Deck 10: Associations Between Categorical Variables

Question 1

Suppose a random sample of 1,220 U.S. adults were asked about their opinion regarding federal spending on infrastructure (i.e. roads and bridges). Respondents were asked whether federal spending on infrastructure was (a) too low, (b) adequate, or (c) too high. Respondents were classified by income level. If we wanted to test whether there was an association between the
Response to the question and income level, would this be a test of homogeneity or of independence?

A) Independence
B) Homogeneity

Answer 1

Independence

Question 2

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of flying than males. $$\begin{array} { l | c | c } & \text { Men } & \text { Women } \\\hline \text { Expressed a fear of flying } & 42 & 112 \\\hline \text { Did not express a fear of flying } & 58 & 98\end{array}$$

-What fraction represents the proportion of people in the study who did not express a fear of flying?

A) $\frac { 156 } { 310 }$

B) $\frac { 154 } { 310 }$

C) $\frac { 210 } { 310 }$

D) $\frac { 100 } { 310 }$

Answer 2

$\frac { 156 } { 310 }$

Question 3

Use the following information to answer the question. A dowsing rod is a "Y" or "L" shaped instrument that some believe can find ground water. Many dowsers today use a pair of simple L-shaped metal rods. One rod is held in each hand, with the short arm of the L held upright, and the long arm pointing forward. When something is found, the rods cross over one
another making an "X" over the found object. Skeptics of dowsing conducted an experiment to see if dowsing rods could find ground water. Five identical 3 foot by 3 foot plots of land were sectioned off and a container of water was buried in one of the plots. Below is a summary of the experiment results and the output for the goodness-of-fit test. $\begin{array}{l|c|c} & \begin{array}{l}\text { Correctly identified } \\\text { location of ground water }\end{array} & \begin{array}{l}\text { Incorrectly identified } \\\text { location of ground water }\end{array} \\\hline \text { Observed } & 23 & 68 \\\hline \text { Expected } & 18.2 & 72.8\end{array}$

$\begin{array} { | l | } \hline \chi ^ { 2 } \text { GOF-Test } \\\chi ^ { 2 } = 1.582417582 \\p = .2084127992 \\\text { d.f. } = 1 \\\text { CNTRB } = \{ 1.26593 \ldots \\\hline\end{array}$

-Test the hypothesis that the dowsing rods worked better at locating ground water than guessing. Using a goodness-of-fit test and a 0.05 level of significance, choose the correct decision regarding
The null hypothesis and conclusion statement.

A) Reject $\mathrm { H } _ { 0 }$ ; There is not enough evidence to conclude that the dowsing rods worked better than guessing.
B) Reject $\mathrm { H } _ { 0 }$ ; There is enough evidence to conclude that the dowsing rods worked better than guessing.
C) Fail to reject $\mathrm { H } _ { 0 }$ ; There is enough evidence to conclude that the dowsing rods worked better than guessing.
D) Fail to reject $\mathrm { H } _ { 0 }$ ; There is not enough evidence to conclude that the dowsing rods worked better than guessing.

Answer 3

Fail to reject $\mathrm { H } _ { 0 }$ ; There is not enough evidence to conclude that the dowsing rods worked better than guessing.

Question 4

Use the following information to answer the question. The data in the top row of the table shows the number of vacation days taken by the respondent in the previous 90 days. The respondents also reported their level of happiness; Very H means very happy, and so on. $\text { Vacation days taken in last } 90 \text { days }$
$\begin{array}{l|c|c|c|c|c|c}&0&1&2&3&4&5+\\\hline\text { Very H } & 9 & 15 & 20 & 25 & 31 & 37 \\\text { Fairly H } & 5 & 8 & 26 & 35 & 48 & 50 \\\text { Not very H } & 6 & 6 & 4 & 2 & 1 & 2 \\\text { Not H } & 5 & 7 & 5 & 3 & 0 & 1\end{array}$

-We wish to test whether happiness is associated with taking vacation days. Choose the statement that is true about the hypothesis test or choose (d) if all the statements are false.

A) A hypothesis test cannot be conducted on this data because some of the observed cell counts are zero and there must be at least one observation in each cell.
B) The sample size is large so the chi-square test is appropriate.
C) The chi-square test is not appropriate because some expected cell counts will be less than 5.
D) All of these statements are false.

Question 5

Suppose a study was conducted to see whether there is an association between marital status and vehicle color. The table below shows the results from the study. Assume all conditions for testing have been met. $\begin{array}{c|c|c|c} & {\text { Vehicle color }} \\& \text { red } & \text { black/white/silver } & \text { blue/green/other } \\\hline \text { Married } & 45 & 22 & 19 \\\hline \text { Unmarried } & 35 & 11 & 18\end{array}$
This was an observational study of randomly chosen vehicle owners. Test the hypothesis that marital status and vehicle color are associated, using a significance level of $0.05$ . Choose the correct decision regarding the null hypothesis and conclusion. Refer to the computer output below.
$\begin{array} { | l | } \hline \chi ^ { 2 } - \text { Test } \\X ^ { 2 } = 1.754774169 \\p = .4158681217 \\\mathrm { df } = 2\\\hline\end{array}$

A) Reject the null hypothesis; marital status and vehicle color are not associated.
B) Fail to reject the null hypothesis; marital status and vehicle color are not associated.
C) Fail to reject the null hypothesis; marital status and vehicle color are associated.
D) Reject the null hypothesis; marital status and vehicle color are associated.

Question 6

Use the following information to answer the question. Bellei Extravecchio Balsamico Tradizionale is a very expensive brand of
balsamic vinegar. A twelve ounce bottle typically costs two-hundred dollars or more. In a blind taste test, a group of food
experts tasted three premium balsamic vinegars, with the most expensive one being Bellei Extravecchio Balsamico
Tradizionale. When asked to pick the most expensive balsamic vinegar, 92 got it right and 76 got it wrong.
If this group were just guessing, how many people (out of 168) would be expected to guess correctly?

