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Statistics

David Freedman, Robert Pisani, Roger Purves

Chapter 28

The Chi-Square Test - all with Video Answers

Educators

Chapter Questions

01:49

Problem 1

You are drawing 100 times at random with replacement from a box. Fill in
the blanks, using the options below.
(a) To test the null hypothesis that the average of the box is $2,$ you would
use
(b) To test the null hypothesis that the box is $\lfloor\mathbb{1}[2][3],$ you would
use
Options (some may not be used):
(i) the one-sample $z$ -test.
(ii) the two-sample z-test.
(iii) the $\chi^{2}$ -test, with a null hypothesis that tells you the contents of the
box (section $1 ) .$
(iv) the $\chi^{2}$ -test for independence (section 4$)$

Bryan Luo
Bryan Luo
Numerade Educator
05:56

Problem 2

As part of a study on the selection of grand juries in Alameda county, the ed-
ucational level of grand jurors was compared with the county distribution:
Could a simple random sample of 62 people from the county show a distribu-
tion of educational level so different from the county-wide one? Choose one
option and explain.
$$
\begin{array}{l}{\text { (i) This is absolutely impossible. }} \\ {\text { (ii) This is possible, but fantastically unlikely. }} \\ {\text { (iv) This is qossible but unlikely-the chance is around } 1 \% \text { or so. }} \\ {\text { (iv) This is quite possible-the chance is around } 10 \% \text { or so. }} \\ {\text { (v) This is nearly certain. }}\end{array}
$$

Heena Haldankar
Heena Haldankar
Numerade Educator
02:03

Problem 3

Each respondent in the Current Population Survey of March 2005 was classi-
fied as employed, unemployed, or outside the labor force. The results for men
in Califomia age $35-44$ can be cross-tabulated by marital status, as follows:
Men of different marital status seem to have different distributions of labor
force status. Or is this just chance variation? (You may assume the datacome
from a simple random sample.)

Bryan Luo
Bryan Luo
Numerade Educator
05:40

Problem 4

(a) Does the histogram in figure 2 represent data, or chance?
(b) There is a block over the interval from 5 to $5.2 .$ What does the area of this
block represent? (Ranges include the left endpoint, but not the right.)
(c) Which chance is larger for 60 throws of a die? Or can this be determined
from figure 2$?$
(i) The chance that the $x^{2}$ -statistic is in the range from 4.8 to 5.0 .
(ii) The chance that the $\chi$ -statistic is in the range from 5.0 to 5.2 .
(d) If $x^{2}=10,$ then $P$ is about
1$\% \quad 10 \% \quad 25 \% \quad 50 \% \quad$ cannot be determined from the figure

Heena Haldankar
Heena Haldankar
Numerade Educator
01:12

Problem 5

An investigator makes a $x^{2}$ -tegt, to see whether the observed frequencies are
too far from the expected frequencies.
(a) If $x^{2}=15,$ the $P$ -value will be bigger with $\longrightarrow$ degrees of freedom
than with $\longrightarrow$ degrees of freedom. Options: $5,10$
(b) If there are 10 degrees of freedom, the $P$ -value will be bigger with
$x^{2}=$ than with $x^{2}=$ Options: $15,20$
No calculations are needed, just look at figure $1 .$

Bryan Luo
Bryan Luo
Numerade Educator
04:57

Problem 6

Someone claims to be rolling a pair of fair dice. To test his claim, you make
him roll the dice 360 times, and you count up the number of times each sum
appears. The results are shown below. (For your convenience, the chance of
throwing each sum with a pair of fair dice is shoun too.) Should you play
craps with individual? Or abserved frequencies too close to the
expected frequencies?

Heena Haldankar
Heena Haldankar
Numerade Educator
04:07

Problem 7

The International Rice Research Institute in the Philippines develops new
lines of rice which combine high yields with resistance to disease and insects.
The technique involves crossing different lines to get a new line which has
the most advantageous combination of genes. Detailed genetic modeling is
required. One project involved breeding new lines to results the "brown plant
hopper"" (an insect): 374 lines were raised, with the results shown below.
According to the IRRI model, the lines are independent: each line has a 25$\%$
chance to be resistant, a 50$\%$ chance to be mixed, and a 25$\%$ chance to be
susceptible. Are the facts consistent with this model?

Bryan Luo
Bryan Luo
Numerade Educator
02:05

Problem 8

Two people are tying to decide whether a die is fair. They roll it 100 times,
with the results shown at the top of the next page. One person wants to make
a z-test, the other wants to make a $x^{2}$ -test. Who is right? Explain briefly.
$$
\approx 3.43, \mathrm{sD} \approx 1.76
$$

Heena Haldankar
Heena Haldankar
Numerade Educator
02:18

Problem 9

Each respondent in the Current Population Survey of March 2005 can be
classified by age and marital status. The table below shows results for women
age $20-29$ in Montana.
Question A. Women of different ages seem to have different
distributions of marital status. Or is this just chance variation?
Question B. If the difference is real, what accounts for it?
(a) Can you answer these questions with the information given? If so, an-
swer them. If not, why not?
(b) Can you answer these questions if the data in the table resulted from a
simple random sample of women age $20-29$ in Montana? If so, answer
them. If not, why not?

Bryan Luo
Bryan Luo
Numerade Educator
08:39

Problem 10

The U.S. has bilateral extradition treaties with many countries. (A person charged with a crime in his home country may escape to the U.S.: if he is captured in the U.S., authonties in his home country may request that he be"extradited," that is, turned over for prosecution under their laws.)
The Senate attached a special rider to the treaty governing extradition to Northern Ireland: fugitives cannot be retumed if they will be discriminated against on the basis of religion. In a leading case, the defense tried to establish discrimination in Northern Ireland's criminal justice system.
One argument was based on 1991 acquital rates for persons charged with terrorist offenses. $^{17}$ According to a defense expert, these rates were significantly
different for Protestants and Catholics: $\chi^{2} \approx 6.2$ on 1 degree of freedom,
$P \approx 1 \% .$ The data are shown below: 8 Protestants out of 15 were acquitted,
compared to 27 Catholics out of $65 .$
(a) Is the calculation of $x^{2}$ correct? If not, can you guess what the mistake
was? (That might be quite difficult.)
(b) What box model did the defense have in mind? Comment brieffy on
the model.

Heena Haldankar
Heena Haldankar
Numerade Educator