Question 1

A San Jose State student collects data from 20 students. He compares the number of classes a student is enrolled in to their GPA. Here are the results of the regression analysis. The conditions for inference are satisfied.

Simple linear regression results:
Dependent Variable: GPA
Sample size: 20
$\text { R-sq }=0.26753742$
$\text { s: } 0.45747$

Simple linear regression results: Dependent Variable: GPA Sample size: 20
R-sq = 0.26753742 s: 0.45747$\begin{array}{|l|r|r|r|r|}
\hline{\text { Coefficient }} & \text { Estimate } &{\text { Std. Err. }} & {\text { T-Stat }} & \text { P-Value } \
\hline \text { Constant } & 3.4246 & 0.16580 & 20.654 & <0.0001 \
\hline \text { No. of Classes } & -0.13940 & 0.054369 & -2.5641 & 0.0195\\hline
\end{array}$

-Is there evidence of a significant relationship between number of classes and GPA? Provide statistical justification for your answer.

Accepted Answer

H₀: ) = 0 There is no evidence of a linear relationship between number of classes and GPA. H_A: ) × 0 There is evidence of a linear relationship between number of classes and GPA. The conditions for inference have been met. t = -2.56; df = 18; p-value = 0.0195; a = 0.05 With a p-value of 1.95% < 5%, I reject H?. We have statistically significantly evidence that there is a linear relationship between number of classes and GPA.

Question 2

Poverty In a study of how the burden of poverty varies among U.S. regions, a random sample of 1000 individuals from each region of the United States recently yielded the information on poverty (based on defining the poverty level as an income below $10,400 for a family of 4 people). The data are provided in the table below. (All the conditions are satisfied - don't worry about checking them.) 
$\begin{array}{|l|c|c|c|c|c|}
\hline& \text { Northwest } & \text { Midwest } & \text { South } & \text { West } & \text { Total } \
\hline \text { Poor } & 112 & 105 & 154 & 113 & 484 \
& (121) & (121) & (121) & (121) & \\hline \text { Not Poor } & \begin{array}{c}
888 \
(879)
\end{array} & \begin{array}{c}
895 \
(879)
\end{array} & \begin{array}{c}
846 \
(879)
\end{array} & \begin{array}{c}
887 \
(879)
\end{array} & 3516 \
\hline \text { Total } & 1000 & 1000 & 1000 & 1000 & 4000 \
\hline
\end{array}$

$\begin{array}{r}
\chi^{2}=\quad 0.67+2.12+9.00+0.53+ \
0.09+0.29+1.24+0.07+ \
=14.01
\end{array}$
$P=0.0029$ 
a. Write appropriate hypotheses.
b. How many degrees of freedom?
c. Suppose the expected values had not been given. Show exactly how to calculate the expected count in the first cell.
d. State your complete conclusion in context.

Accepted Answer

a. $\mathrm { H } _ { 0 }$ : Poverty percentages are the same for the geographic regions in the Unites States.
$\mathrm { H } _ { \mathrm { A } }$ : Poverty percentages differ by geographic region in the United States.
b. degrees of freedom $= ( 4 - 1 ) ( 2 - 1 ) = 13$
c. $\frac { 1000 } { 4 000 } ( 484 ) = 121$
d. Since the $P$-value of $0.0029$ is so low, there is strong evidence that poverty percentages differ by geographic location in the United States. We will reject the null hypothesis and conclude that poverty is associated with geographic location in the United States. Specifically, the South contains a greater proportion of people below the poverty level than expected.

Question 3

Suppose that after the study described in #5 we want to see if there's evidence that the exercise program's effectiveness in lowering blood pressure depends on how high the person's initial blood pressure was. We should do a A) ç² test of independence B) ç² goodness-of-fit test C) matched pairs t-test D) linear regression t-test E) 2-sample t-test

Accepted Answer

A linear regression t-test is appropriate when we want to investigate the relationship between two continuous variables, such as the effectiveness of the exercise program and initial blood pressure. This test can help us determine if there is a significant linear relationship between the two variables, and whether the effectiveness of the exercise program changes based on initial blood pressure levels. The other options - ç₂ test of independence, ç₂ goodness-of-fit test, matched pairs t-test, and 2-sample t-test - are not appropriate for investigating this type of relationship between continuous variables.