A) Not enough information given to calculate expected value.
B) 84
C) 70
D) 56

Question 7

Use the following information to answer the question. A dowsing rod is a "Y" or "L" shaped instrument that some believe can find ground water. Many dowsers today use a pair of simple L-shaped metal rods. One rod is held in each hand, with the short arm of the L held upright, and the long arm pointing forward. When something is found, the rods cross over one
another making an "X" over the found object. Skeptics of dowsing conducted an experiment to see if dowsing rods could find ground water. Five identical 3 foot by 3 foot plots of land were sectioned off and a container of water was buried in one of the plots. Below is a summary of the experiment results and the output for the goodness-of-fit test. $\begin{array}{l|c|c} & \begin{array}{c}\text { Correctly identified } \\\text { location of ground water }\end{array} & \begin{array}{l}\text { Incorrectly identified } \\\text { location of ground water }\end{array} \\\hline \text { Observed } & 21 & 57 \\\hline \text { Expected } & 15.6 & 62.4\end{array}$


$\begin{array} { l } \chi ^ { 2 } \text { GOF-Test } \\\chi ^ { 2 } = 2.336538462 \\p = 1263700849 \\\text { d.f. } = 1 \\\text { CNTRB } = \{ 1.86923 \ldots\end{array}$

-Suppose a random sample of 1,220 U.S. adults were asked about their opinion regarding federal spending on infrastructure (i.e. roads and bridges). Respondents were asked whether federal spending on infrastructure was (a) too low, (b) adequate, or (c) too high. Respondents were classified by income level. Choose the correct hypotheses to test whether there is an association
Between the response to the question and income level.

A) $\mathrm { H } _ { 0 }$ : There is no difference between the proportions of U.S. adults who responded (a), (b) or (c) to the opinion question.
$\mathrm { H } _ { \mathrm { a } }$ : There is a difference between the proportions of U.S. adults who responded (a), (b) or (c) to the opinion question.

B) $\mathrm { H } _ { 0 }$ : Among U.S. adults, opinions about federal spending on infrastructure and income level are independent.
$\mathrm { H } _ { \mathrm { a } }$ : Among U.S. adults, opinions about federal spending on infrastructure and income level are associated.

C) $\mathrm { H } _ { 0 }$ : Among U.S. adults, opinions about federal spending on infrastructure and income level are associated.
$\mathrm { H } _ { \mathrm { a } }$ : Among U.S. adults, opinions about federal spending on infrastructure and income level are independent.
D) None of these

Question 8

Choose the statement that is not true about the chi-square test or choose (d) if all the statements are true.

A) To conduct a chi-square test you must have a large enough sample. This condition is met if each expected value is 5 or more.
B) The test statistic is the same for a test of homogeneity or a test for independence, it is chi-square $$\left( x ^ { 2 } \right)$$
C) The conclusion of a chi-square test tells only whether the variables under study are associated, not how they are associated.
D) All of these statements are true.

Question 9

Use the following information to answer the question. Lambda olive oil is touted as the "World's most expensive olive oil."
twelve ounce bottle typically costs fifty dollars or more. In a blind taste test, a group of food experts tasted three premium olive oils, one of which was Lambda olive oil. When asked to pick the Lambda olive oil, 84 got it right and 87 got it wrong.

-Calculate the observed value of the chi-square statistic. Round to the nearest hundredth.

A) 19.18
B) 12.79
C) 6.39
D) 23.68

Question 10

Choose the statement that is not true about Fisher's Exact Test or choose (d) if all the statements are true.

A) Fisher's Exact Test can be used for tables with more than two rows or columns.
B) When sample size is small resulting in expected cell counts that are less than 5, Fisher's Exact Test is one option that can be used to conduct a hypothesis test.
C) With Fisher's Exact Test an exact p-value can be calculated instead of using an approximation for the p-value as is the case with the chi-square test.
D) All of these statements are true.

Question 11

Suppose a researcher was interested in learning more about parents' concerns when their children go away to college. The researcher asks the parents of 900 randomly selected freshman at a private college and the parents of 1,020 randomly selected freshman at a public college to rate their level of concern with the following statement: We are (a) not at all concerned (b) somewhat concerned or (c) very concerned about the potential pressure to drink alcohol that our child will be exposed to while at college. If we wanted to test whether there was an association between the response to the question and the type of college that the freshman was attending, would this be a test of homogeneity or of independence?

A) Independence
B) Homogeneity

Question 12

The following table shows the results from a study to see if a home remedy ointment for mosquito bites worked better than a placebo. Each participant was randomly assigned to receive the home remedy ointment or the placebo ointment. "Improvement" means there were no symptoms of itching after three minutes.
$\begin{array} { l c c c } & \text { Home } & & \\ & \text { Remedy } & \text { Placebo } & \text { Total } \\ \text { No Improvement } & 2 & 4 & 6 \\ \text { Improvement } & 6 & 5 & 11 \\ \text { Total } & 8 & 9 & 17 \end{array}$
The alternative hypothesis is that the home remedy ointment leads to improvement (in this case, less itching). The p-value for a one-tailed Fisher's exact test with these data is $0.618$ . Suppose the study had turned out differently, as in the following table.
$\begin{array} { l c c c } & \text { Home } & & \\ \text { No Improvement } & 6 & 0 & 6 \\ \text { Improvement } & 2 & 9 & 11 \\ \text { Total } & 8 & 9 & 17 \end{array}$
Would Fisher's Exact Test have led to a p-value larger or smaller than 0.618?

A) The p-value would be smaller.
B) The p-value would be larger.

Question 13

Choose the statement that is not true about the chi-square distribution or choose (d) if all the statements are true.

A) The lower the degrees of freedom, the more skewed to the right the chi-square distribution will be.
B) Values for the chi-square statistic (on the horizontal axis) can be negative, positive, or zero.
C) The shape of the chi-square distribution depends on the degrees of freedom.
D) All of these statements are true about the chi-square distribution.

Question 14

Use the following information to answer the question. A dowsing rod is a "Y" or "L" shaped instrument that some believe can find ground water. Many dowsers today use a pair of simple L-shaped metal rods. One rod is held in each hand, with the short arm of the L held upright, and the long arm pointing forward. When something is found, the rods cross over one
another making an "X" over the found object. Skeptics of dowsing conducted an experiment to see if dowsing rods could find ground water. Five identical 3 foot by 3 foot plots of land were sectioned off and a container of water was buried in one of the plots. Below is a summary of the experiment results and the output for the goodness-of-fit test. $\begin{array}{l|c|c} & \begin{array}{c}\text { Correctly identified } \\\text { location of ground water }\end{array} & \begin{array}{l}\text { Incorrectly identified } \\\text { location of ground water }\end{array} \\\hline \text { Observed } & 21 & 57 \\\hline \text { Expected } & 15.6 & 62.4\end{array}$


$\begin{array} { l } \chi ^ { 2 } \text { GOF-Test } \\\chi ^ { 2 } = 2.336538462 \\p = 1263700849 \\\text { d.f. } = 1 \\\text { CNTRB } = \{ 1.86923 \ldots\end{array}$

-Choose the correct null and alternative hypothesis.

