Question 1

Perform a chi-square independence test using the critical value approach, provided the conditions for using the test aremet. Be sure to state the hypotheses and the significance level, to obtain the expected frequencies, to obtain the criticalvalue, to compute the value of the test statistic, and to state your conclusion.
-Tests for adverse reactions to a new drug yielded the results given in the table. At the 0.05 significance level, do the data provide sufficient evidence to conclude that an association exists between the treatment (drug or placebo)and the reaction (whether or not headaches were experienced)?  $\begin{array} { r | r c }  & \text { Drug } & \text { Placebo } \ \hline \text { Headaches } & 11 & 7 \ \text { No headaches } & 73 & 91 \end{array}$

Accepted Answer

The answer of Perform a chi-square independence test using the...

Question 2

Suppose that you wish to perform a chi-square test of independence. True or false, if the two variables are not associated, then the test statistic $\chi ^ { 2 } = \Sigma \frac { ( \mathrm { O } - \mathrm { E } ) ^ { 2 } } { \mathrm { E } }$ has approximately a chi-square distribution with rc - 1 degrees of freedom, where $r$ and $c$ are the number of possible values for the two variables under consideration? If the statement is false, explain why.

Accepted Answer

The answer of Suppose that you wish to perform a...

Question 3

The contingency table below shows political party affiliation cross classified by age group for a certain population. What information would be given by the marginal distribution of age group? How would you obtain this marginal distribution? What information would be given by the marginal distribution of political party affiliation? How would you obtain this marginal distribution?

Accepted Answer

The answer of The contingency table below shows political party...

Question 4

Suppose that you wish to perform a chi-square test of independence. Give an example of sample data for which assumption 1 (all expected frequencies are 1 or greater)is violated. You should present the sample data in the form of a contingency table.

Accepted Answer

The answer of Suppose that you wish to perform a...

Question 5

$x ^ { 2 }$ -value.
-For a $\chi ^ { 2 }$-curve with 27 degrees of freedom, find the $\chi ^ { 2 }$-value having area $0.01$ to its right.

Accepted Answer

The $\chi^2$-value with an area of 0.01 to its right for a $\chi^2$-curve with 27 degrees of freedom is $46.963$. This value is typically found using a $\chi^2$ distribution table or statistical software, where you look up the critical value corresponding to the right-tail probability of 0.01 for 27 degrees of freedom.

Question 6

Use the contingency table to solve the problem.
-The following contingency table shows the popular votes cast in the 1984 presidential election cross classified by region and political party. Data are in thousands, rounded to the nearest thousand.  What percentage of the voters were in the Midwest?

Accepted Answer

The answer of Use the contingency table to solve the...

Question 7

$\text { True or false, the total area under a } \chi ^ { 2 } \text {-curve is equal to } 1 \text { ? }$

Accepted Answer

The total area under a chi-square (χ^2) distribution curve is equal to 1, as it is a probability distribution.

Question 8

What is the null hypothesis for the chi-square test of independence? What assumptions are required for this test?

Accepted Answer

The answer of What is the null hypothesis for the...

Question 9

Find the specified conditional or marginal distribution for the given contingency table. Round to the nearest tenth of apercent if needed.
-The contingency table below shows the income level for a random sample of adults cross classified by sex. 
Find the marginal distribution of the variable "income bracket".

Accepted Answer

The answer of Find the specified conditional or marginal distribution...

Question 10

The table below shows the conditional distributions of the variable blood type for men and for women for a certain population. Is there association between the two variables blood type and sex? How can you tell?

Accepted Answer

The answer of The table below shows the conditional distributions...

Question 11

Perform the indicated goodness-of-fit test. Be sure to state the hypotheses and the significance level, to obtain the criticalvalue, to compute the value of the test statistic, and to state your conclusion.
-You roll a die 48 times with the following results. $$\begin{array} { c | r | r | r | r | r | r }  \text { Number } & 1 & 2 & 3 & 4 & 5 & 6 \ \hline \text { Frequency } & 3 & 12 & 3 & 3 & 14 & 13 \end{array}$$ Do the data provide sufficient evidence to conclude that the die is loaded (i.e., that the six numbers are not equally likely)? Perform the hypothesis test at the 0.05 level of significance.

Accepted Answer

The answer of Perform the indicated goodness-of-fit test. Be sure...

Question 12

A $\chi ^ { 2 }$-curve with 24 degrees of freedom resembles a normal curve more closely than a $\chi ^ { 2 }$-curve with 23 degrees of freedom?

Accepted Answer

As the degrees of freedom increase, the shape of the $\chi^2$-distribution becomes more symmetrical and starts to resemble the normal distribution more closely. Therefore, a $\chi^2$-curve with 24 degrees of freedom is closer to a normal curve than one with 23 degrees of freedom.

