PSYC FPX 4700 Assessment 4 Anova Chi Square Tests and Regression

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Capella University

PSYC FPX 4700 Statistics for the Behavioral Sciences

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Date

PSYC FPX 4700 Assessment 4 Anova Chi Square Tests and Regression

PSYC FPX 4700 Assessment 4 focuses on three important statistical methods: ANOVA, chi-square tests, and regression analysis. ANOVA is used to compare means across groups, chi-square tests analyze categorical frequency data and relationships between categorical variables, and regression examines whether one variable can predict another. In this assessment, students apply these methods using statistical tables, JASP, and Microsoft Excel and then interpret and report the findings using appropriate statistical and APA conventions.

The assessment requires students to complete all problems in the provided Word document. For calculations, statistical procedures, and interpretations, the work should be clearly shown so that each final answer can be identified easily. Answers may be distinguished from calculations by using highlighting, bold formatting, or a different font color.

ANOVA in PSYC FPX 4700 Assessment 4

Analysis of variance, commonly called ANOVA, is a statistical method used to determine whether the means of multiple groups differ significantly. A one-way ANOVA is appropriate when a study includes one categorical independent variable with two or more groups or levels and one continuous dependent variable.

ANOVA does not simply determine whether sample means are numerically different. Instead, it evaluates whether the observed differences are large enough, relative to the variability within the groups, to provide evidence of a statistically significant difference in the population.

Problem Set 4.1: ANOVA Critical Values

The first part of the assessment examines how the number of levels of an independent variable affects the critical value of the F distribution and statistical power.

For a one-way ANOVA, the numerator degrees of freedom are calculated as:

df₁ = k − 1

The denominator degrees of freedom are calculated as:

df₂ = N − k

where k represents the number of levels of the independent variable and N represents the total sample size.

In an F-distribution table, the numerator degrees of freedom and denominator degrees of freedom are used to identify the appropriate critical F value. Separate tables are commonly provided for significance levels such as α = .05 and α = .01.

Assume that the sample contains 24 participants (N = 24). The appropriate critical F values should be identified for each value of k.

Number of Levels (k)	Critical F at α = .05	Critical F at α = .01
k = 2	Determine from F table	Determine from F table
k = 4	Determine from F table	Determine from F table
k = 6	Determine from F table	Determine from F table
k = 8	Determine from F table	Determine from F table

As the number of independent-variable levels changes, both degrees of freedom and the corresponding critical F value change. Students should use the specified F-distribution table rather than estimating the values.

Increasing the number of groups can influence statistical power because power depends on several factors, including sample size, effect size, within-group variability, significance level, and the structure of the groups. A larger number of levels does not automatically guarantee greater statistical power. The critical value should therefore be interpreted together with the calculated F statistic and other characteristics of the study.

Problem Set 4.2: One-Way ANOVA Using JASP

This problem requires students to conduct a one-way ANOVA in JASP using the stress.jasp dataset.

The dataset contains information about the amount of fat, measured in grams, consumed during a buffet-style lunch by professional bodybuilders experiencing high, moderate, or low levels of stress.

To complete the analysis, open the stress.jasp dataset in JASP and select ANOVA from the toolbar. Under the Classical ANOVA procedure, place Fat grams consumed in the Dependent Variable field and Stress level in the Fixed Factors field.

Select Descriptive statistics so that JASP produces descriptive information for the groups. The resulting output should then be copied and pasted into the Word document.

The purpose of this analysis is to determine whether mean fat consumption differs across the stress-level groups.

Problem Set 4.3: One-Way ANOVA Using Microsoft Excel

The next problem requires a one-way ANOVA using Microsoft Excel.

Use the following data:

High Stress	Moderate Stress	Low Stress
10	9	9
7	4	4
8	7	6
12	6	5
6	8	7

Enter the data into a new Excel worksheet. Place High in cell A1, Moderate in cell B1, and Low in cell C1.

From the Excel toolbar, select Data Analysis and choose ANOVA: Single Factor. For the input range, use:

$A$1:$C$6

Select Labels in First Row and run the analysis. Excel will generate the ANOVA table in a new worksheet.