A) H0: The dowsing rods work no better at locating ground water than guessing. Ha: The dowsing rods work better at locating ground water than guessing.
B) H0: The dowsing rods correctly identify the location of ground water 50% of the time. Ha: The dowsing rods correctly identify the location of ground water more than 50% of the
Time.
C) H0: The dowsing rods work better at locating ground water than guessing. Ha: The dowsing rods work no better at locating ground water than guessing.
D) H0: The dowsing rods correctly identify the location of ground water 20% of the time. Ha: The dowsing rods correctly identify the location of ground water more than 20% of the
Time.

Question 15

The table below shows the gender and the percentage of each gender that spent different amounts at a local toy store. The data was taken from a random sample of single shoppers collected over five consecutive Saturdays at the toy store. Choose the reason(s) why you cannot do a chi-square test with this data. $$\begin{array} { | l | c | c | c | } \hline & \$ 0 - \$ 20 & > \$ 20 - \$ 50 & > \$ 50 \\\hline \text { Male shopper } & 18 \% & 49 \% & 33 \% \\\hline \text { Female shopper } & 20 \% & 62 \% & 18 \% \\\hline\end{array}$$

A) There is not enough information to convert the percentages to counts.
B) The samples were not collected randomly.
C) The data are from the entire population, not a sample, so inference is unnecessary.
D) All of these.

Question 16

Choose the statement that is not true about the chi-square test or choose (d) if all the statements are true.

A) The test statistic is the same for a test of homogeneity or a test for independence, it is chi-square
$$\left( x ^ { 2 } \right)$$
B) The conclusion of a chi-square test tells whether the variables under study are associated and how they are associated.
C) To conduct a chi-square test you must have a large enough sample. This condition is met if each expected value is 5 or more.
D) All of these statements are true.

Question 17

Use the following information to answer the question. The data in the top row of the table shows the number of vacation days taken by the respondent in the previous 90 days. The respondents also reported their level of happiness; Very H means very happy, and so on. $\text { Vacation days taken in last } 90 \text { days }$
$\begin{array}{l|c|c|c|c|c|c}&0&1&2&3&4&5+\\\hline\text { Very H } & 10 & 14 & 19 & 27 & 30 & 37 \\\text { Fairly H } & 6 & 7 & 28 & 33 & 45 & 49 \\\text { Not very H } & 5 & 7 & 6 & 3 & 0 & 1 \\\text { Not H } & 6 & 6 & 5 & 2 & 1 & 2\end{array}$

-The following table shows the data after merging categories so that there are two column categories (0-1 vacation days and 2 or more vacation days), and two row categories (happy and unhappy). Expected values for each cell are also shown in parentheses. Test the hypothesis that there is an Association between happiness and number of vacation days taken in the last 90 days, using a Significance level of 0.05. State the value of the test statistic rounded to two decimal places and state Whether the p-value is closer to zero or one. $\begin{array}{l|l|l} & 0-1 \text { vacation days } & 2+\text { vacation days } \\\hline \text { Happy } & 84 & 221 \\& (104) & (201) \\\hline \text { Unhappy } & 35 & 9 \\& (15.003) & (28.997)\end{array}$

A) $\chi ^ { 2 } = 5.03$ ; The $p$ -value will be close to zero.
B) $\chi ^ { 2 } = 46.28$ ; The p-value will be close to one.
C) $\chi ^ { 2 } = 5.03$ ; The $p$ -value will be close to one.
D) $\chi ^ { 2 } = 46.28$ ; The p-value will be close to zero.

Question 18

Use the following information to answer the question. The data in the top row of the table shows the number of vacation days taken by the respondent in the previous 90 days. The respondents also reported their level of happiness; Very H means very happy, and so on. $\text { Vacation days taken in last } 90 \text { days }$
$\begin{array}{l|c|c|c|c|c|c}&0&1&2&3&4&5+\\\hline\text { Very H } & 9 & 15 & 20 & 25 & 31 & 37 \\\text { Fairly H } & 5 & 8 & 26 & 35 & 48 & 50 \\\text { Not very H } & 6 & 6 & 4 & 2 & 1 & 2 \\\text { Not H } & 5 & 7 & 5 & 3 & 0 & 1\end{array}$

-The following table shows the data after merging categories so that there are two column categories (0-1 vacation days and 2 or more vacation days), and two row categories (happy and unhappy). Expected values for each cell are also shown in parentheses. Test the hypothesis that there is an Association between happiness and number of vacation days taken in the last 90 days, using a Significance level of 0.05. State the value of the test statistic rounded to two decimal places and state Whether the p-value is closer to zero or one. $\begin{array}{l|l|l} & 0-1 \text { vacation days } & 2+\text { vacation days } \\\hline \text { Happy } & 37 & 272 \\& (53.701) & (255.3) \\\hline \text { Unhappy } & 24 & 18 \\& (7.2991) & (34.701)\end{array}$

A) $\chi ^ { 2 } = 4.22$ ; The p-value will be close to zero.
B) $\chi ^ { 2 } = 52.54$ ; The $p$ -value will be close to zero.
C) $\chi ^ { 2 } = 4.22$ ; The $p$ -value will be close to one.
D) $\chi ^ { 2 } = 52.54$ ; The p-value will be close to one.

Question 19

A health foods store owner is thinking about carrying some new products and is interested in her customer's opinions. The shop owner decides to randomly sample 202 customers and ask them
Whether they have (a) heard about the health benefits of coconut milk and (b) whether they have heard of the health benefits of Quinoa flour. She also asked each respondent how often they typically make purchases during a month. The table below shows the results from the study.
Assume all conditions for testing have been met. $\begin{array}{|l|c|c|c|}\hline & 1-2 \text { purchases/mo. } & \text { 3+ purchases/mo. } & \text { 5+ purchases/mo. } \\\hline \text { Coconut milk, yes? } & 57 & 21 & 11 \\\hline \text { Coconut milk, no? } & 12 & 11 & 10 \\\hline \text { Quinoa flour, yes? } & 19 & 25 & 10 \\\hline \text { Quinoa flour, no? } & 10 & 8 & 8 \\\hline\end{array}$

Test the hypothesis that how the respondents answered the questions is associated with number of monthly purchases, using a significance level of $0.05$ . Choose the correct decision regarding the null hypothesis and conclusion. Refer to the computer output below.
$\begin{array}{l}\chi^{2}-\text { Test } \\\qquad \begin{array}{l}X^{2}=19.43852896 \\p=.0034836969 \\\mathrm{df}=6\end{array}\end{array}$

A) Fail to reject the null hypothesis; how the respondents answered the questions and number of monthly purchases are associated.
B) Reject the null hypothesis; how the respondents answered the questions and number of monthly purchases are not associated.
C) Fail to reject the null hypothesis; how the respondents answered the questions and number of monthly purchases are not associated.
D) Reject the null hypothesis; how the respondents answered the questions and number of monthly purchases are associated.

