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PSYC FPX4700 Assessment 4 ANOVA, Chi-Square Tests, and Regression

ANOVA, Chi-Square Tests, and Regression

Complete the following problems within this Word document. Do not submit other files. Show your work for problem sets that require calculations. Ensure that your answer to each problem is clearly visible. You may want to highlight your answer or use a different type color to set it apart.

ANOVA

Problem Set 4.1: Critical Value

Criterion: Explain the relationship between k and power based on calculated k values.

Instructions: Read the following and answer the questions. 

Work through the following and write down what you see in the F-table. This will help familiarize you with the table. 

The F-table: The degrees of freedom for the numerator (k − 1) are across the columns; the degrees of freedom for the denominator (Nk) are across the rows in the table. A separate table is included for a .05 and .01 level of significance. 

Increasing the levels of the independent variable (k):

Suppose we have a sample size of 24 participants (N = 24). Record the critical values given the following values for k:

 

.05

.01

k = 2

k = 4

k = 6

k = 8

4.30

3.10

2.60

2.37

7.03

4.25

3.45

3.06

As k increases (from 1 to 8), does the critical value increase or decrease? Based on your answer, explain how k is related to power.

PSYC FPX4700 Assessment 4 ANOVA, Chi-Square Tests, and Regression

As the number of groups (k) increases from 1 to 8, the critical F-value shows a decreasing trend. This implies that the F-statistic needed to reject the null hypothesis decreases with the increase in groups. The reason behind this is that when the number of groups is more, the difference in means among them can be better clarified by the treatment effect, which reduces the likelihood of finding a significant difference by chance alone.

On the other hand, the decreasing critical F-value implies that as the number of groups (k) increases, the power of the test to detect a significant difference between group means also decreases. Power refers to the probability of accurately rejecting the null hypothesis when it is actually false, and it is affected by various factors, including sample size, effect size, and significance level. As k rises, the degrees of freedom for the denominator also decrease, which minimizes the overall data variability and may decrease the test’s power. As a result, the relationship between k and power is not straightforward and is influenced by other factors such as sample size and effect size.

Problem Set 4.2: One-way ANOVA in JASP

Criterion: Calculate an ANOVA in JASP.

Data: Use the dataset stress.jasp.The dataset stress.jasp is a record of the amount of fat (in grams) consumed in a buffet-style lunch among professional bodybuilders under conditions of high, moderate, and low stress.

Instructions: Complete the steps below.

  1. Download stress.jasp. Double-click the icon to open the dataset in JASP. 
  2. In the Toolbar, click ANOVA. In the menu that appears, under Classical, select ANOVA.
  3. Select Fat grams consumed and then click the upper Arrow to send it over to the Dependent Variable box.
  4. Select Stress level and then click the lower Arrow to send it over to the Fixed factors box.
  5. Check the Descriptive statistics box. 
  6. Copy and paste the output below.

ANOVA

ANOVA – Fat Grams Consumed 

Cases

Sum of Squares

df

Mean Square

F

p

Stress Level

 

15.600

 

2

 

7.800

 

1.773

 

0.212

 

Residuals

 

52.800

 

12

 

4.400

   

 

 
 

Note.  Type III Sum of Squares

 

Descriptives

Descriptives – Fat Grams Consumed 

Stress Level

N

Mean

SD

SE

Coefficient of variation

High

 

5

 

8.600

 

2.408

 

1.077

 

0.280

 

Low

 

5

 

6.200

 

1.924

 

0.860

 

0.310

 

Moderate

 

5

 

6.800

 

1.924

 

0.860

 

0.283

 
 

 

Problem Set 4.3: One-way ANOVA in Excel

Criterion: Calculate an ANOVA in Excel.

Instructions: Use the data from the table below to complete the following steps:

  1. Open Excel to an empty sheet.
  2. Enter the data from this table.

Stress Levels

High 

Moderate

Low

10

9

9

7

4

4

8

7

6

12

6

5

6

8

7

 

  1. In Row 1, enter High in cell A1, Moderate in cell B1, and Low in cell B1.
  2. In the toolbar, click Data Analysis, select Anova: Single Factor, and click OK.
  3. In Input Range: $A$1:$C$6, put a check next to Labels in First Row, click OK.
  4. Results will appear in a new sheet to the left; copy and paste the input below.