Question 4

For the scenario described below, simply name the procedure that is appropriate to answer the question. For example,
1-proportion z-interval or chi-square goodness of fit test. Do NOT carry out the procedure.
-A flower pot manufacturer is testing his clay pots to ensure that the thickness of the sides are made to proper specifications. The sides are designed to be 4 mm thick. In a random sample of 25 pots, it is found that the average thickness if 4.3 mm. Does this provide statistically significant evidence that the manufacturing process is out of alignment?

Accepted Answer

The answer of For the scenario described below, simply name...

Question 5

College admissions According to information from a college admissions office, 62% of the students there attended public high schools, 26% attended private high schools, 2% were home schooled, and the remaining students attended schools in other countries. Among this college's Honors Graduates last year there were 47 who came from public schools, 29 from private schools, 4 who had been home schooled, and 4 students from abroad. Is there any evidence that one type of high school might better equip students to attain high academic honors at this college? Test an appropriate hypothesis and state your conclusion.

Accepted Answer

The answer of College admissions According to information from a...

Question 6

Peanut M&Ms According to the Mars Candy Company, peanut M&M's are 12% brown,
15% yellow, 12% red, 23% blue, 23% orange, and 15% green. On a Saturday when you have
run out of statistics homework, you decide to test this claim. You purchase a medium bag
of peanut M&M's and find 39 browns, 44 yellows, 36 red, 78 blue, 73 orange, and 48 greens. Test an appropriate hypothesis and state your conclusion.

Accepted Answer

The answer of Peanut M&Ms According to the Mars Candy...

Question 7

Cheater? A group of curious college students decide to test the integrity of their fellow collegians. In order to see if students will cheat, when given an opportunity, they decide to use chocolately M&M's. They tell each student that a discerning palette will be able to tell the difference in flavor between and red and a yellow candy. The blindfolded subjects are given two piles of candy to test. But the experimenter turns his back so that the subject thinks that they have a window of opportunity to take a quick peak. Unbeknownst to the subjects, there is another helper who is hidden and secretly watching to see who cheats. Here is their data. $\begin{array}{|l|l|l|}
\hline  & \text { Cheated } & \text { Did not cheat } \
\hline \text { Male } & 6 & 2 \
\hline \text { Female } & 3 & 7 \
\hline
\end{array}$ 
a. What is the probability that a subject cheated?
b. If a subject was a male, what are the chances that they cheated?
c. Using your answers to (a) and (b), does it appear that cheating and gender are
independent?
d. A statistics student in the group decides she wants to run a Chi-square test for independence. Why would this not be an advisable choice?
e. An argument begins. The girls are suggesting that the guys cheated more than girls; and
that this difference is larger than can be explained by chance variation. Of course, the guys insist that with a small sample size like this, anything could happen. Fortunately, a randomization machine is discovered. The 18 observations are randomly placed into the 4 categories randomly. This procedure is repeated 1000 times and the number of male
cheaters is counted each time. A graph is below. What does this graph tell you about the claims of the two groups?

Accepted Answer

The answer of Cheater? A group of curious college students...

Question 8

A sports analyst was interested in finding out how well a football team's winning percentage (stated as a proportion) can be predicted based upon points scored and points allowed. She selects a random sample of 15 football teams. Each team played 10 games. She decided to use the point differential, points scored minus points allowed as the predictor variable. The data are shown in the table below, and regression output is given afterward.

$\begin{array}{|c|c|}
\hline\text { Point Diff }&\text { Win Pct }\
\hline-75 & .100 \
\hline-60 & .300 \
\hline-40 & .200 \
\hline-33 & .300 \
\hline-21 & .400 \
\hline-15 & .500 \
\hline 5 & .500 \
\hline 14 & .700 \
\hline 20 & .600 \
\hline 28 & .700 \
\hline 31 & .800 \
\hline 45 & .900 \
\hline 56 & .700 \
\hline 72 & .600 \
\hline 80 & 1.000 \
\hline
\end{array}$

$\text { Regression Analysis: Win Pct versus Point Diff }$
$\begin{array}{lrrrr}
\text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \
\text { Constant } & 0.51802 & 0.03071 & 16.87 & 0.000 \
\text { Point Diff } & 0.0049500 & 0.0006682 & 7.41 & 0.000
\end{array}$
$S=0.117511 \quad \text { R-Sq }=80.8 \% \quad \text { R-Sq }(a d j)=79.48$ 
Is there evidence of an association between Point Differential and Winning Percentage? Test an appropriate hypothesis and state your conclusion in the proper context.