Question 20

The following table shows the results from a study to see if a home remedy ointment for mosquito bites worked better than a placebo. Each participant was randomly assigned to receive the home remedy ointment or the placebo ointment. "Improvement" means no symptoms of itching after three minutes. $\begin{array}{lccc} & \text { Home } & & \\& \text { Remedy } & \text { Placebo } & \text { Total } \\\text { No Improvement } & 2 & 4 & 6 \\\text { Improvement } & 6 & 5 & 11 \\\text { Total } & 8 & 9 & 17\end{array}$
The alternative hypothesis is that the home remedy ointment leads to improvement (in this case, less itching). The p-value for a one-tailed Fisher's exact test with these data is $0.618$ . Suppose the study had turned out differently, as in the following table.
$\begin{array} { l c c c } & \begin{array} { c } \text { Home } \\ \text { Remedy } \end{array} & \text { Placebo } & \text { Total } \\ \text { No Improvement } & 0 & 6 & 6 \\ \text { Improvement } & 8 & 3 & 11 \\ \text { Total } & 8 & 9 & 17 \end{array}$ Would Fisher's Exact Test have led to a p-value larger or smaller than 0.618?

A) The p-value would be smaller.
B) The p-value would be larger.

Question 21

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of heights than males. $$\begin{array} { l | c | c } & \text { Men } & \text { Women } \\\hline \text { Expressed a fear of heights } & 68 & 109 \\\hline \text { Did not express a fear of heights } & 94 & 89\end{array}$$

-What fraction represents the proportion of people in the study who expressed a fear of heights?

A) $\frac { 177 } { 360 }$
B) $\frac { 198 } { 360 }$
C) $\frac { 183 } { 360 }$
D) $\frac { 162 } { 360 }$

Question 22

Use the following information to answer the question. The data in the top row of the table shows the number of vacation days taken by the respondent in the previous 90 days. The respondents also reported their level of happiness; Very H means very happy, and so on. $\text { Vacation days taken in last } 90 \text { days }$
$\begin{array}{l|c|c|c|c|c|c}&0&1&2&3&4&5+\\\hline\text { Very H } & 10 & 14 & 19 & 27 & 30 & 37 \\\text { Fairly H } & 6 & 7 & 28 & 33 & 45 & 49 \\\text { Not very H } & 5 & 7 & 6 & 3 & 0 & 1 \\\text { Not H } & 6 & 6 & 5 & 2 & 1 & 2\end{array}$

-We wish to test whether happiness is associated with taking vacation days. Choose the statement that is true about the hypothesis test or choose (d) if all the statements are false.

A) A hypothesis test cannot be conducted on this data because some of the observed cell counts are zero and there must be at least one observation in each cell.
B) The chi-square test is not appropriate because some expected cell counts will be less than 5.
C) The sample size is large so the chi-square test is appropriate.
D) All of these statements are false.

Question 23

Choose the statement that is not true about Fisher's Exact Test or choose (d) if all the statements are true.

A) When sample size is small resulting in expected cell counts that are less than 5, Fisher's Exact Test is one option that can be used to conduct a hypothesis test.
B) With Fisher's Exact Test an exact p-value can be calculated instead of using an approximation for the p-value as is the case with the chi-square test.
C) Fisher's Exact Test cannot be used for tables with more than two rows or columns.
D) All of these statements are true.

Question 24

A health foods store owner is thinking about carrying some new products and is interested in her customer's opinions. The shop owner decides to randomly sample 202 customers and ask them whether they have (a) heard about the health benefits of coconut milk and (b) whether they have heard of the health benefits of Quinoa flour. She also asked each respondent how often they typically make purchases during a month. The table below shows the results from the study.
Assume all conditions for testing have been met. $\begin{array}{|l|c|c|c|}\hline & 1-2 \text { purchases/mo. } & \text { 3+ purchases/mo. } & \text { 5+ purchases/mo. } \\\hline \text { Coconut milk, yes? } & 57 & 21 & 11 \\\hline \text { Coconut milk, no? } & 12 & 11 & 10 \\\hline \text { Quinoa flour, yes? } & 19 & 25 & 10 \\\hline \text { Quinoa flour, no? } & 10 & 8 & 8 \\\hline\end{array}$
Test the hypothesis that how the respondents answered the questions is associated with number of monthly purchases, using a significance level of $0.05$ . Choose the correct decision regarding the null hypothesis and conclusion. Refer to the computer output below.
$\begin{array} { | l | } \hline \chi ^ { 2 } - \text { Test } \\X ^ { 2 } = 19.43852896 \\p = .0034836969 \\\mathrm { df } = 6\\\hline \end{array}$

A) Reject the null hypothesis; how the respondents answered the questions and number of monthly purchases are associated.
B) Fail to reject the null hypothesis; how the respondents answered the questions and number of monthly purchases are not associated.
C) Fail to reject the null hypothesis; how the respondents answered the questions and number of monthly purchases are associated.
D) Reject the null hypothesis; how the respondents answered the questions and number of monthly purchases are not associated.

Question 25

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers
carried out to determine if females expressed a greater fear of heights than males.
How many categorical variables are summarized in the table?