Question 13

$x ^ { 2 }$ -value.
-For a $\chi ^ { 2 }$-curve with 23 degrees of freedom, find $\chi _ { 0.01 } ^ { 2 }$.

Accepted Answer

The $\chi^2$ value for 23 degrees of freedom at the 0.01 significance level is 41.638.

Question 14

When performing a chi-square goodness-of-fit test, why do large values of the test statistic suggest that the variable does not have the distribution stated in the null hypothesis? Refer to the formula for the test statistic in your explanation.

Accepted Answer

The answer of When performing a chi-square goodness-of-fit test, why...

Question 15

Perform the indicated goodness-of-fit test. Be sure to state the hypotheses and the significance level, to obtain the criticalvalue, to compute the value of the test statistic, and to state your conclusion.
-A firm that manufactures engines for racing boats gathers data from all of its plants. The table below gives the distribution of the life span of the engines (in numbers of months)for the entire firm.  
$\begin{array}{lc}
\text { Months } & \text { Relative Frequency } \
 \text { Under } 60 & 0.017 \
70-79 & 0.201 \
80-89 & 0.353 \
90-99 & 0.331 \
100-109 & 0.098
\end{array}$
 
$\text { A random sample of } 448 \text { engines from the firm's Ohio plant gives the following results. }$ $\begin{array} { l r } 
\text { Months } & \text { Number of engines } \
 \text { Under } 60 & 22 \
70 - 79 & 131 \
80 - 89 & 46 \
90 - 99 & 154 \
100 - 109 & 95
\end{array}$  
At the 5% significance level, do the data provide evidence that the distribution of the life spans of the engines manufactured at the Ohio plant is different from the firm's overall distribution?

Accepted Answer

The answer of Perform the indicated goodness-of-fit test. Be sure...

Question 16

Perform a chi-square independence test using the critical value approach, provided the conditions for using the test aremet. Be sure to state the hypotheses and the significance level, to obtain the expected frequencies, to obtain the criticalvalue, to compute the value of the test statistic, and to state your conclusion.
-A researcher performed a study to determine whether an association exists between political party affiliation and income. She obtained the following sample data.  
At the 10% significance level, do the data provide sufficient evidence to conclude that an association exists between political party affiliation and income?

Accepted Answer

The answer of Perform a chi-square independence test using the...

Question 17

Perform the indicated goodness-of-fit test. Be sure to state the hypotheses and the significance level, to obtain the criticalvalue, to compute the value of the test statistic, and to state your conclusion.
-The U.S. Department of Defense provides data on the distribution of military personnel on active duty by branch of service. Below is a table giving the percentage distribution for 1985. 
$\begin{array}{lr} \text { Branch of Service } & \text { Percentage } \ \text { Army } & 36.3 \ \text { Navy } & 26.5 \ \text { Marine Corps } & 9.2 \ \text { Air Force } & 28.0 \end{array}$ 
A random sample of 325 military personnel currently on active duty gave the following statistics. 
$\begin{array}{lr} \text { Branch of Service } & \text { Frequency } \  \text { Army } & 118 \ \text { Navy } & 92 \ \text { Marine Corps } & 31 \ \text { Air Force } & 84 \end{array}$ 
At the 5% significance level, do the data provide evidence that the current distribution of military personnel on active duty differs from the 1985 distribution?

Accepted Answer

The answer of Perform the indicated goodness-of-fit test. Be sure...

Question 18

You have been using the chi-square test of independence to test for association between
two variables. Give an example of a pair of variables for which you would expect to find
no association. Give an example of a pair of variables for which you would expect to find
association, but not a causal relationship. Finally, give an example of a pair of variables for
which you would expect to find a causal relationship.

Accepted Answer

The answer of You have been using the chi-square test...

Question 19

Perform the indicated goodness-of-fit test using the P-value method. Be sure to state the hypotheses and the significancelevel , to compute the value of the test statistic, to obtain the P-value, and to state your conclusion.
-A firm that manufactures engines for racing boats gathers data from all of its plants. The table below gives the distribution of the life span of the engines (in numbers of months)for the entire firm. $\begin{array}{l} \begin{array} { c c }  \frac { \text { Months } } { \text { Under 60 } } & \text { Relative Frequency } \ 70 - 79 & 0.017 \ 80 - 89 & 0.201 \ 90 - 99 & 0.353 \ 100 - 109 & 0.331 \ & 0.098 \end{array}\ \text { A random sample of } 448 \text { engines from the firm's Ohio plant gives the following results. }\ \begin{array} { l r }  \frac { \text { Months } } { \text { Under } 60 } & \text { Number of engine } \ 70 - 79 & 22 \ 80 - 89 & 131 \ 90 - 99 & 46 \ 100 - 109 & 154 \ \hline \end{array} \end{array}$ At the 5% significance level, do the data provide evidence that the distribution of the life spans of the engines manufactured at the Ohio plant is different from the firm's overall distribution?