Accepted Answer

The answer of A sports analyst was interested in finding...

Question 9

Suppose you were asked to analyze each of the situations described below. (NOTE: DO NOT DO THESE PROBLEMS!) For each, indicate which inference procedure you would use (from the list), the test statistic (z, t, or ç₂ ), and, if t or ç₂ , the number of degrees of freedom. $\begin{array}{l}\begin{array}{|l|l|l|l|}
\hline & \text { Type } & z, t, \text { or } \chi^{2} & d f \
\hline \text { a. } & & & \
\hline \text { b. } & & & \
\hline \text { c. } & & & \
\hline \text { d. } & & & \
\hline \text { e. } & & & \
\hline \text { f. } & & & \
\hline \text { g. } & & & \
\hline \text { h. } & & & \
\hline
\end{array}
&\begin{array} { l  }
 \text {1. proportion, 1 sample}\
 \text {2. difference of proportions, 2 samples}\
 \text {3. mean, 1 sample}\
 \text {4. mean of differences, matched pairs}\
 \text {5. difference of means, independent samples}\
 \text {6. goodness of fit}\
 \text {7. homogeneity}\
 \text {8. independence}\
 \text {9. regression, inference for )}\
\end{array}\end{array}$ 
a. Doctors offer small candies to sixty teenagers, recording the number of candies consumed by each. One hour later they test the blood sugar level for each person. Is there any evidence that high blood sugar levels in teenagers are related to the amount of candy eaten?
b. Which takes less time to travel to work -- car or train? We select a random sample of 45 businessmen and compare their travel time to work for both types of commute.
c. An orthodontist wonders if soda in the diet may be a factor in loose cement on children's braces. She checks the cement bonds of 40 randomly selected patients who do not drink soda, and 40 patients who do drink soda.
d. Forty people complaining of allergies take an antihistamine. They report that their discomfort subsided in an average of 18 minutes; the standard deviation was 4 minutes. The manufacturer wants a 95% confidence interval for the "relief time".
e. A health professional selected a random sample of 100 patients from each of four major hospital emergency rooms to see if the primary reasons for emergency room visits are similar in all four major hospitals. The primary reasons were categorized as accident, illegal activity, illness, or other.
f. A policeman believes that more than 40% of older drivers speed on highways, but a confidential survey found that 49 of 88 randomly selected older drivers admitted speeding on highways at least once. Is this strong evidence that the policeman was wrong?
g. According to United Nations Population Division, the age distribution of the Commonwealth of Australia is: 21% less than 15 years of age, 67% between 15 and 65 years of age, and 12% are over 65 years old. A random sample of 210 residents of Canberra revealed 40 were less than 15 years of age, 145 were between 15 and 65 years of age, and 25 were over 65 years old. Are the ages of Canberra residents unusual in any way?
h. Among a random sample of college-age students, 6% of the 473 men said they had been adopted, compared to only 4% of the 552 women. Does this indicate a significant difference between adoption rates of males and females in college-age students?

Accepted Answer

The answer of Suppose you were asked to analyze each...

Question 10

A college admissions counselor was interested in finding out how well high school grade point averages (HS GPA) predict first-year college GPAs (FY GPA). A random sample of data from first-year students was reviewed to obtain high school and first-year college GPAs. The data are shown below:

$\begin{array}{|l|l|l|l|l|l|l|l|l|}
\hline \text {HS GPA }& 3.82 & 3.90 & 3.20 & 3.40 & 3.88 & 3.50 & 3.60 & 3.70 \
\hline \text {FY GPA} & 3.75 & 3.45 & 2.60 & 2.95 & 3.50 & 2.76 & 3.10 & 3.40 \
\hline
\end{array}$