A) Zero
B) Three
C) Two
D) Four

Question 26

Use the following information to answer the question. A dowsing rod is a "Y" or "L" shaped instrument that some believe can find ground water. Many dowsers today use a pair of simple L-shaped metal rods. One rod is held in each hand, with the short arm of the L held upright, and the long arm pointing forward. When something is found, the rods cross over one another making an "X" over the found object. Skeptics of dowsing conducted an experiment to see if dowsing rods could find ground water. Five identical 3 foot by 3 foot plots of land were sectioned off and a container of water was buried in one of the plots. Below is a summary of the experiment results and the output for the goodness-of-fit test. $\begin{array}{l|c|c} & \begin{array}{l}\text { Correctly identified } \\\text { location of ground water }\end{array} & \begin{array}{l}\text { Incorrectly identified } \\\text { location of ground water }\end{array} \\\hline \text { Observed } & 21 & 57 \\\hline \text { Expected } & 15.6 & 62.4\end{array}$

$\begin{array} { | l | } \hline \chi ^ { 2 } \text { GOF-Test } \\\chi ^ { 2 } = 2.336538462 \\p = .1263700849 \\\text { d.f. } = 1 \\\text { CNTRB } = \{ 1.86923 \ldots \\\hline\end{array}$



-Test the hypothesis that the dowsing rods worked better at locating ground water than guessing. Using a goodness-of-fit test and a 0.05 level of significance, choose the correct decision regarding the null hypothesis and conclusion statement.

A) Reject $\mathrm { H } _ { 0 }$ ; There is not enough evidence to conclude that the dowsing rods worked better than guessing.
B) Fail to reject $\mathrm { H } _ { 0 }$ ; There is enough evidence to conclude that the dowsing rods worked better than guessing.
C) Reject $\mathrm { H } _ { 0 }$ ; There is enough evidence to conclude that the dowsing rods worked better than guessing.
D) Fail to reject $\mathrm { H } _ { 0 }$ ; There is not enough evidence to conclude that the dowsing rods worked better than guessing.

Question 27

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers
carried out to determine if females expressed a greater fear of flying than males.
Find the expected number of women who should express a fear of flying, if the variables are independent. Round to the nearest whole number.

A) 104
B) 11
C) 81
D) 50

Question 28

Use the following information to answer the question. Lambda olive oil is touted as the "World's most expensive olive oil."
twelve ounce bottle typically costs fifty dollars or more. In a blind taste test, a group of food experts tasted three premium
olive oils, one of which was Lambda olive oil. When asked to pick the Lambda olive oil, 84 got it right and 87 got it wrong.
If this group were just guessing, how many people (out of 171) would be expected to guess correctly?

A) 57
B) Not enough information given to calculate expected value.
C) 114
D) 86

Question 29

Suppose a researcher was interested in learning more about parents' concerns when their children start elementary school. The researcher asks the parents of 800 randomly selected first graders in rural school district and the parents of 950 randomly selected first graders in an urban school district to rate their level of concern with the following statement: We are (a) not at all concerned
(b) somewhat concerned or (c) very concerned about the nutrition level of school lunches. If we
Wanted to test whether there was an association between the response to the question and the type
Of school district that the first grader was attending, would this be a test of homogeneity or of
Independence?

A) Independence
B) Homogeneity

Question 30

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers
carried out to determine if females expressed a greater fear of flying than males.
How many categorical variables are summarized in the table?

A) Zero
B) Two
C) Four
D) Three

Question 31

Use the following information to answer the question. Bellei Extravecchio Balsamico Tradizionale is a very expensive brand of balsamic vinegar. A twelve ounce bottle typically costs two-hundred dollars or more. In a blind taste test, a group of food experts tasted three premium balsamic vinegars, with the most expensive one being Bellei Extravecchio Balsamico
Tradizionale. When asked to pick the most expensive balsamic vinegar, 92 got it right and 76 got it wrong.

-Calculate the observed value of the chi-square statistic. Round to the nearest hundredth.

A) 23.14
B) 34.71
C) 41.21
D) 11.57

Question 32

Suppose a random sample of 1,220 U.S. adults were asked about their opinion regarding federal spending on public education. Respondents were asked whether federal spending on public
Education was (a) too low, (b) adequate, or (c) too high. Respondents were classified by income
Level. If we wanted to test whether there was an association between the response to the question
And income level, would this be a test of homogeneity or of independence?

A) Independence
B) Homogeneity

Question 33

The table below shows the gender and the percentage of each gender that spent different amounts at a local hardware store. The data was taken from a random sample of single shoppers collected over five consecutive Saturdays at the hardware store. Choose the reason(s) why you cannot do a Chi-square test with this data. $$\begin{array} { | l | c | c | c | } \hline & \$ 0 - \$ 20 & > \$ 20 - \$ 50 & > \$ 50 \\\hline \text { Male shopper } & 27 \% & 42 \% & 31 \% \\\hline \text { Female shopper } & 46 \% & 35 \% & 19 \% \\\hline\end{array}$$

A) The data are from the entire population, not a sample, so inference is unnecessary.
B) There is not enough information to convert the percentages to counts.
C) The samples were not collected randomly.
D) All of these.

Question 34

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of heights than males. $$\begin{array} { l | c | c } & \text { Men } & \text { Women } \\\hline \text { Expressed a fear of heights } & 68 & 109 \\\hline \text { Did not express a fear of heights } & 94 & 89\end{array}$$

-Find the expected number of men who should express a fear of heights, if the variables are independent. Round to the nearest whole number.

A) 93
B) 80
C) 97
D) 87

Question 35

Use the following information to answer the question. A dowsing rod is a "Y" or "L" shaped instrument that some believe can find ground water. Many dowsers today use a pair of simple L-shaped metal rods. One rod is held in each hand, with the short arm of the L held upright, and the long arm pointing forward. When something is found, the rods cross over one another making an "X" over the found object. Skeptics of dowsing conducted an experiment to see if dowsing rods could find ground water. Five identical 3 foot by 3 foot plots of land were sectioned off and a container of water was buried in one of the plots. Below is a summary of the experiment results and the output for the goodness-of-fit test. $\begin{array}{l|c|c} & \begin{array}{c}\text { Correctly identified } \\\text { location of ground water }\end{array} & \begin{array}{l}\text { Incorrectly identified } \\\text { location of ground water }\end{array} \\\hline \text { Observed } & 23 & 68 \\\hline \text { Expected } & 18.2 & 72.8\end{array}$

$\begin{array} { | l | } \hline \chi ^ { 2 } \text { GOF-Test } \\\chi ^ { 2 } =1.582417582 \\p = .2084127992 \\\text { d.f. } = 1 \\\text { CNTRB } = \{ 1.26593 \ldots \\\hline\end{array}$


-Choose the correct null and alternative hypothesis.

