WEEK 4 Problems

Chapters 13 & 15

Chapters:

Packet Pages:

Ch. 13: The Pearson Correlation Coefficient

Ch. 15: Non-parametric Statistical Tests: Chi-Square

1-9

10-17

Refer to the Resource Page on CANVAS for:

StatSheets

Table 6, Critical Values of r

Table 10, Critical Values of Chi-Square

Online calculator/statistical app links & tutorials

Chapter 13

The Pearson Correlation Coefficient

Ch. 13-

Defining Key Terms

Provide definitions for the following key terms:

COEFFFICIENT OF DETERMINATION/EFFECT SIZE (r2):

DIRECT/POSITIVE RELATIONSHIP:

INVERSE/NEGATIVE RELATIONSHIP:

OUTCOME VARIABLE:

OUTLIER:

PARTIAL CORRELATION:

PEARSON CORRELATION COEFFICIENT (and possible values):

PERFECT RELATIONSHIP:

PREDICTOR VARIABLE:

WEEK 4 CH 13 & 15 Problems

Page 1 of 17

Ch. 13- Looking Up Critical Values of r (rcv) in Table 6 & Marking the

Rare & Common Zones

For each of the following studies, look up the critical value of r (Table 6) that separates the rare zone

from the common zone & mark on the sampling distribution of the r ratio. (see textbook pp. 496499). There is also an on-line calculator for finding the critical values.

Marking Rare & Common Zones

The common zone is the middle section

of the distribution, the section centered

around zero where it would be common

to find the r-value for a sample if there

was no relationship between variables.

As the r-value increases (moves farther

away from zero), the results are more

likely to be statistically significant,

meaning a significant relationship.

Example:

A developmental psychologist wanted to determine if there were a relationship between the age at

which children started to walk and their intelligence at age 16. She went to a pediatricianâ€™s office and

randomly selected 10 charts of 16-year-old girls. In the charts, she found the age (in months) at which

each girl started walking and then she gave each girl a standard IQ test. Use alpha=.05 two-tailed test.

Answer:

STEP 1: For N=10 pairs of data, Use Equation 13.1: df = N-2. For this study df = 10-2 = 8.

STEP 2: Use Table 6 (Critical Values of r) to find cutoffs for Rare & Common Zones. Or use

online calculator.

rcv = +/- .632

-.632

WEEK 4 CH 13 & 15 Problems

+.632

Page 2 of 17

Find the critical value of r & mark the common & rare zones on the sampling distribution:

1. For a two-tailed hypothesis test with alpha set at Î± = .05 and df =7?

2. For a two-tailed hypothesis test with alpha set at Î± = .05 and df =27?

3. For a two-tailed hypothesis test with alpha set at Î± = .05 and N =33?

WEEK 4 CH 13 & 15 Problems

Page 3 of 17

Ch. 13-

Reading Research Studies: Pearson Correlation Coefficient

Use the table and excerpt from the research article to answer the questions below. A brief â€œstatistical

guideâ€ summary is included to assist with understanding.

Statistical Guide:

A correlation coefficient indicates the strength and direction of a relationship between two variables.

The most widely used is the Pearson r. When it is positive in value, the relationship is direct (i.e., those

with high scores on one variable tend to have high scores on the other variable and those with low

scores on one variable tend to have low scores on the other). In a direct relationship, the closer r is to

1.00, the stronger the relationship; the closer to 0.00, the weaker. When the value of Pearson r is

negative, the relationship is inverse (i.e., those with high scores on one variable tend to have low scores

on the other one). In an inverse relationship, the closer r is to -1.00, the stronger the relationship; the

closer it is to 0.00, the weaker. For the table below, the lower the probability (p-value), the more

significant the relationship. The p-values are provided in the key below the table.

CORRELATES OF ALCOHOL AND TOBACCO USE

Excerpt from the Research Article

From intact families, 321 adolescents participated in the present studyâ€¦ the age range for all being

12-16 years. There was no significant age difference between the sexes. Participants were asked to

estimate how often (a) they, (b) their mothers, (c) their fathers, and (d) their best friends smoked tobacco

and used alcoholic beverages. The estimations were made on a 5-point scale with anchors of 0: never, 1:

almost never, 2: occasionally, 3: often, 4: very often.

The present study corroborates previous research indicating both parental and peer influence

on tobacco and alcohol consumption among adolescents. The study suggests that the influence of

peers might be greater than that of parents. However, correlations as such do not suggest causal

relationships. There are certainly other social psychological factors that may contribute to the

explanation of or co-vary with adolescent alcohol and tobacco consumption.