$\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline\text { HS GPA} & 4.00 & 3.30 & 3.50 & 3.80 & 3.87 & 4.00 & 3.20 & 3.82 \
\hline\text { FY GPA }& 3.90 & 2.70 & 3.00 & 3.00 & 3.10 & 3.77 & 2.80 & 3.54 \
\hline
\end{array}$

Dependent variable is: $ \quad $ FY GPA
No Selector
$ \mathrm{R} $ squared $ =75.4 \% \quad \mathrm{R} $ squared (adjusted) $ =73.6 \% $
$ s=0.2118 $ with $ 16-2=14 $ degrees of freedom

$ \begin{array}{llrrr}\text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \ \text { Regression } & 1.92283 & 1 & 1.92283 & 42.9 \ \text { Residual } & 0.627867 & 14 & 0.044848 & \ & & & & \ \text { Variable } & \text { Coefticient } & \text { s.e. of Coeft } & \text { t-ratio } & \text { prob } \ \text { Constant } & -1.56410 & 0.7306 & -2.14 & 0.0504 \ \text { HS GPA } & 1.30527 & 0.1993 & 6.55 & \leq 0.0001\end{array} $

-Is there evidence of an association between high school and first-year college GPAs? Test an appropriate hypothesis and state your conclusion in the proper context.

Accepted Answer

The answer of A college admissions counselor was interested in...

Question 11

Height and weight Is the height of a man related to his weight? The regression analysis from a sample of 26 men is shown. (Show work. Don't write hypotheses. Assume the assumptions for inference were satisfied.) $$\begin{array} { | l c c | } 
\hline \begin{array} { l } 
\text { Dependent variable is: } \
\text { Variable }
\end{array} & \text { Coefficient } & \begin{array} { c } 
\text { Weight(lbs) } \
\text { s.e. of coeff }
\end{array} \
\text { Constant } & - 449.61 & 93.28 \
\text { Height(inches) } & 8.737 & 1.312 \
\hline
\end{array}$$ 
a. How many degrees of freedom?
b. What is the value of the t statistic?
c. What is the P-value?
d. State your conclusion in context.

Accepted Answer

The answer of Height and weight Is the height of...

Question 12

For the scenario described below, simply name the procedure that is appropriate to answer the question. For example,
1-proportion z-interval or chi-square goodness of fit test. Do NOT carry out the procedure.
-A national marketing campaign is conducted to improve the name-brand recognition with potential customers. The company sells household products, primarily to the female head of the household. The current market analysis shows that 23% of their customer base recognizes the name of their product. When the marketing campaign is finished, 135
people out of 500 randomly polled recognize the brand name. Does this provide statistically significant evidence that there has been an increase in name-brand recognition?

Accepted Answer

The answer of For the scenario described below, simply name...

Question 13

A college admissions counselor was interested in finding out how well high school grade point averages (HS GPA) predict first-year college GPAs (FY GPA). A random sample of data from first-year students was reviewed to obtain high school and first-year college GPAs. The data are shown below: $\begin{array}{|l|l|l|l|l|l|l|l|l|}
\hline \text {HS GPA }& 3.82 & 3.90 & 3.20 & 3.40 & 3.88 & 3.50 & 3.60 & 3.70 \
\hline \text {FY GPA} & 3.75 & 3.45 & 2.60 & 2.95 & 3.50 & 2.76 & 3.10 & 3.40 \
\hline
\end{array}$

$\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline\text { HS GPA} & 4.00 & 3.30 & 3.50 & 3.80 & 3.87 & 4.00 & 3.20 & 3.82 \
\hline\text { FY GPA }& 3.90 & 2.70 & 3.00 & 3.00 & 3.10 & 3.77 & 2.80 & 3.54 \
\hline
\end{array}$

Dependent variable is: $ \quad $ FY GPA
No Selector
$ \mathrm{R} $ squared $ =75.4 \% \quad \mathrm{R} $ squared (adjusted) $ =73.6 \% $
$ s=0.2118 $ with $ 16-2=14 $ degrees of freedom

$ \begin{array}{llrrr}\text { Source } & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F-ratio } \ \text { Regression } & 1.92283 & 1 & 1.92283 & 42.9 \ \text { Residual } & 0.627867 & 14 & 0.044848 & \ & & & & \ \text { Variable } & \text { Coefticient } & \text { s.e. of Coeft } & \text { t-ratio } & \text { prob } \ \text { Constant } & -1.56410 & 0.7306 & -2.14 & 0.0504 \ \text { HS GPA } & 1.30527 & 0.1993 & 6.55 & \leq 0.0001\end{array} $

-Create and interpret a 95% confidence interval for the slope of the regression line.