A) $\mathrm { H } _ { 0 }$ : The dowsing rods correctly identify the location of ground water $20 \%$ of the time. $\mathrm { H } _ { \mathrm { a } }$ : The dowsing rods correctly identify the location of ground water more than $20 \%$ of the time.
B) $\mathrm { H } _ { 0 }$ : The dowsing rods correctly identify the location of ground water $50 \%$ of the time. $\mathrm { H } _ { \mathrm { a } }$ : The dowsing rods correctly identify the location of ground water more than $50 \%$ of the time.
C) $\mathrm { H } _ { 0 }$ : The dowsing rods work no better at locating ground water than guessing. $\mathrm { H } _ { \mathrm { a } }$ : The dowsing rods work better at locating ground water than guessing.
D) $\mathrm { H } _ { 0 }$ : The dowsing rods work better at locating ground water than guessing. $\mathrm { H } _ { \mathrm { a } }$ : The dowsing rods work no better at locating ground water than guessing.

Question 36

Suppose a study was conducted to see whether there is an association between marital status and breast cancer remission. The table below shows the results from the study. Assume all conditions for testing have been met. $\begin{array}{l|c|c|c} & {\text { Patient was in remission }} & \text { after treatment } \\& \text { 1 year } & 2 \text { years } & 3 \text { or more years } \\\hline \begin{array}{l}\text { Married (at the } \\\text { time of treatment) }\end{array} & 40 & 20 & 15 \\\hline \begin{array}{l}\text { Unmarried (at the } \\\text { time of treatment) }\end{array} & 33 & 18 & 17\end{array}$
This was an observational study of randomly chosen patients who had received similar chemotherapy treatments. Test the hypothesis that marital status and remission from breast cancer are associated, using a significance level of $0.05$ . Choose the correct decision regarding the null hypothesis and conclusion. Refer to the computer output below.
$\begin{array} { | l | } \hline \chi ^ { 2 } - \text { Test } \\X ^ { 2 } = .5601810006 \\p = .7557153459 \\\mathrm { df } = 2 \\\hline\end{array}$

A) Reject the null hypothesis; marital status and cancer remission are not associated.
B) Reject the null hypothesis; marital status and cancer remission are associated.
C) Fail to reject the null hypothesis; marital status and cancer remission are associated.
D) Fail to reject the null hypothesis; marital status and cancer remission are not associated.

Question 37

Use the following information to answer the question. A dowsing rod is a "Y" or "L" shaped instrument that some believe can find ground water. Many dowsers today use a pair of simple L-shaped metal rods. One rod is held in each hand, with the short arm of the L held upright, and the long arm pointing forward. When something is found, the rods cross over one another making an "X" over the found object. Skeptics of dowsing conducted an experiment to see if dowsing rods could find ground water. Five identical 3 foot by 3 foot plots of land were sectioned off and a container of water was buried in one of the plots. Below is a summary of the experiment results and the output for the goodness-of-fit test. $\begin{array}{l|c|c} & \begin{array}{c}\text { Correctly identified } \\\text { location of ground water }\end{array} & \begin{array}{l}\text { Incorrectly identified } \\\text { location of ground water }\end{array} \\\hline \text { Observed } & 23 & 68 \\\hline \text { Expected } & 18.2 & 72.8\end{array}$

$\begin{array} { l } \chi ^ { 2 } \text { GOF-Test } \\\chi ^ { 2 } = 1.582417582 \\p = .2084127992 \\\text { d.f. } = 1 \\\text { CNTRB } = \{ 1.26593 \ldots\end{array}$

-Of the following statements, which one is not true about the chi-square statistic and p-value? Choose (d) if all statements are true.

A) The smaller the chi-square statistic, the larger the p-value.
B) On the chi-square distribution, the p-value is represented by the area under the curve to the right of the chi-square statistic.
C) Under the assumption that the null is true, the p-value is the probability that the chi-square statistic will be as big as or bigger than the observed value.
D) All of these statements are true.

Question 38

Use the following information to answer the question. A dowsing rod is a "Y" or "L" shaped instrument that some believe can find ground water. Many dowsers today use a pair of simple L-shaped metal rods. One rod is held in each hand, with the short arm of the L held upright, and the long arm pointing forward. When something is found, the rods cross over one another making an "X" over the found object. Skeptics of dowsing conducted an experiment to see if dowsing rods could find ground water. Five identical 3 foot by 3 foot plots of land were sectioned off and a container of water was buried in one of the plots. Below is a summary of the experiment results and the output for the goodness-of-fit test. $\begin{array}{l|c|c} & \begin{array}{c}\text { Correctly identified } \\\text { location of ground water }\end{array} & \begin{array}{l}\text { Incorrectly identified } \\\text { location of ground water }\end{array} \\\hline \text { Observed } & 21 & 57 \\\hline \text { Expected } & 15.6 & 62.4\end{array}$

$\begin{array} { | l | } \hline \chi ^ { 2 } \text { GOF-Test } \\\chi ^ { 2 } = 2.336538462 \\p = .1263700849 \\\text { d.f. } = 1 \\\text { CNTRB } = \{ 1.86923 \ldots \\\hline\end{array}$

-Of the following statements, which one is not true about the chi-square statistic and p-value? Choose (d) if all statements are true.

A) Under the assumption that the null is true, the p-value is the probability that the chi-square statistic will be as big as or bigger than the observed value.
B) The larger the chi-square statistic, the smaller the p-value.
C) On the chi-square distribution, the p-value is represented by the area under the curve to the right of the chi-square statistic.
D) All of these statements are true.

Question 39

Suppose a random sample of 1,220 U.S. adults were asked about their opinion regarding federal spending on public education. Respondents were asked whether federal spending on public education was (a) too low, (b) adequate, or (c) too high. Respondents were classified by income Level. Choose the correct hypotheses to test whether there is an association between the response to the question and income level.

A) $\mathrm { H } _ { 0 } :$ There is no difference between the proportions of U.S. adults who responded (a), (b) or (c) to the opinion question.
$\mathrm { H } _ { \mathrm { a } }$ : There is a difference between the proportions of U.S. adults who responded (a), (b) or (c) to the opinion question.

B) $\mathrm { H } _ { 0 }$ : Among U.S. adults, opinions about federal spending on education and income level are associated.
$\mathrm { H } _ { \mathrm { a } }$ : Among U.S. adults, opinions about federal spending on education and income level are independent.