The resulting output should be copied into the Word document and clearly identified as the ANOVA analysis.

Problem Set 4.4: Reporting One-Way ANOVA Results in APA Style

ANOVA results should be reported in a concise format that identifies the statistical test, degrees of freedom, F statistic, p value, and statistical conclusion.

The null hypothesis for a one-way ANOVA is that the population means are equal across all groups. In general form:

H₀: μ₁ = μ₂ = μ₃ = …

The alternative hypothesis states that at least one population mean differs from another.

An APA-style ANOVA statement should include the test statistic, degrees of freedom, and p value. For example, the structure can be written as:

A one-way ANOVA was conducted to examine whether mean fat consumption differed by stress level. The analysis produced a statistically significant/non-significant effect of stress level, F(df₁, df₂) = X.XX, p = .XXX.

The actual values should be taken directly from the Excel or JASP output rather than estimated.

A p value below the selected alpha level, commonly .05, indicates that the null hypothesis is rejected. A p value greater than .05 generally indicates insufficient evidence to reject the null hypothesis.

Problem Set 4.5: Interpreting ANOVA Results

Drakou et al. (2006) examined life satisfaction among sport coaches and investigated whether life satisfaction differed according to variables such as sex, age, marital status, and education.

Independent Variable	Group	M	SD	F	p
Sex	Men	3.99	0.51	0.68	.409
	Women	3.94	0.49
Age	20s	3.85	0.42	3.04	.029
	30s	4.03	0.52
	40s	3.97	0.57
	50s	4.02	0.50
Marital Status	Single	3.85	0.48	12.46	< .001
	Married	4.10	0.50
	Divorced	4.00	0.35
Education	High school	3.92	0.48	0.82	.536
	Postsecondary	3.85	0.54
	University degree	4.00	0.51
	Master’s	4.00	0.59

Using α = .05, a result is considered statistically significant when the p value is less than .05.

The results indicate that age and marital status produced statistically significant findings. Age had an F statistic of 3.04 with p = .029, while marital status had an F statistic of 12.46 with p < .001.

Sex was not statistically significant because p = .409, which is greater than .05. Education was also not statistically significant because p = .536.

The independent variables contain the following numbers of levels:

• Sex: 2 levels

• Age: 4 levels

• Marital status: 3 levels

• Education: 4 levels

A significant omnibus ANOVA indicates that differences exist somewhere among the group means, but it does not by itself identify exactly which groups differ. When an independent variable has more than two levels, a post hoc analysis may be needed to identify specific group differences.

Problem Set 4.6: Tukey HSD Post Hoc Test in JASP

When an ANOVA produces a statistically significant result involving multiple groups, a post hoc procedure can help determine which specific pairs of groups differ.

Tukey’s Honestly Significant Difference, or Tukey HSD, is commonly used for pairwise comparisons following ANOVA.

Using the stress.jasp dataset, repeat the initial ANOVA procedure. Place Fat grams consumed in the Dependent Variable field and Stress level in the Fixed Factors field.

Select Descriptive statistics and then select Post-Hoc Tests. Move Stress level into the post hoc analysis field and select the Standard and Tukey options.

Remove unnecessary post hoc options and review the JASP output. Copy and paste the resulting output into the Word document.

The post hoc output will be used in the next problem to determine which stress-level groups differ significantly.

Problem Set 4.7: Interpreting the Tukey HSD Test

The Tukey HSD results should be interpreted at the specified .05 significance level.

Each pairwise comparison should be examined to determine whether the p value is below .05. A statistically significant comparison indicates that the corresponding two stress-level groups have significantly different mean fat-consumption scores.

The interpretation should identify the specific pair or pairs of groups that are significantly different and then explain the result in practical terms. For example, if the high-stress and low-stress groups show a significant difference, the conclusion should state that mean fat consumption differs significantly between those two stress groups.

The exact significant comparisons should be taken directly from the JASP Tukey output.

Chi-Square Tests in PSYC FPX 4700

Chi-square procedures are designed primarily for categorical frequency data. Unlike ANOVA, which compares means of continuous outcomes, chi-square tests compare observed and expected frequencies or examine whether categorical variables are associated.