Accepted Answer

The answer of A college admissions counselor was interested in...

Question 14

As part of a survey, students in a large statistics class were asked whether or not they ate breakfast that morning. The data appears in the following table: $$\begin{array}{l}
\quad\quad\quad\quad\quad\quad\quad\quad\quad\text { Breakfast }\
\begin{array} { c c | c c | c } 
& & \text { Yes } & \text { No } & \text { Total } \
\hline { \text { Sex } } & \text { Male } & 66 & 66 & 132 \
& \text { Female } & 125 & 74 & 199 \
\hline & \text { Total } & 191 & 140 & 331
\end{array}
\end{array}$$ 
Is there evidence that eating breakfast is independent of the student's sex? Test an appropriate hypothesis. Give statistical evidence to support your conclusion.

Accepted Answer

The answer of As part of a survey, students in...

Question 15

Carnivores A random sample of some of the heaviest carnivores on Earth was reviewed to determine if there is an association between the length (in meters) and weight (in kilograms) of these carnivores. Here are the scatterplot, the residuals plot, a histogram of the residuals, and the regression analysis of the data. Use this information to analyze the association between the length and weight of these carnivores.

The regression equation is 
Weight $ =-668+485 $ Length
$\begin{array}{lrrrr}
\text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \
\text { Constant } & -668.3 & 232.8 & -2.87 & 0.024 \
\text { Length } & 485.21 & 90.63 & 5.35 & 0.001
\end{array}$
$S=194.794 \quad R-S q=80.48 \quad R-S q(a d j)=77.68$

a. Is there an association? Write appropriate hypotheses. 
b. Are the assumptions for regression satisfied? Explain. 
c. What do you conclude?
d. Create a 98% confidence interval for the true slope. 
e. Explain in context what your interval means.

Accepted Answer

The answer of Carnivores A random sample of some of...

Question 16

Voter registration A random sample of 337 college students was asked whether or not they were registered to vote. We wonder if there is an association between a student's sex and whether the student is registered to vote. The data are provided in the table below
(expected counts are in parentheses). (All the conditions are satisfied - don't worry about checking them.) $$\begin{array} { | l | c | c | c | } 
\hline & \text { Men } & \text { Women } & \text { Total } \
\hline \text { Registered } & 104 ( 102 ) & 147 ( 149 ) & 251 \
\hline \text { Not Registered } & 33 ( 35 ) & 53 ( 51 ) & 86 \
\hline \text { Total } & 137 & 200 & 337 \
\hline
\end{array}$$ 
 $$\text { The calculated statistic is } ? ^ { 2 } = 0.249 \text {. }$$ 
a. Write appropriate hypotheses.
b. Suppose the expected values had not been given. Show exactly how to calculate the expected number of men who are registered to vote.
c. Show how to calculate the component of ç? for the first cell. d. How many degrees of freedom are there?
e. Find the P-value for this test.
f. State your complete conclusion in context.

Accepted Answer

The answer of Voter registration A random sample of 337...

Question 17

In the study "The Role of Sports as a Social Determinant for Children," student respondents in grades 4 through 6 were asked what they would most like to do at school: make good grades, be popular or be good at sports. Results delineated by type of school district are reported below. $$\begin{array} { | l | c | c | c | } 
\hline & \text { Rural } & \text { Suburban } & \text { Urban } \
\hline \text { Make Good Grades } & 57 & 87 & 24 \
\hline \text { Be Popular } & 50 & 42 & 6 \
\hline \text { Be Good at Sports } & 42 & 22 & 5 \
\hline
\end{array}$$ 
Source: Chase, M.A and Dummer, G.M. (1992), "The Role of Sports as a Social Determinant for
Children," Research Quarterly for Exercise and Sport, 63, 418-424.
Is there evidence that type of school district and personal school goals are independent? Test an appropriate hypothesis. Give Statistical evidence to support your conclusion.