 

Anova: Single Factor

     
       

SUMMARY

     

Groups

Count

Sum

Average

Variance

  

High 

5

43

8.6

5.8

  

Moderate

5

34

6.8

3.7

  

Low

5

31

6.2

3.7

  
       
       

ANOVA

      

Source of Variation

SS

df

MS

F

P-value

F crit

Between Groups

15.6

2

7.8

1.772727

0.211576

3.885294

Within Groups

52.8

12

4.4

   
       

Total

68.4

14

 

 

 

 

 

Problem Set 4.4: One-way ANOVA Results in APA Style

Criterion: Report ANOVA results in APA format.

Data: Use the results from Problem Set 4.4.

Instructions: Complete the following:

  1. State the null hypothesis. _______ No relationship between stress and fat consumption exists.
  2. Report your results in APA format (as you might see them reported in a journal article). _  

A one-way ANOVA was performed with high, medium, and low-stress levels as the independent variable and fat consumption as the dependent variable. The analysis did not yield sufficient evidence to conclude a relationship between stress and fat consumption. Participants in the medium and low-stress groups reported lower fat consumption levels than those in the high-stress group. Interestingly, there was no significant difference in fat consumption between the medium and low-stress groups, suggesting that stress may not be a crucial factor in determining fat consumption levels._______

PSYC FPX4700 Assessment 4 ANOVA, Chi-Square Tests, and Regression

Problem Set 4.5: Interpret ANOVA Results 

Criterion: Interpret the results of an ANOVA.

Instructions: Read the following and answer the question. 

Data: Life satisfaction among sport coaches. Drakou et al. (2006) tested differences in life satisfaction among sport coaches. They tested differences by sex, age, marital status, and education. The results of each test in the following table are similar to the way in which the data were given in their article.

 

Independent Variables

Life Satisfaction

M

SD

F

p

Sex

  

0.68

.409

Men

3.99

0.51

   

Women

3.94

0.49

   

Age

  

3.04

.029

 

20s

3.85

0.42

   

30s

4.03

0.52

   

40s

3.97

0.57

   

50s

4.02

0.50

   

Marital status

  

12.46

.000

 

Single

3.85

0.48

   

Married

4.10

0.50

   

Divorced

4.00

0.35

   

Education

  

0.82

.536

 

High school

3.92

0.48

   

Postsecondary

3.85

0.54

   

University degree

4.00

0.51

   

Masters

4.00

0.59

   

 

  1. Which factors were significant at a .05 level of significance? _______ 

According to the provided results, age and marital status were found to be significant factors at a .05 level of significance, with p-values less than .05. In contrast, sex and education were not significant, as their p-values were greater than .05. These findings suggest that age and marital status may play a crucial role in the outcome variable, while sex and education may not have a significant effect.

  1. State the number of levels for each factor. __The number of levels for sex =2, for age = 4, for marital status= 3 and education =04___

 

Problem Set 4.6: Tukey HSD Test in JASP

Criterion: Calculate post hoc analyses in JASP.

Data: Use stress.jasp data from Problem Set 4.2.

Instructions: Complete the steps below. (Note: The first 7 steps below are repeated from Problem Set 4.2). 

  1. Download stress.jasp. Double-click the icon to open the dataset in JASP. 
  2. In the Toolbar, click ANOVA. In the menu that appears, under Classical, select ANOVA.
  3. Select Fat grams consumed and then click the upper Arrow to send it over to the Dependent Variable box.
  4. Select Stress level and then click the lower Arrow to send it over to the Fixed factors box.
  5. Check the Descriptive statistics box. 
  6. Select Post-Hoc Tests. In the menu that appears, select Stress level and then click the Arrow to move it from the left to the right box.
  7. Check Standard and Tukey and uncheck any other boxes in the Post-Hoc area. 
  8. Copy and paste the output below.