WEEK 4 CH 13 & 15 Problems

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RESEARCH STUDY QUESTIONS:

1. For boys, which variable correlates most highly with their tobacco habits?

2. For girls, which variable correlates most highly with their alcohol habits?

3. Is the relationship between boysâ€™ alcohol habits and their fathersâ€™ tobacco habits strong?

Explain.

4. Is the relationship between boysâ€™ alcohol habits and their friendsâ€™ alcohol habits statistically

significant? If yes, at what probability level?

5. For the question above, should the null hypothesis for the relationship be rejected?

WEEK 4 CH 13 & 15 Problems

Page 5 of 17

Ch. 13-

Computing a Pearson Correlation Coefficient

For the study below, go thru steps of hypothesis testing to conduct a Pearson Correlation

Coefficient r-test & determine effect size (known as â€œcoefficient of determinationâ€™, represented by

r2.) Answer questions in the table. (see text pp. 493-518, Ch. 13 StatSheet & online calculators).

EXAMPLE A.

Study: The data in the table are from 10 high school seniors. Each senior took both a Logical

Reasoning Test and a Creativity Test. The investigator wanted to know if there is a

relationship between logical reasoning and creativity.

QUESTIONS:

a.

HYPOTHESES: List the Null & Alternative Hypotheses:

Null Hyp: There is no relationship between Logical Reasoning & Creativity Scores.

Alt Hyp: There is a relationship between Logical Reasoning & Creativity Scores.

b.

SET THE DECISION RULE: Using the decision rule of alpha=.05, find the degrees of freedom

for this study (Equation 13.1), & find the critical values of r (using Table 6 or online

calculator). Mark the common & rare zones on the distribution below.

rCV= -.632

Rare Zone

WEEK 4 CH 13 & 15 Problems

rCV= .632

Remember, the df

formula for this

test is df=N-2.

With N=10 pairs

for this study, that

means the df=8.

Use Table 6 to find

alpha=.05. , df=8,

critical values = +/.632.

Rare Zone

Page 6 of 17

EXAMPLE A (continuedâ€¦)

c.

CALCULATE THE STATISTIC: What is the calculated or computed r-test statistic for this

problem? (Equations 13.2; & on-line calculator)

The statistic was calculated using Ch. 13 Calculators found on WEEK 7 RESOURCE PAGE.

The Calculated Pearson r statistic = .353, falling short of the critical value of .632 set by the

Decision Rule.

INTERPRET THE RESULTS

(see pp. 506+)

d.

WAS THE NULL HYPOTHESIS REJECTED?: Is the Pearson Correlation result statistically

significant?

No, the Null Hypothesis is not rejected for this problem. The calculated r-value does not

meet the Decision Criteria of .632 needed for significance. The statistic fell in the Common

Zone.

e.

HOW BIG IS THE EFFECT?: Calculate the Effect Size (Coefficient of Determination) using r2

(Equation 13.3; on-line calculator) & interpret:

Since the calculated r=.353 for this problem, the

Effect Size is calculated with Equation 13.3:

r2 = (r) 2 x 100

r2 = (.353) 2 x 100 = 12.46%, medium

f.

CONCLUSION?: What conclusion or specific statement can be made about the relationship

between the pretest and final course grade in statistics?

The calculated r-value was not in the rare zone, meaning it was not statistically significant.

There is not enough evidence to conclude there is a relationship between Logical

Reasoning and Creativity Scores. Possibly a larger sample size might be studied in the

future.

WEEK 4 CH 13 & 15 Problems

Page 7 of 17

Ch. 13-

Computing a Pearson Correlation Coefficient

For the study below, go through the steps of hypothesis testing to conduct a Pearson

Correlation Coefficient r-test & determine effect size (known as the â€œcoefficient of

determinationâ€™, represented by r2.) Answer the questions in the table below. (see textbook

pp. 493-518, Ch. 13 StatSheet & online calculators).

Research Study:

The following data represent data from 10 graduate students. Each student took a screening

pretest before starting their statistics class. They each completed the class; their final grades are

shown in the table below. The question being considered is: Can a screening test predict

success in a statistics class? â€¦ or Is there a significant relationship between screening test

performance and final grade in the class?

STUDENT

PRETEST

SCORE (x)

FINAL

COURSE

GRADE (y)

1

18

9

2

14

13

3

7

15

4

6

4

5

10

3

6

8

2

7

8

2

8

8

2

9

8

2

10

5

8

QUESTIONS:

a.

HYPOTHESES: List the Null & Alternative Hypotheses:

WEEK 4 CH 13 & 15 Problems

Page 8 of 17

b.