Accepted Answer

The answer of In the study "The Role of Sports...

Question 18

Of the 23 first year male students at State U. admitted from Jim Thorpe High School, 8 were offered baseball scholarships and 7 were offered football scholarships. The University admissions committee looked at the students' composite ACT scores (shown in table), wondering if the University was lowering their standards for athletes. Assuming that this group of students is representative of all admitted students, what do you think?

$\quad $$\quad $$\text {Composite ACT Score } $
$\begin{array}{|c|c|c|}
\hline\text { Baseball }&\text { Non-athletes }& \text {Football }\
\hline 25 & 21 & 22 \
22 & 27 & 21 \
19 & 29 & 24 \
25 & 26 & 27 \
24 & 30 & 19 \
25 & 27 & 23 \
24 & 26 & 17 \
23 & 23 & \
\hline
\end{array}$

Boxplots:

$\text {Analysis of Variance Table}$
$\begin{array}{|c|c|r|c|c|c|}
\hline &\text { Sums of }&&\text { Mean }&&\text {  P-}  \
 \text { Source}&\text { Squares } &d f &\text { Squares } &\text {  F-ratio }&\text { value }\
\hline Team & 71.00 & 2 & 35.50 & 4.56 & 0.023 \
\hline Error & 155.61 & 20 & 7.78 & & \
\hline Total & 226.61 & 22 & & & \
\hline
\end{array}$

$\text {Means and Std Deviations}$
$\begin{array}{|l|r|r|r|}
\hline  \text {Level }&  \text {Number} &  \text {Mean }&  \text {Std Dev} \
\hline  \text {Baseball} & 8 & 23.3750 & 2.06588 \
\hline  \text {Football }& 7 & 21.8571 & 3.28778 \
\hline  \text {Non Athlete} & 8 & 26.1250 & 2.94897 \
\hline
\end{array}$

Normal Probability Plot:

-Test an appropriate hypothesis and state your conclusion

Accepted Answer

The answer of Of the 23 first year male students...

Question 19

Several volunteers engage in a special exercise program intended to lower their blood pressure. We measure each person's initial blood pressure, lead them through the exercises daily for a month, then check blood pressures again. To see if the program lowered blood pressure significantly we should do a A) matched pairs t-test B) ç² test of homogeneity C) 2-sample t-test D) linear regression t-test E) ç² goodness-of-fit test

Accepted Answer

A matched pairs t-test is appropriate because we are measuring each person's blood pressure before and after the exercise program, and we are testing whether there is a significant difference between the two measurements. This type of test is appropriate when dealing with dependent/paired samples.

Question 20

Wingspan A person's wingspan is the distance from fingertip to fingertip when their arms are fully extended. The longer a person's wingspan, the taller they tend to be. Regression analysis was executed on 24 individuals to see if height in inches can be used to predict wingspan (also in inches). The conditions for inference were deemed to be reasonably satisfied.

Dependent Variable: Wingspan
Sample size: 24
R-sq = 0.8026696 s: 2.1512606 $$\begin{array} { | l | r | r | r | r | } 
\hline \text { Coefficient } & { \text { Estimate } } &  { \text { Std. Err. } } &  { \text { T-Stat } } & \text { P-Value } \
\hline \text { Constant } & - 13.024544 & 8.5719795 & - 1.5194324 & 0.1429 \
\hline \text { Height } & 1.1909246 & 0.125893 & 9.459816 & < 0.0001 \
\hline
\end{array}$$ 
a. Write the equation of the regression line. Make sure to define all the variables in your equation.
b. Interpret the slope of the regression equation in context. c. Interpret the value of s in context.
d. Find and interpret a 95% confidence interval for slope.
e. Is the relationship between wingspan and height a strong relationship? Why? Give two reasons to justify your answer.
^

Accepted Answer

The answer of Wingspan A person's wingspan is the distance...

Quiz 7: Scatterplots, Association, and Correlation

Suppose that after the study described in #5 we want to see if there's evidence that the exercise program's effectiveness in lowering blood pressure depends on how high the person's initial blood pressure was. We should do a