Note: You will use these results for Problem Set 4.7.

ANOVA

ANOVA – Fat Grams Consumed 

Cases

Sum of Squares

df

Mean Square

F

p

Stress Level

 

15.600

 

2

 

7.800

 

1.773

 

0.212

 

Residuals

 

52.800

 

12

 

4.400

   

 

 
 

Note.  Type III Sum of Squares

 

Descriptives

Descriptives – Fat Grams Consumed 

Stress Level

N

Mean

SD

SE

Coefficient of variation

High

 

5

 

8.600

 

2.408

 

1.077

 

0.280

 

Low

 

5

 

6.200

 

1.924

 

0.860

 

0.310

 

Moderate

 

5

 

6.800

 

1.924

 

0.860

 

0.283

 
 

 

Post Hoc Tests

Standar

Post Hoc Comparisons – Stress Level 

 

95% CI for Mean Difference

 
  

Mean Difference

Lower

Upper

SE

t

ptukey 

High

 

Low

 

2.400

 

-1.139

 

5.939

 

1.327

 

1.809

 

0.208

 

 

 

Moderate

 

1.800

 

-1.739

 

5.339

 

1.327

 

1.357

 

0.393

 

Low

 

Moderate

 

-0.600

 

-4.139

 

2.939

 

1.327

 

-0.452

 

0.894

 
 

Note.  P-value and confidence intervals adjusted for comparing a family of 3 estimates (confidence intervals corrected using the tukey method).

 

Problem Set 4.7: Tukey HSD Interpretation

Criterion: Interpret Tukey HSD results from JASP output.

Data: Use your output from Problem Set 4.6.

Instructions: Identify where significant differences exist at the .05 level between the stress levels.

There are no significant differences at the 0.05 level of significance, as all the obtained p-values are less than or equal to the chosen significance level of 0.05. 

PSYC FPX4700 Assessment 4 ANOVA, Chi-Square Tests, and Regression

Chi-Square Tests

Problem Set 4.8: Critical Values

Criterion: Explain changes in critical value based on calculations.

Instructions: Read the following and answer the questions. 

The chi-square table. The degrees of freedom for a given test are listed in the column to the far left; the level of significance is listed in the top row to the right. These are the only two values you need to find the critical values for a chi-square test.

Work through the following exercise and write down what you see in the chi-square table. This will help familiarize you with the table. 

Increasing k and α in the chi-square table:
  1. Record the critical values for a chi-square test, given the following values for k at each level of significance:
 

.10

.05

.01

k = 10

14.68

16.919

23.209

k = 16

22.307

26.296

37.155

k = 22

29.593

33.92

48.814

k = 30

39.369

43.773

63.167

Note: Because there is only one k given, assume this is a goodness-of-fit test and compute the degrees of freedom as (k − 1).

  1. As the level of significance increases (from .01 to .10), does the critical value increase or decrease? Explain. _____________ 

As the level of significance increases from .01 to .10, the critical value decreases. This happens because a higher level of significance implies an increase in the probability of rejecting the null hypothesis when it is true, leading to a higher probability of making a Type I error. Thus, the critical value must be lowered to maintain the same Type I error rate. A lower critical value implies that rejecting the null hypothesis becomes easier, and the test becomes less conservative.

  1. As k increases (from 10 to 30), does the critical value increase or decrease? Explain your answer as it relates to the test statistic. _________ 

The critical value also increases as the number of categories or groups being compared (k) increases from 10 to 30. This is because an increase in k leads to an increase in degrees of freedom for the chi-square distribution, causing the distribution to become more dispersed. Consequently, the critical value becomes larger for a given level of significance. Thus, when k increases, a larger observed test statistic is required to reject the null hypothesis and indicate that there is a significant difference among the categories or groups being compared.

Problem Set 4.9: Parametric Tests

Criterion: Identify parametric tests. 

Instructions: Based on the scale of measurement for the data, identify if a test is parametric or nonparametric. 

