=110$ with a standard deviation of $\sigma=50$. The population distribution is approximately normal.
a.Do the data from this sample provide evidence for a significant increase in REM activity during exams? Use a one-tailed test with $\alpha=.01$.
b.Compute Cohen's $d$ to estimate the size of the effect.
c.Write a sentence describing the outcome of the hypothesis test and the measure of effect size as it would appear in a research report

Lucas Finney

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Problem 14

There is some evidence indicating that people with visible tattoos are viewed more negatively than people without visible tattoos (Resenhoeft, Villa, \& Wiseman, 2008). In a similar study, a researcher first obtained overall ratings of attractiveness for a woman with no tattoos shown in a color photograph. On a 7-point scale, the woman received an average rating of $\mu=4.9$, and the distribution of ratings was normal with a standard deviation of $\sigma$ $=0.84$. The researcher then modified the photo by adding a tattoo of a butterfly on the woman's left arm. The modified photo was then shown to a sample of $n=16$ students at a local community college and the students used the same 7-point scale to rate the attractiveness of the woman. The average score for the photo with the tattoo was $M=4.2$.
a.Do the data indicate a significant difference in rated attractiveness when the woman appeared to have a tattoo? Use a two-tailed test with $\alpha=.05$.
b.Compute Cohen's $d$ to measure the size of the effect.
c.Write a sentence describing the outcome of the hypothesis test and the measure of effect size as it would appear in a research report.

Victor Salazar

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04:04

Problem 15

A psychologist is investigating the hypothesis that children who grow up as the only child in the household develop different personality characteristics than those who grow up in larger families. A sample of $n=30$ only children is obtained and each child is given a standardized personality test. For the general population, scores on the test from a normal distribution with a mean of $\mu=50$ and a standard deviation of $\sigma=15$. If the mean for the sample is $M=58$, can the researcher conclude that there is a significant difference in personality between only children and the rest of the population? Use a two-tailed test with $\alpha=.05$.

Sheryl Ezze

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Problem 16

A researcher is testing the hypothesis that consuming a sports drink during exercise improves endurance. A sample of $n=50$ male college students is obtained and each student is given a series of three endurance tasks and asked to consume 4 ounces of the drink during each break between tasks. The overall endurance score for this sample is $M=53$. For the general population, without any sports drink, the scores for this task average $\mu=50$ with a standard deviation of $\sigma=12$.
a.Can the researcher conclude that endurance scores with the sports drink are significantly higher than scores without the drink? Use a one-tailed test with $\alpha=.05$.
b.Can the researcher conclude that endurance scores with the sports drink are significantly different than scores without the drink? Use a two-tailed test with $\alpha=.05$.
c.You should find that the two tests lead to different conclusions. Explain why.

Rashmi Sinha

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06:30

Problem 17

Montarello and Martins (2005) found that fifth-grade students completed more mathematics problems correctly when simple problems were mixed in with their regular math assignments. To further explore this phenomenon, suppose that a researcher selects a standardized mathematics achievement test that produces a normal distribution of scores with a mean of $\mu=100$ and a standard deviation of $\sigma=18$. The researcher modifies the test by inserting a set of very easy problems among the standardized questions, and gives the modified test to a sample of $n=36$ students. If the average test score for the sample is $M=104$, is this result sufficient to conclude that inserting the easy questions improves student performance? Use a one-tailed test with $\alpha=.01$.

Ahmad Reda

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03:08

Problem 18

Researchers have often noted increases in violent crimes when it is very hot. In fact, Reifman, Larrick, and Fein (1991) noted that this relationship even extends to baseball. That is, there is a much greater chance of a batter being hit by a pitch when the temperature increases. Consider the following hypothetical data. Suppose that over the past 30 years, during any given week of the major-league season, an average of $\mu=12$ players are hit by wild pitches. Assume that the distribution is nearly normal with $\sigma=3$. For a sample of $n=4$ weeks in which the daily temperature was extremely hot, the weekly average of hit-by-pitch players was $M$ $=15.5$. Are players more likely to get hit by pitches during hot weeks? Set alpha to .05 for a one-tailed test.

Heena Haldankar

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Problem 19

Briefly explain how increasing sample size influences each of the following. Assume that all other factors are held constant.
a.The size of the $z$-score in a hypothesis test.
b. The size of Cohen's $d$.
c.The power of a hypothesis test.

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Problem 20

Explain how the power of a hypothesis test is influenced by each of the following. Assume that all other factors are held constant.
a.Increasing the alpha level from .01 to .05 .
b. Changing from a one-tailed test to a two-tailed test.

Melissa A

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10:44

Problem 21

A researcher is investigating the effectiveness of a new medication for lowering blood pressure for individuals with systolic pressure greater than 140 . For this population, systolic scores average $\mu=160$ with a standard deviation of $\sigma=20$, and the scores form a normal-shaped distribution. The researcher plans to select a sample of $n=25$ individuals, and measure their systolic blood pressure after they take the medication for 60 days. If the researcher uses a two-tailed test with $\alpha=.05$,
a. What is the power of the test if the medication has a 5-point effect?
b. What is the power of the test if the medication has a 10-point effect?

Ahmad Reda

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10:44

Problem 22

A researcher is evaluating the influence of a treatment using a sample selected from a normally distributed population with a mean of $\mu=80$ and a standard deviation of $\sigma=20$. The researcher expects a 12-point treatment effect and plans to use a two-tailed hypothesis test with $\alpha=.05$.
a.Compute the power of the test if the researcher uses a sample of $n=16$ individuals. (See Example 8.6.)
b.Compute the power of the test if the researcher uses a sample of $n=25$ individuals.

Ahmad Reda

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10:44

Problem 23

A researcher expects a treatment to increase scores by 5 points. The regular population, without treatment, averages $\mu=40$ with a standard deviation of $\sigma=8$, and the scores form a normal distribution. If the researcher uses a one-tailed test with $\alpha=.01$,
a. What is the power of the test for a sample of $n=16$ ?
b. What is the power of the test for a sample of $n=64$ ?

Ahmad Reda

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