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Statistics Learning From Data

Thomas H. Short, Roxy Peck

Chapter 14

Learning from Categorical Data - all with Video Answers

Educators

Section 1

Chi-Square Tests for Univariate Categorical Data

01:33

Problem 1

A particular cell phone case is available in a choice of four different colors. A store sells all four colors. To test the hypothesis that sales are equally divided among the four colors, a random sample of 100 purchases is identified.
a. If the resulting $X^{2}$ value were $6.4,$ what conclusion would you reach when using a test with significance level $0.05 ?$
b. What conclusion would be appropriate at significance level 0.01 if $X^{2}=15.3 ?$
c. If there were six different colors rather than just four, what would you conclude if $X^{2}=13.7$ and a test with $\alpha=0.05$ was used?

Vanna Tran
Vanna Tran
Numerade Educator
06:06

Problem 2

What is the approximate $P$ -value for the following values of $X^{2}$ and df?
a. $X^{2}=6.62, \mathrm{df}=3$
b. $X^{2}=16.97, \mathrm{df}=10$
c. $X^{2}=30.19, \mathrm{df}=17$

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
01:13

Problem 3

14.3 The authors of the paper "Is It Really About Me? Message Content in Social Awareness Streams" (Computer Supported Cooperative Work 2010) studied a random sample of 350 Twitter users. For each Twitter user in the sample, the tweets sent during a particular time period were analyzed and the Twitter user was classified into one of the following categories based on the type of messages they usually sent:
The accompanying table gives the observed counts for the five categories (approximate values from a graph in the paper).
Carry out a hypothesis test to determine if there is convincing evidence that the proportions of Twitter users falling into each of the five categories are not all the same. Use a significance level of $\alpha=0.05 .$ (Hint: See Example 14.3.)

Tyler Moulton
Tyler Moulton
Numerade Educator
06:19

Problem 4

The article "In Bronx, Hitting Home Runs Is a Breeze" (USA TODAY, June 2, 2009) included a classification of 87 home runs hit at the new Yankee Stadium according to the direction that the ball was hit, resulting in the accompanying data.
a. Assume that this sample of 87 home runs is representative of home runs hit at Yankee Stadium. Carry out a hypothesis test to determine if there is convincing evidence that the proportions of home runs hit are not the same for all five directions.
b. Write a few sentences describing how the observed counts for the five directions differ from what would have been expected if the proportion of home runs were the same for all five directions.

Gus Steppen
Gus Steppen
Numerade Educator
02:09

Problem 5

In $2014,$ the University of Houston carried out a study for the Texas Lottery Commission ("Demographic Survey of Texas Lottery Players," www.uh.edu/class/hobby/_docs /Texas\%20Lottery\%20Study\%202014.pdf, retrieved May 27 2017). The accompanying table gives the age distribution for a representative sample of 375 Texas Lottery players.

Christopher Stanley
Christopher Stanley
Numerade Educator
06:06

Problem 6

What is the approximate $P$ -value for the following values of $X^{2}$ and df?
a. $X^{2}=14.44, \mathrm{df}=6$
b. $X^{2}=16.91, \mathrm{df}=9$
c. $X^{2}=32.32, \mathrm{df}=20$

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
01:22

Problem 7

Packages of mixed nuts made by a certain company contain four types of nuts. The percentages of nuts of Types $1,2,3,$ and 4 are advertised to be $40 \%, 30 \%, 20 \%,$ and $10 \%,$ respectively. A random sample of nuts is selected, and each one is categorized by type.
a. If the sample size is $n=200$ and the resulting test statistic value is $X^{2}=19.0,$ what conclusion would be appropriate for a significance level of $0.001 ?$
b. If the random sample had consisted of only 40 nuts, would you use the chi-square goodness-of-fit test? Explain your reasoning.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
01:57

Problem 8

The authors of the paper "Talking Smack: Verbal Aggression in Professional Wrestling" (Communication Studies [2008]: 242-258) analyzed the content of 36 hours of televised professional wrestling. Each instance of verbal aggression was classified according to the type of aggression, resulting in the frequency table at the bottom of the page. Assume that this sample of 804 verbal aggression incidents is representative of all verbal aggression incidents in televised professional wrestling. Carry out a hypothesis test to determine if there is convincing evidence that the proportions of verbal aggression incidents are not the same for all five types of aggression. Use a significance level of 0.01 .

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:37

Problem 9

The "Global Automotive 2016 Color Popularity Report" (Axalta Coating Systems, www.axaltacs.com, retrieved May 27,2017 ) included data on the colors for a sample of new cars sold in North America. The report stated that $25 \%$ of the cars in the sample were white, $21 \%$ were black, $16 \%$ were grey, $11 \%$ were silver, $10 \%$ were red, and $17 \%$ were some other color. Suppose that these percentages were based on a random sample of 1200 new cars sold in North America. Is there convincing evidence that the proportions of new cars sold are not the same for all of the six color categories?