  1. A researcher measures the proportion of schizophrenic patients born in each season. Nonparametric__
  2. A researcher measures the average age that schizophrenia is diagnosed among male and female patients. Parametric___
  3. A researcher tests whether frequency of Internet use and social interaction are independent. Nonparametric___
  4. A researcher measures the amount of time (in seconds) that a group of teenagers uses the Internet for school-related and non-school-related purposes. Parametric____

Problem Set 4.10: Chi-Square Analysis in JASP

Criterion: Use JASP for chi-square analysis. 

Data: Use the dataset yummy.jasp. Tandy’s ice cream shop serves chocolate, vanilla, and strawberry ice cream. Tandy wants to plan for the future years. She knows that on average she expects to purchase 100 cases of chocolate, 75 cases of vanilla, and 25 cases of strawberry (4:3:1). This year, she purchased 133 cases of chocolate, 82 cases of vanilla, and 33 cases of strawberry. The dataset yummy.jasp is a record of ice cream sales this year. 

Instructions: Complete the steps below.

  1. Download Yummy.jasp. Double-click the icon to open the dataset in JASP. 
  2. In the Toolbar, click Frequencies. In the menu that appears, under Classical, select Multinomial Test.
  3. Select Flavor and then click the upper Arrow to send it over to the Factor box.
  4. Select Frequency and then click the lower Arrow to send it over to the Counts box.
  5. Click the circle next to Expected Proportions (χ2). 
  6. Fill in 4 for Chocolate, 1 for strawberry, and 3 for Vanilla.
  7. Select the box for Descriptives and click the circle next to Proportions.
  8. Copy and paste the output to the Word document.
  9. Answer this: Was Tandy’s distribution of proportions the same as expected?

Based on the test results, the chi-square statistic is calculated to be 2.083 with 2 degrees of freedom, and the associated p-value is 0.353. Comparing the p-value with the chosen significance level, we find that the p-value is higher than the significance level, indicating insufficient evidence to reject the null hypothesis. The null hypothesis states that there is no significant difference between the actual distribution of ice cream flavors and the distribution expected based on Tandy’s projections. Therefore, we can conclude that there is no significant difference between the two distributions, and Tandy’s proportions are not significantly different from the expected distribution.

Multinomial Test

Multinomial Test 

 

χ²

df

p

H₀ (a)

 

2.083

 

2

 

0.353

 
 

Descriptives 

Flavor

Observed

Expected: H₀ (a)

Chocolate

 

0.536

 

0.500

 

Strawberry

 

0.133

 

0.125

 

Vanilla

 

0.331

 

0.375

 
 

 

Regression

Problem Set 4.11: Analysis of Regression in JASP

Criterion: Use JASP to complete an analysis of regression to determine if the variable age is a predictor of the variable life satisfaction. 

Data: Use the dataset satisfaction.jasp. This dataset is a record of responses to a survey which asked participants of various ages to rate their level of life satisfaction on a 1–10 scale, with 1 being “very dissatisfied” and 10 being “completely satisfied”:

Instruction: Complete steps below.

  1. Download satisfaction.jasp. Double-click the icon to open the dataset in JASP. 
  2. In the Toolbar, click Regression. In the menu that appears, under Classical, select Linear regression.
  3. Select Life Satisfaction and then click the upper Arrow to send it over to the Dependent box.
  4. Set Method to “Enter.” 
  5. Select Age and then click the Arrow to send it over to the Covariates box.
  6. Under Statistics, select Descriptives, Estimates, and Model Fit and deselect all other boxes. 
  7. Copy and paste the output to this Word document.

Linear Regression

Model Summary – Life Satisfaction 

Model

R

Adjusted R²

RMSE

H₀

 

0.000

 

0.000

 

0.000

 

1.491

 

H₁

 

0.479

 

0.230

 

0.133

 

1.388

 
 

 

ANOVA 

Model

 

Sum of Squares

df

Mean Square

F

p

H₁

 

Regression

 

4.592

 

1

 

4.592

 

2.384

 

0.161

 

 

 

Residual

 

15.408

 

8

 

1.926

   

 

 

 

 

Total

 

20.000

 

9

     

 

 
 

Note.  The intercept model is omitted, as no meaningful information can be shown.

