ANOVA, ChiSquare 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 Ftable. This will help familiarize you with the table.
The Ftable: The degrees of freedom for the numerator (k − 1) are across the columns; the degrees of freedom for the denominator (N − k) 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, ChiSquare Tests, and Regression
As the number of groups (k) increases from 1 to 8, the critical Fvalue shows a decreasing trend. This implies that the Fstatistic 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 Fvalue 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: Oneway 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 buffetstyle lunch among professional bodybuilders under conditions of high, moderate, and low stress.
Instructions: Complete the steps below.
 Download stress.jasp. Doubleclick the icon to open the dataset in JASP.
 In the Toolbar, click ANOVA. In the menu that appears, under Classical, select ANOVA.
 Select Fat grams consumed and then click the upper Arrow to send it over to the Dependent Variable box.
 Select Stress level and then click the lower Arrow to send it over to the Fixed factors box.
 Check the Descriptive statistics box.
 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: Oneway ANOVA in Excel
Criterion: Calculate an ANOVA in Excel.
Instructions: Use the data from the table below to complete the following steps:
 Open Excel to an empty sheet.
 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 
 In Row 1, enter High in cell A1, Moderate in cell B1, and Low in cell B1.
 In the toolbar, click Data Analysis, select Anova: Single Factor, and click OK.
 In Input Range: $A$1:$C$6, put a check next to Labels in First Row, click OK.
 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  Pvalue  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: Oneway 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:
 State the null hypothesis. _______ No relationship between stress and fat consumption exists.
 Report your results in APA format (as you might see them reported in a journal article). _
A oneway ANOVA was performed with high, medium, and lowstress 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 lowstress groups reported lower fat consumption levels than those in the highstress group. Interestingly, there was no significant difference in fat consumption between the medium and lowstress groups, suggesting that stress may not be a crucial factor in determining fat consumption levels._______
PSYC FPX4700 Assessment 4 ANOVA, ChiSquare 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 
 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 pvalues less than .05. In contrast, sex and education were not significant, as their pvalues 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.
 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).
 Download stress.jasp. Doubleclick the icon to open the dataset in JASP.
 In the Toolbar, click ANOVA. In the menu that appears, under Classical, select ANOVA.
 Select Fat grams consumed and then click the upper Arrow to send it over to the Dependent Variable box.
 Select Stress level and then click the lower Arrow to send it over to the Fixed factors box.
 Check the Descriptive statistics box.
 Select PostHoc Tests. In the menu that appears, select Stress level and then click the Arrow to move it from the left to the right box.
 Check Standard and Tukey and uncheck any other boxes in the PostHoc area.
 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. Pvalue 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 pvalues are less than or equal to the chosen significance level of 0.05.
PSYC FPX4700 Assessment 4 ANOVA, ChiSquare Tests, and Regression
ChiSquare 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 chisquare 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 chisquare test.
Work through the following exercise and write down what you see in the chisquare table. This will help familiarize you with the table.
Increasing k and α in the chisquare table:
 Record the critical values for a chisquare 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 goodnessoffit test and compute the degrees of freedom as (k − 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.
 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 chisquare 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.
 A researcher measures the proportion of schizophrenic patients born in each season. Nonparametric__
 A researcher measures the average age that schizophrenia is diagnosed among male and female patients. Parametric___
 A researcher tests whether frequency of Internet use and social interaction are independent. Nonparametric___
 A researcher measures the amount of time (in seconds) that a group of teenagers uses the Internet for schoolrelated and nonschoolrelated purposes. Parametric____
Problem Set 4.10: ChiSquare Analysis in JASP
Criterion: Use JASP for chisquare 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.
 Download Yummy.jasp. Doubleclick the icon to open the dataset in JASP.
 In the Toolbar, click Frequencies. In the menu that appears, under Classical, select Multinomial Test.
 Select Flavor and then click the upper Arrow to send it over to the Factor box.
 Select Frequency and then click the lower Arrow to send it over to the Counts box.
 Click the circle next to Expected Proportions (χ2).
 Fill in 4 for Chocolate, 1 for strawberry, and 3 for Vanilla.
 Select the box for Descriptives and click the circle next to Proportions.
 Copy and paste the output to the Word document.
 Answer this: Was Tandy’s distribution of proportions the same as expected?
Based on the test results, the chisquare statistic is calculated to be 2.083 with 2 degrees of freedom, and the associated pvalue is 0.353. Comparing the pvalue with the chosen significance level, we find that the pvalue 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.
 Download satisfaction.jasp. Doubleclick the icon to open the dataset in JASP.
 In the Toolbar, click Regression. In the menu that appears, under Classical, select Linear regression.
 Select Life Satisfaction and then click the upper Arrow to send it over to the Dependent box.
 Set Method to “Enter.”
 Select Age and then click the Arrow to send it over to the Covariates box.
 Under Statistics, select Descriptives, Estimates, and Model Fit and deselect all other boxes.
 Copy and paste the output to this Word document.
Linear Regression
Model Summary – Life Satisfaction  

Model  R  R²  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.
 Open Excel and work in a new sheet.
 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.
 Go to the tool bar, click Data Analysis, and select Regression.
 Put a check next to Labels and Confidence Level.
 In Input Y Range: $B$1:$B$11, In Input X Range: $A$1:$A$11
 Select Ok. Your data will appear in a new Sheet to the left.
 Copy and paste the output to this document.
PSYC FPX4700 Assessment 4 ANOVA, ChiSquare 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.
 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 signedrank 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.
 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 MannWhitney 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 MannWhitney U test is a suitable alternative to the ttest.
 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 KruskalWallis 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.
 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 ranksum 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.
 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.