SET THE DECISION RULE: Using the decision rule of alpha=.05, find the degrees of freedom for

this study (Equation 13.1), & find the critical values of r (using Table 6). Use the critical values

to mark the common & rare zones on the distribution below.

c.

CALCULATE THE STATISTIC: What is the calculated or computed r-test statistic for this

problem? (Equations 13.2; & on-line calculator)

INTERPRET THE RESULTS

(see pp. 506+)

d.

WAS THE NULL HYPOTHESIS REJECTED?: Is the Pearson Correlation result statistically

significant?

e.

HOW BIG IS THE EFFECT?: Calculate the Effect Size (Coefficient of Determination) using r2

(Equation 13.3; on-line calculator) & interpret:

f.

CONCLUSION?: What conclusion or specific statement can be made about the relationship

between the pretest and final course grade in statistics?

WEEK 4 CH 13 & 15 Problems

Page 9 of 17

Chapter 15

Non-parametric Statistical Tests: Chi-Square

Ch. 15-

Defining Key Terms

Provide definitions for the following key terms:

CHI-SQUARE GOODNESS-OF-FIT TEST:

CHI-SQUARE TEST OF INDEPENDENCE:

CONTINGENCY TABLE:

MANNâ€“WHITNEY U TEST:

NONPARAMETRIC TEST:

PARAMETRIC TEST:

SINGLE-SAMPLE TEST:

SPEARMAN RANK-ORDER CORRELATION COEFFICIENT:

WEEK 4 CH 13 & 15 Problems

Page 10 of 17

Ch. 15- Understanding Concepts

(see textbook pp. 567-8, 581-2, 572-4).

Provide a brief answer to the following questions:

QUESTIONS:

1. When would you use a non-parametric test?

ANSWER:

2. What do we mean when we say the non-parametric test is less powerful? Why is it less

powerful?

ANSWER:

3. What parametric test is the Chi-Square Goodness-of-Fit test most similar to and why?

ANSWER:

4. Which parametric test is the Chi-Square Test of Independence most similar to?

ANSWER:

5. What do the calculated expected frequencies tell you?

ANSWER:

WEEK 4 CH 13 & 15 Problems

Page 11 of 17

Ch. 15-

Computing a Chi-Square Goodness-of-Fit Test

For the study below, go through the steps of hypothesis testing to conduct a Chi-Square Goodness-ofFit Test. Answer the questions below. (see text pp. 568-581, Ch. 15 StatSheet & online calculators).

EXAMPLE A.

A researcher investigated whether gender roles are becoming more equal, with men and women

sharing child rearing responsibility, earning equal pay and sharing chores. For the study, a

researcher collected data at a local grocery store in late morning one day and observed who was

shopping with children; noting if the person was a male or a female. For the 42 people observed, 14

were men with children and 28 were women with children. From the data collected, can the

researcher say that men and women, at least as far as this aspect of child rearing goes, are equal?

NOTE: We are using the Chi-Square Goodness-of-Fit Test because comparing an observed value to

an expected value for a nominal level variable. This test is similar to a Single-Sample z or t Test,

since drawing a single sample and comparing the sample data to expected population data.

1.List the Hypotheses:

Null Hyp: Distribution of gender in sample is same as population

Alt Hyp: Distribution of gender in sample differs from population

Note: if men & women shared equally, then half the people with

children in the store would be women, & half men. As half of 42 is

21, we expect that 21 of the people with children would be men &

21 would be women. We compare observed with expected for

this test.

Using standard level of alpha=.05, 2 tailed test. Go to Table 10,

â€œCritical Values of Chi Squareâ€, for df=k-1 (where k = # of

categories).

2.Set the Decision Rule:

df= 2-1 = 1

10

WEEK 4 CH 13 & 15 Problems

Critical Value= 3.841. If the calculated

the Null Hypothesis.

â‰¥ 3.841, then reject

Page 12 of 17

METHOD 1: Use Equation 15.3 to calculate the Chi-Square value.

Take the observed frequencies of 14 & 28. We already know the

expected frequencies of 21 and 21.

3.Calculate the Test Statistic:

(the calculations are performed for each cell/category & then summed

to obtain the chi-square value).

METHOD 2: Use online Calculator for Chapter 15.

4.Interpret the Results:

Using the decision rule, we are rejecting the null hypothesis

because the calculated test statistic of 4.66 is greater than the

critical value of 3.841.

The observed distribution of gender differs from what would be

expected if men and women split responsibilities equally.

WEEK 4 CH 13 & 15 Problems

Page 13 of 17

Ch. 15-

Computing a Chi-Square Goodness-of-Fit Test

For the study below, go through the steps of hypothesis testing to conduct a Chi-Square

Goodness-of-Fit Test. Answer the questions in the table below. (see textbook pp. 568-581, Ch.