Two commonly used chi-square procedures are the goodness-of-fit test and the test of independence.

A chi-square goodness-of-fit test determines whether observed frequencies differ from expected frequencies based on a specified distribution.

A chi-square test of independence evaluates whether two categorical variables are statistically associated.

Problem Set 4.8: Chi-Square Critical Values

The chi-square distribution uses degrees of freedom and the selected significance level to determine the appropriate critical value.

For a chi-square goodness-of-fit test, degrees of freedom are calculated as:

df = k − 1

where k represents the number of categories.

The relevant critical values should be obtained from the appropriate chi-square distribution table.

k	α = .10	α = .05	α = .01
10	Determine from χ² table	Determine from χ² table	Determine from χ² table
16	Determine from χ² table	Determine from χ² table	Determine from χ² table
22	Determine from χ² table	Determine from χ² table	Determine from χ² table
30	Determine from χ² table	Determine from χ² table	Determine from χ² table

As the significance level increases from .01 to .10, the critical chi-square value generally decreases. A smaller significance level requires stronger evidence before the null hypothesis is rejected, resulting in a more extreme critical value.

As the number of categories increases, degrees of freedom also increase. With greater degrees of freedom, the critical value of the chi-square distribution generally increases for a given significance level.

Therefore, the chi-square critical value should always be interpreted in relation to both the degrees of freedom and the selected alpha level.

Problem Set 4.9: Identifying Parametric and Nonparametric Tests

Choosing an appropriate statistical test depends on the research question, variables, measurement scale, distribution of the data, and study design.

A researcher who measures the proportion of patients with schizophrenia born during each season is working with categorical frequency data. A chi-square goodness-of-fit approach would generally be appropriate.

A researcher comparing the average age at which schizophrenia is diagnosed among male and female patients is examining a continuous variable across two groups. A parametric comparison, such as an independent-samples t test, may be appropriate when relevant assumptions are satisfied.

A researcher examining whether frequency of Internet use and social interaction are statistically independent is working with categorical variables. A chi-square test of independence would generally be appropriate.

A researcher measuring the amount of time teenagers spend using the Internet is working with a continuous ratio-level variable. A parametric procedure may be appropriate if the relevant assumptions are satisfied.

The choice between parametric and nonparametric methods should therefore be based on the characteristics of the data rather than simply on the topic being studied.

Problem Set 4.10: Chi-Square Analysis Using JASP

This problem requires a chi-square goodness-of-fit analysis using the yummy.jasp dataset.

Tandy’s ice cream shop sells chocolate, vanilla, and strawberry flavors. Historical expectations indicate that the shop expects the flavors to occur in a ratio of 4:3:1.

The expected purchases are:

• Chocolate: 100 cases

• Vanilla: 75 cases

• Strawberry: 25 cases

During the current year, the observed purchases were:

• Chocolate: 133 cases

• Vanilla: 82 cases

• Strawberry: 33 cases

Open yummy.jasp in JASP and select Frequencies from the toolbar. Under Classical, choose Multinomial Test.

Move Flavor to the Factor field and Frequency to the Counts field. Select Expected Proportions (χ²) and enter the expected proportions based on the 4:3:1 ratio:

Flavor	Expected Proportion
Chocolate	4
Vanilla	3
Strawberry	1

Select Descriptives and Proportions, then copy the JASP output into the Word document.

The final interpretation should determine whether the observed distribution is consistent with the expected distribution. This conclusion should be based on the chi-square test statistic, degrees of freedom, p value, and selected significance level.

Problem Set 4.11: Linear Regression Analysis Using JASP

Regression analysis examines whether one or more predictor variables can statistically explain or predict variation in an outcome variable.

In simple linear regression, one predictor is used to estimate a continuous dependent variable. The general form of a simple regression equation is:

Y = b₀ + b₁X + ε

where Y is the outcome, X is the predictor, b₀ is the intercept, b₁ is the regression coefficient, and ε represents error.

For this problem, use the satisfaction.jasp dataset to determine whether age predicts life satisfaction.