Bon Zapata
Bon Zapata
Numerade Educator
01:44

Problem 10

A popular urban legend is that more babies than usual are born during certain phases of the lunar cycle, especially near the full moon. The paper "The Effect of the Gravitation of the Moon on Frequency of Births" (Environmental Health Insights [2010]: 65-69) classified a random sample of 1007 births at a large hospital in Japan according to lunar phase. In each lunar cycle (27.32 days), the moon moves 360 degrees relative to the earth. To determine lunar phase, the researchers divided the 360 degrees in one lunar cycle into 12 phases of 30 degrees. The sample data are summarized in the accompanying frequency table.
The researchers concluded that the frequency of births is not related to lunar cycle. Carry out a chi-square goodness-of-fit test to determine if the data are consistent with the researchers' claim. Use a significance level of 0.05 for your test.

James Kiss
James Kiss
Numerade Educator
06:06

Problem 11

What is the approximate $P$ -value for the following values of $X^{2}$ and $\mathrm{df}$ ?
a. $X^{2}=34.52, \mathrm{df}=13$
b. $X^{2}=39.25, \mathrm{df}=16$
c. $X^{2}=26.00, \mathrm{df}=19$

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
01:14

Problem 12

For which of the $X^{2}$ and df pairs in the previous exercise would the null hypothesis be rejected if a significance level of $\alpha=0.01$ were used?

Tyler Moulton
Tyler Moulton
Numerade Educator
03:22

Problem 13

Think about how you would answer the following question.
Next Wednesday's meeting has been moved forward two days. What day is the meeting now that it has been rescheduled?
This question is ambiguous, as "moved forward" can be interpreted in two different ways. Would you have answered Monday or Friday? The authors of the paper "Even Abstract Motion Influences the Understanding of Time" (Metaphor and Symbol [2011]: $260-271$ ) wondered if the answers Monday and Friday would be provided an equal proportion of the time. Each student in a random sample of students at Stanford University was asked this question, and the responses are summarized in the following table.
The authors of the paper used a chi-square goodness-of-fit test to test the null hypothesis $H_{0}: p_{1}=0.50, p_{2}=0.50,$ where $p_{1}$ is the proportion who would respond Monday, and $p_{2}$ is the proportion who would respond Friday. They reported $X^{2}=11.00$ and $P$ -value $<0.001$. What conclusion can be drawn from this test?

Nick Johnson
Nick Johnson
Numerade Educator
20:20

Problem 14

A certain genetic characteristic of a particular plant can appear in one of three forms (phenotypes). A researcher has developed a theory, according to which the hypothesized proportions are $p_{1}=0.25, p_{2}=0.50$, and $p_{3}=0.25$. A random sample of 200 plants yields $X^{2}=4.63$.
a. Carry out a test of the null hypothesis that the theory is correct, using level of significance $\alpha=0.05$.
b. Suppose that a random sample of 300 plants had resulted in the same value of $X^{2}$. How would your analysis and conclusion differ from those in Part (a)?

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator
20:20

Problem 15

A certain genetic characteristic of a particular plant can appear in one of three forms (phenotypes). A researcher has developed a theory, according to which the hypothesized proportions are $p_{1}=0.25, p_{2}=0.50,$ and $p_{3}=0.25 .$ A random sample of 200 plants yields $X^{2}=4.63$.
a. Carry out a test of the null hypothesis that the theory is correct, using level of significance $\alpha=0.05$.
b. Suppose that a random sample of 300 plants had resulted in the same value of $X^{2}$. How would your analysis and conclusion differ from those in Part (a)?
Mendel's laws of inheritance imply that $p_{1}=\frac{9}{16}, p_{2}=\frac{3}{16}$, $p_{3}=\frac{3}{16},$ and $p_{4}=\frac{1}{16} .$ Are the data from this experiment consistent with Mendel's laws? Use a 0.01 significance level.

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator
05:32

Problem 16

Suppose that each observation in a random sample of 100 fatal bicycle accidents in 2015 was classified according to the day of the week on which the accident occurred. Data consistent with information on the Insurance Institute for Highway Safety website (www.iihs.org/iihs/topics/t/generalstatistics/fatalityfacts/overview-of-fatality-facts) are given in the following table.

Jon Southam
Jon Southam
Numerade Educator
02:43

Problem 17

Birds use color to select and avoid certain types of food. The authors of the article "Colour Avoidance in Northern Bobwhites: Effects of Age, Sex, and Previous Experience" (Animal Behaviour [1995]: 519-526) studied the pecking behavior of 1 -day-old bobwhites. In an area painted white, they inserted four pins with different colored heads. The color of the pin chosen on the bird's first peck was noted for each of 33 bobwhites, resulting in the accompanying table.
Do these data provide evidence of a color preference? Test using $\alpha=0.01$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:49

Problem 18

The authors of the paper "External Factors and the Incidence of Severe Trauma: Time, Date, Season and Moon" (Injury [2014]: S93-S99) classified admissions to hospitals in Germany according to season. They wondered if severe trauma injuries were more common in some seasons than others. For purposes of this exercise, assume that there were 1200 trauma cases in the sample and that the sample is representative of severe trauma injuries in Germany. The data in the accompanying table are consistent with summary quantities given in the paper. Do these data support the theory that the proportion of severe trauma cases is not the same for the four seasons? Test the relevant hypotheses using a significance level of 0.05 .

Teresa Wray
Teresa Wray
Numerade Educator