15 StatSheet & online calculators).

STUDY:

Vallone studied attitudes about smoking on a college campus. She hypothesized that 60% of

undergraduates would want a smoke-free campus. She surveyed 200 undergraduates and found

that 90 preferred a smoke-free campus.

Hint: create a table to organize the data.

Observed Frequency

Expected Frequency

Smoke

Free

90

100

Non-Smoke

Free

110

100

N=200

QUESTIONS:

a.

LIST THE HYPOTHESES:

b. SET THE DECISION RULE: Use the decision rule of alpha=.05, 2-tailed test. Use Table 10 to find

the critical value.

c.

CALCULATE THE TEST STATISTIC:

d. INTERPRET THE RESULTS:

WEEK 4 CH 13 & 15 Problems

Page 14 of 17

Ch. 15-

Computing a Chi-Square Test of Independence

For the study below, go through the steps of hypothesis testing to conduct a Chi-Square Test

of Independence. Answer the questions in the table below. (see textbook pp. 581-597, Ch.

15 StatSheet & online calculators).

NOTE: We are using the Chi-Square Test of Independence to determine whether two groups differ

on a categorical dependent variable.

EXAMPLE A.

The researcher in the previous example was intrigued so did a larger study to see if a difference

existed in gender roles between small towns that had colleges and small towns without colleges.

The researcher drew random samples of both kinds of towns, went to grocery stores during the

day, and for those people shopping with kids, noted the gender of the adult. The results are in the

table below.

Shopping with Kids

Observed

Male

College Towns

130

Non-College Towns

100

230

1. List the Hypotheses:

H0= There is no relationship

between the variables.

H1= There is a relationship

between the variables.

Female

370

400

770

500

500

Null Hyp: The two variables are independent of each other (not

related).

Alt Hyp: The two variables are not independent of each other

(are related).

Using standard level of alpha=.05, 2 tailed test. Use df formula

from Equation 15.4: df= (R-1) x (C-1).

df = (2-1) (2-1) = 1

Go to Table 10, â€œCritical Values of Chi Squareâ€, for df=1 &

alpha=.05

Critical Value= 3.841. If the calculated X2 â‰¥ 3.841, then reject the

Null Hypothesis.

WEEK 4 CH 13 & 15 Problems

Page 15 of 17

3. Calculate the Test Statistic:

METHOD 1: Use Eq. 15.3 & 15.5 to calculate expected frequencies

& Chi-Square value.

X2 = (130-115)2+ (370-385)2+ (100-115)2+ (400-385)2= 5.08

115

385

115

385

METHOD 2: Use online Calculator for Chapter 15.

Using the decision rule, we are rejecting the null hypothesis

because the calculated test statistic of 5.08 is greater than the

critical value of 3.841.

4. Interpret the Results:

A difference exists between the two populations in terms of the

percentages of men who take children shopping: It is more

Using the

rule, we

arethan

rejecting

the null hypothesis

common

in decision

college towns

(26%)

in noncollege

towns (20%).

Usingthe

thecalculated

decision rule,

we are rejecting

null than the

because

test statistic

of 5.08 isthe

greater

hypothesis

critical

value of because

3.841. the calculated test statistic of

5.08 is greater than the critical value of 3.841.

A difference exists between the two populations in terms of the

A difference

exists

between

the two

populations

percentages

of men

who

take children

shopping:

It is more

in

terms

of

the

percentages

of

men

who

take towns (20%).

common in college towns (26%) than in noncollege

children shopping: It is more common in college

towns (26%) than in noncollege towns (20%).

WEEK 4 CH 13 & 15 Problems

Page 16 of 17

Ch. 15-

Computing a Chi-Square Test of Independence

For the study below, go through the steps of hypothesis testing to conduct a Chi-Square Test of

Independence. Answer the questions in the table below. (see textbook pp. 581-597, Ch. 15

StatSheet & online calculators).

STUDY:

Harris & Sanborn were interested in gender differences in movie preference. They surveyed 50

men & 50 women & asked them whether they had seen a horror movie in the last month. Of the

men, 35 had seen a horror movie. Of the women, 20 had seen a horror movie.

Hint: create tables to organize the data.

QUESTIONS:

a.

LIST THE HYPOTHESES:

b. SET THE DECISION RULE: Use the decision rule of alpha=.05, 2-tailed test. Use Table 10 to find

the critical value.

c.

CALCULATE THE TEST STATISTIC:

d. INTERPRET THE RESULTS:

WEEK 4 CH 13 & 15 Problems

Page 17 of 17

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