 

Coefficients 

Model

 

Unstandardized

Standard Error

Standardized

t

p

H₀

 

(Intercept)

 

7.000

 

0.471

   

14.849

 

< .001

 

H₁

 

(Intercept)

 

4.667

 

1.574

   

2.966

 

0.018

 

 

 

Age

 

0.094

 

0.061

 

0.479

 

1.544

 

0.161

 
 

 

Problem Set 4.12: Analysis of Regression in Excel

Criterion: Use Excel to complete an analysis of regression.

Data: Use the data from this table

X (Age in Years)

Y (Life Satisfaction)

18

6

18

8

26

7

28

5

32

9

19

8

21

5

20

6

25

7

42

9

 

Instructions: Complete the following steps. 

  1. Open Excel and work in a new sheet.
  2. Enter the data from the table in Problem Set 4.11. Cell A1 will be X. Cell B1 will be Y. Then, enter the data below.
  3. Go to the tool bar, click Data Analysis, and select Regression.
  4. Put a check next to Labels and Confidence Level.
  5. In Input Y Range: $B$1:$B$11, In Input X Range: $A$1:$A$11
  6. Select Ok. Your data will appear in a new Sheet to the left.
  7. Copy and paste the output to this document.

PSYC FPX4700 Assessment 4 ANOVA, Chi-Square Tests, and Regression

Problem Set 4.13: Identify Tests for Ordinal Data 

Criterion: Identify tests for ordinal data. 

Instructions: Read the following and answer the questions.  

Identify the appropriate nonparametric test for each of the following examples and explain why a nonparametric test is appropriate.  

  1. A researcher measures fear as the time it takes to walk across a presumably scary portion of campus. The times (in seconds) that it took a sample of 12 participants were 8, 12, 15, 13, 12, 10, 6, 10, 9, 15, 50, and 52. _____ 

The Wilcoxon signed-rank test is a suitable nonparametric test for this example due to the small sample size and the fact that the data is measured on an ordinal scale. In addition, the assumption of normality is not satisfied. 

  1. Two groups of participants were given 5 minutes to complete a puzzle. The participants were told that the puzzle would be easy. In truth, in one group, the puzzle had a solution (Group Solution), and in the second group, the puzzle had no solution (Group No Solution). The researchers measured stress levels and found that frustration levels were low for all participants in Group Solution and for all but a few participants who showed strikingly high levels of stress in Group No Solution. ________ 

The Mann-Whitney U test is a nonparametric test used to compare two independent groups when the assumption of normality is not met. Since the data in this example is measured on an ordinal scale and the sample size is small, the Mann-Whitney U test is a suitable alternative to the t-test.

  1. A researcher measured student scores on an identical assignment to see how well students perform for different types of professors. In Group Adviser, their professor was also their adviser; in Group Major, their professor taught in their major field of study; in Group Nonmajor, their professor did not teach in their major field of study. Student scores were ranked in each class, and the differences in ranks were compared. __________

The Kruskal-Wallis test is the suitable nonparametric test for this scenario. This is because the data is measured on an ordinal scale, the groups are independent, and the normality assumption is not fulfilled.

  1. A researcher has the same participants rank two types of advertisements for the same product. Differences in ranks for each advertisement were compared. _____

Based on the characteristics of the data, the most suitable nonparametric test for this example is the Wilcoxon rank-sum test. The data is measured on an ordinal scale, and the two samples being compared are independent. Additionally, the assumption of normality is not met.

  1. A professor measures student motivation before, during, and after a statistics course in a given semester. Student motivation was ranked at each time in the semester, and the differences in ranks were compared. ________ 

The Friedman test is the appropriate nonparametric test for this example. The reason for this is that the data is measured on an ordinal scale, the participants are measured three times (before, during, and after the course), and the assumption of normality is not met.

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