Accepted Answer

The answer of Perform the indicated goodness-of-fit test using the...

Question 20

A chi-square goodness-of-fit test is to be performed. The relative frequencies for the null
hypothesis and the sample size are given. True or false, the assumptions for using a chi-square
goodness-of-fit test are satisfied? Sample size: $\mathrm { n } = 50$
Relative frequencies: $0.23,0.01,0.33,0.13,0.3$

Accepted Answer

One of the assumptions for using a chi-square goodness-of-fit test is that the expected frequency for each category must be at least 5. Given the sample size of 50 and the relative frequencies provided, the expected frequencies for these categories would be 11.5, 0.5, 16.5, 6.5, and 15, respectively. The second category, with an expected frequency of 0.5, does not meet this requirement, thus the assumptions for using a chi-square goodness-of-fit test are not satisfied.

Quiz 13: Chi-Square Procedures

Suppose that you wish to perform a chi-square test of independence. Give an example of sample data for which assumption 1 (all expected frequencies are 1 or greater)is violated. You should present the sample data in the form of a contingency table.

$x ^ { 2 }$ -value. -For a $\chi ^ { 2 }$ -curve with 27 degrees of freedom, find the $\chi ^ { 2 }$ -value having area $0.01$ to its right.

Use the contingency table to solve the problem. -The following contingency table shows the popular votes cast in the 1984 presidential election cross classified by region and political party. Data are in thousands, rounded to the nearest thousand. What percentage of the voters were in the Midwest?

$\text { True or false, the total area under a } \chi ^ { 2 } \text {-curve is equal to } 1 \text { ? }$

What is the null hypothesis for the chi-square test of independence? What assumptions are required for this test?

The table below shows the conditional distributions of the variable blood type for men and for women for a certain population. Is there association between the two variables blood type and sex? How can you tell?

A $\chi ^ { 2 }$ -curve with 24 degrees of freedom resembles a normal curve more closely than a $\chi ^ { 2 }$ -curve with 23 degrees of freedom?

$x ^ { 2 }$ -value. -For a $\chi ^ { 2 }$ -curve with 23 degrees of freedom, find $\chi _ { 0.01 } ^ { 2 }$ .

When performing a chi-square goodness-of-fit test, why do large values of the test statistic suggest that the variable does not have the distribution stated in the null hypothesis? Refer to the formula for the test statistic in your explanation.

Quiz 13: Chi-Square Procedures

Suppose that you wish to perform a chi-square test of independence. Give an example of sample data for which assumption 1 (all expected frequencies are 1 or greater)is violated. You should present the sample data in the form of a contingency table.

x2x ^ { 2 }x2 -value. -For a χ2\chi ^ { 2 }χ2 -curve with 27 degrees of freedom, find the χ2\chi ^ { 2 }χ2 -value having area 0.010.010.01 to its right.

Use the contingency table to solve the problem. -The following contingency table shows the popular votes cast in the 1984 presidential election cross classified by region and political party. Data are in thousands, rounded to the nearest thousand. What percentage of the voters were in the Midwest?

True or false, the total area under a χ2-curve is equal to 1 ? \text { True or false, the total area under a } \chi ^ { 2 } \text {-curve is equal to } 1 \text { ? } True or false, the total area under a χ2-curve is equal to 1 ?

What is the null hypothesis for the chi-square test of independence? What assumptions are required for this test?

The table below shows the conditional distributions of the variable blood type for men and for women for a certain population. Is there association between the two variables blood type and sex? How can you tell?

A χ2\chi ^ { 2 }χ2 -curve with 24 degrees of freedom resembles a normal curve more closely than a χ2\chi ^ { 2 }χ2 -curve with 23 degrees of freedom?

x2x ^ { 2 }x2 -value. -For a χ2\chi ^ { 2 }χ2 -curve with 23 degrees of freedom, find χ0.012\chi _ { 0.01 } ^ { 2 }χ0.012​ .

When performing a chi-square goodness-of-fit test, why do large values of the test statistic suggest that the variable does not have the distribution stated in the null hypothesis? Refer to the formula for the test statistic in your explanation.

$x ^ { 2 }$ -value. -For a $\chi ^ { 2 }$ -curve with 27 degrees of freedom, find the $\chi ^ { 2 }$ -value having area $0.01$ to its right.

$\text { True or false, the total area under a } \chi ^ { 2 } \text {-curve is equal to } 1 \text { ? }$

A $\chi ^ { 2 }$ -curve with 24 degrees of freedom resembles a normal curve more closely than a $\chi ^ { 2 }$ -curve with 23 degrees of freedom?

$x ^ { 2 }$ -value. -For a $\chi ^ { 2 }$ -curve with 23 degrees of freedom, find $\chi _ { 0.01 } ^ { 2 }$ .