C) $\mathrm { H } _ { 0 }$ : Among U.S. adults, opinions about federal spending on education and income level are independent.
$\mathrm { H } _ { \mathrm { a } }$ : Among U.S. adults, opinions about federal spending on education and income level are associated.
D) None of these

Question 40

Use the following information to answer the question. The data in the top row of the table shows the number of days for which the respondent participated in an outdoor activity for at least thirty minutes in the previous 60 days. The respondents also reported their level of happiness; Very H means very happy, and so on.

-Suppose you want to test the hypothesis that happiness level is associated with level of daily outdoor activity. Can a chi-square test for independence be used to test the hypothesis?
Explain why or why not.

Question 41

Use the following information to answer the question. Almas caviar is a very expensive Iranian brand of Beluga sturgeon caviar. A one pound gold-plated tin typically costs $12,000 or more. In a blind taste test, a group of food experts tasted four premium Beluga sturgeon caviars, with the most expensive one being Almas caviar. When asked to pick the most expensive caviar, 141 got it right and 79 got it wrong.

-If this group were just guessing, how many people (out of 220) would be expected to guess correctly?

Question 42

Suppose a researcher was interested in learning more about high school seniors' concerns about the future. The researcher asks the 750 randomly selected female high school seniors and the 800 randomly selected male high school seniors to rate their level of concern with the following statement:
I am (a) not at all concerned (b) somewhat concerned or (c) very concerned about getting accepted into a two or four year college. If we wanted to test whether there was an association between the response to the question and the gender of the high school senior, would this be a test of homogeneity or independence?

Question 43

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of public speaking then males. $$\begin{array} { l | c | c } & \text { Men } & \text { Women } \\ \hline \text { Expressed a fear of public speaking } & 112 & 79 \\ \hline \text { Did not express a fear of public speaking } & 91 & 82 \end{array}$$

-Suppose a goodness-of-fit test is used to test the claim that obesity rates in the elderly have changed since the time of the Egyptian mummies. The p-value is calculated to be 0.00023. Describe the value of the chi-square test statistic (is it likely to be large?
small?
) and the decision regarding the null hypothesis that there is no difference in obesity rates for mummies and modern-day elderly people.

Question 44

Use the following information to answer the question. The data in the top row of the table shows the number of days for which the respondent participated in an outdoor activity for at least thirty minutes in the previous 60 days. The respondents also reported their level of happiness; Very H means very happy, and so on.

-The following table shows the data after merging categories so that there are two column categories (0-2 days of outdoor activity of at least thirty minutes and 3 or more days of outdoor activity of at least thirty minutes), and two row categories (happy and unhappy). Expected values for each cell are also shown in parenthesis. Test the hypothesis that there is an association between happiness and with level of daily outdoor activity in the last 60 days, using a significance level of 0.05. State the value of the test statistic rounded to two decimal places, state whether the p-value is closer to zero or one. $$\begin{array} { l | l | l } & \begin{array} { l } 0 - 2 \text { days of } \\ \text { outdoor activity } \end{array} & \begin{array} { l } \text { 3+ days of } \\ \text { outdoor activity } \end{array} \\ \hline \text { Happy } & 71 & 133 \\ & ( 87.787 ) & ( 116.21 ) \\ \hline \text { Unhappy } & 34 & 6 \\ & ( 17.213 ) & ( 22.787 ) \end{array}$$

Question 45

Use the following information to answer the question. There are those who believe that identical twins have a psychic connection. Skeptics of this belief conducted an experiment to see if identical twins had some kind of psychic connection. A twin was placed at random behind on of four similar doors and three other non-related people were placed behind the other three doors. The other twin was asked to identify which door their sibling was behind. The results of the experiment are summarized below, along with the output for the goodness-of-fit test. $\begin{array}{l|c|c} & \begin{array}{c} \text { Correctly identified the door } \\ \text { concealing the twin } \end{array} & \begin{array}{c} \text { Incorrectly identified the } \\ \text { door concealing the twin } \end{array} \\ \hline \text { Observed } & 51 & 70 \\ \hline \text { Expected } & 30.25 & 90.75 \end{array}$

-In the context of this description, state the correct null and alternative hypothesis to test the claim that identical twins have a psychic connection.

Question 46

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of public speaking then males. $$\begin{array} { l | c | c } & \text { Men } & \text { Women } \\ \hline \text { Expressed a fear of public speaking } & 112 & 79 \\ \hline \text { Did not express a fear of public speaking } & 91 & 82 \end{array}$$

-Calculate the percentage of people in the study that did not express a fear of public speaking. Round to the nearest tenth of a percent.

Question 47

An eco-friendly home improvement store owner is thinking about carrying some new products and is interested in her customer's opinions. The shop owner decides to randomly sample 397 customers and ask them whether they have (a) heard about bricks made from recycled materials and (b) whether they have heard of landscaping materials made from recycled glass. She also asked each respondent how often they typically make purchases during a month. The table below shows the results from the study. Assume all conditions for testing have been met. $$\begin{array} { | l | c | c | c | } \hline & 1 - 2 \text { purchases/mo. } & \text { 3+ purchases/mo. } & \text { 5+ purchases/mo. } \\ \hline \text { Recycled bricks, yes? } & 82 & 42 & 72 \\ \hline \text { Recycled bricks, no? } & 44 & 32 & 12 \\ \hline \text { Landscaping glass, yes? } & 23 & 7 & 44 \\ \hline \text { Landscaping glass, no? } & 11 & 18 & 10 \\ \hline \end{array}$$ Test the hypothesis that how the respondents answered the questions is associated with number of monthly purchases, using a significance level of 0.05. State the correct decision regarding the null hypothesis and write a conclusion using a full sentence. Refer to the output below.

Question 48

The following table shows the results from a study to see if a home remedy salve for bee stings worked better than a placebo. Each participant was randomly assigned to receive the home remedy salve or a placebo ointment. "Improvement" means that there were no symptoms of pain after fifteen minutes. Home $\begin{array} { c c c } \text { Remedy } & \text { Placebo } & \text { Total } \\ 2 & 1 & 3 \\ 6 & 7 & 13 \\ 8 & 8 & 16 \end{array}$ The alternative hypothesis is that the home remedy salve leads to improvement (in this case, no pain). The p-value for a one-tailed Fisher's Exact Test with these data is $0.500$ . Suppose the study had turned out differently, as in the following table. $\begin{array} { l c c c } & \begin{array} { c } \text { Home } \\ \text { Remedy } \end{array} & \text { Placebo } & \text { Total } \\ \text { No Improvement } & 0 & 3 & 3 \\ \text { Improvement } & 8 & 5 & 13 \\ \text { Total } & 8 & 8 & 16 \end{array}$ Would Fisher's Exact Test have led to a p-value larger or smaller than $0.500$ ?
Explain.

Question 49

Use the following information to answer the question. Suppose that you are a researcher and that you want to research whether a home remedy that recommends applying a salve of tobacco and saliva to cuts or scrapes to absorb toxins really works. You decide to conduct a study to see whether there is an association between the treatment and the outcome. A positive outcome means that the patient reported lower pain levels and the wound had healed without infection in less than 14 days. The table below shows the results from your study. Assume all conditions for testing have been met. $$\begin{array} { c | c | c | } \hline & \text { Home remedy } & \text { Placebo } \\ \hline \text { Positive outcome } & 52 & 23 \\ \hline \begin{array} { c } \text { Not positive } \\ \text { outcome } \end{array} & 35 & 33 \end{array}$$

-Test the hypothesis that the treatment is associated with the outcome, using a significance level of 0.05. State the correct decision regarding the null hypothesis and write a conclusion using a full sentence. Refer to the TI-83 output below.

Question 50

Use the following information to answer the question. Almas caviar is a very expensive Iranian brand of Beluga sturgeon caviar. A one pound gold-plated tin typically costs $12,000 or more. In a blind taste test, a group of food experts tasted four premium Beluga sturgeon caviars, with the most expensive one being Almas caviar. When asked to pick the most expensive caviar, 141 got it right and 79 got it wrong.

-Calculate the observed value of the chi-square statistic by hand. Show all your work. Round to the nearest hundredth.

Question 51

Suppose a random sample of 1,105 adults were asked about their opinion regarding salaries of full-time employees at a local state funded university. Respondents were asked whether salaries at the university were (a) too low, (b) adequate, or (c) too high. Respondents were also classified by income level. State the correct hypothesis to test whether there is an association between the response to the questions and income level.

Question 52

Suppose a random sample of 1,105 adults were asked about their opinion regarding salaries of full-time employees at a local state funded university. Respondents were asked whether salaries at the university were (a) too low, (b) adequate, or (c) too high. Respondents were also classified by income level. If we wanted to test whether there was an association between the response to the question and income level, would this be a test of homogeneity or of independence?

Question 53

Compare the sampling distributions of the normally distributed test statistic z and the chi-square test statistic. How do the sampling distributions differ in shape and range of values?

Question 54

Use the following information to answer the question. Suppose that you are a researcher and that you want to research whether a home remedy that recommends applying a salve of tobacco and saliva to cuts or scrapes to absorb toxins really works. You decide to conduct a study to see whether there is an association between the treatment and the outcome. A positive outcome means that the patient reported lower pain levels and the wound had healed without infection in less than 14 days. The table below shows the results from your study. Assume all conditions for testing have been met. $$\begin{array} { c | c | c | } & \text { Home remedy } & \text { Placebo } \\ \hline \text { Positive outcome } & 52 & 23 \\ \hline \begin{array} { c } \text { Not positive } \\ \text { outcome } \end{array} & 35 & 33 \end{array}$$

-From a purely scientific point-of-view, would you suggest the tobacco and saliva salve to a co-worker who expressed an interest in home remedies?
Explain why or why not.

Question 55

Describe at least one advantage and one disadvantage of combining categories as was done in the previous question?

Question 56

Use the following information to answer the question. There are those who believe that identical twins have a psychic connection. Skeptics of this belief conducted an experiment to see if identical twins had some kind of psychic connection. A twin was placed at random behind on of four similar doors and three other non-related people were placed behind the other three doors. The other twin was asked to identify which door their sibling was behind. The results of the experiment are summarized below, along with the output for the goodness-of-fit test. $\begin{array}{l|c|c} & \begin{array}{c} \text { Correctly identified the door } \\ \text { concealing the twin } \end{array} & \begin{array}{c} \text { Incorrectly identified the } \\ \text { door concealing the twin } \end{array} \\ \hline \text { Observed } & 51 & 70 \\ \hline \text { Expected } & 30.25 & 90.75 \end{array}$

-The table below shows the gender and the percentage of each gender that spent different amounts at a local book store. The data was taken from a random sample of single shoppers collected over five consecutive Saturdays at the book store. Can a chi-square test be done with this data?
Explain why or why not. $$\begin{array} { | l | c | c | c | } \hline & \$ 0 - \$ 20 & > \$ 20 - \$ 50 & > \$ 50 \\ \hline \text { Male shopper } & 36 \% & 42 \% & 22 \% \\ \hline \text { Female shopper } & 67 \% & 21 \% & 12 \% \\ \hline \end{array}$$

Question 57

Use the following information to answer the question. There are those who believe that identical twins have a psychic connection. Skeptics of this belief conducted an experiment to see if identical twins had some kind of psychic connection. A twin was placed at random behind on of four similar doors and three other non-related people were placed behind the other three doors. The other twin was asked to identify which door their sibling was behind. The results of the experiment are summarized below, along with the output for the goodness-of-fit test. $\begin{array}{l|c|c} & \begin{array}{c} \text { Correctly identified the door } \\ \text { concealing the twin } \end{array} & \begin{array}{c} \text { Incorrectly identified the } \\ \text { door concealing the twin } \end{array} \\ \hline \text { Observed } & 51 & 70 \\ \hline \text { Expected } & 30.25 & 90.75 \end{array}$

-Test the hypothesis that identical twins have a psychic connection and could locate their twin behind the door more often than would be expected by guessing. Using a goodness-of-fit test and a 0.05 level of significance, state the correct decision regarding the null hypothesis and summarize your conclusion using a complete sentence.

Question 58

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of public speaking then males. $$\begin{array} { l | c | c } & \text { Men } & \text { Women } \\ \hline \text { Expressed a fear of public speaking } & 112 & 79 \\ \hline \text { Did not express a fear of public speaking } & 91 & 82 \end{array}$$

-Describe the categorical variables that are summarized in the table. Describe one association that could be tested based on the information from the table.

Question 59

Use the following table to answer the question. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a greater fear of public speaking then males. $$\begin{array} { l | c | c } & \text { Men } & \text { Women } \\ \hline \text { Expressed a fear of public speaking } & 112 & 79 \\ \hline \text { Did not express a fear of public speaking } & 91 & 82 \end{array}$$

-Find the expected number of men who should express a fear of public speaking, if the variables are independent. Round to the nearest hundredth.