The dataset contains participants’ ages and life-satisfaction ratings ranging from 1 to 10.

Open the dataset in JASP and select Regression from the toolbar. Under Classical, select Linear Regression.

Place Life Satisfaction in the Dependent Variable field. Set the analysis method to Enter and place Age in the Covariates field.

Under Statistics, select:

• Descriptives

• Estimates

• Model Fit

Deselect unnecessary statistical options and copy the relevant JASP output into the Word document.

The interpretation should consider the regression coefficient, p value, model fit information, and direction of the relationship. A statistically significant predictor indicates evidence that age is associated with variation in life satisfaction within the analyzed sample.

Problem Set 4.12: Linear Regression Analysis Using Excel

The following data can be used to examine the relationship between age and life satisfaction.

Age in Years (X)	Life Satisfaction (Y)
18	6
18	8
26	7
28	5
32	9
19	8
21	5
20	6
25	7
42	9

Enter X in cell A1 and Y in cell B1, followed by the corresponding observations.

Select Data Analysis and choose Regression. Select Labels and Confidence Level.

If the worksheet contains the headers and ten observations, the data ranges should correspond to the actual worksheet dimensions. For example:

Input Y Range: $B$1:$B$11

Input X Range: $A$1:$A$11

Select OK to generate the regression output.

The resulting analysis should be interpreted in terms of the direction and strength of the relationship between age and life satisfaction. The regression coefficient indicates the direction of the association, while the p value helps determine whether the predictor makes a statistically significant contribution to the model.

The R² value is also important because it indicates the proportion of variation in the outcome that is explained by the predictor within the regression model.

Problem Set 4.13: Selecting Nonparametric Tests for Ordinal or Non-Normal Data

Nonparametric tests are useful when data are ordinal, ranked, substantially non-normal, or do not meet the assumptions required for a particular parametric procedure.

Scenario 1

A researcher measures fear using the amount of time, in seconds, participants take to walk across a frightening portion of campus. The recorded times for 12 participants are:

8, 12, 15, 13, 12, 10, 6, 10, 9, 15, 50, and 52 seconds.

The presence of unusually large values, particularly 50 and 52 seconds, suggests potential skewness and outliers. A nonparametric one-sample procedure such as the Wilcoxon signed-rank test may be appropriate when the research question involves comparing the sample to a specified median and the assumptions for a parametric alternative are not satisfied.

Scenario 2

Two independent groups receive different types of puzzles. One group receives a puzzle with a solution, while the other receives a puzzle with no solution. Stress scores are highly uneven, with several unusually high observations in one group.

A Mann-Whitney U test, also called the Wilcoxon rank-sum test, may be appropriate for comparing two independent groups when the outcome is ordinal or substantially non-normal and the assumptions of an independent-samples t test are not appropriate.

Scenario 3

Students are assigned to three independent professor conditions: Adviser, Major, and Nonmajor. Student assignment scores are ranked within each class.

A Kruskal-Wallis H test is appropriate when comparing three or more independent groups using ordinal or ranked data. It serves as a nonparametric alternative to a one-way ANOVA.

Scenario 4

The same participants rank two advertisements for the same product, and researchers compare the rankings.

The Wilcoxon signed-rank test is appropriate for comparing two related or paired measurements when the data are ordinal or when a paired parametric test is not appropriate.

Scenario 5

A professor measures student motivation before, during, and after a statistics course using the same participants. Motivation is ranked at each of the three time points.

The Friedman test is appropriate because the study involves three or more related measurements from the same participants and uses ranked data. It is the nonparametric counterpart to a repeated-measures ANOVA in this type of design.

Frequently Asked Questions About PSYC FPX 4700 Assessment 4

What is ANOVA used for?

ANOVA is used to determine whether statistically significant differences exist among group means. A one-way ANOVA is commonly used when there is one categorical independent variable with two or more levels and one continuous dependent variable.

What does the F statistic represent in ANOVA?

The F statistic compares variability between groups with variability within groups. A larger F statistic can provide stronger evidence that group means differ, but the F value must be interpreted using its degrees of freedom and corresponding p value.

What is the difference between ANOVA and a chi-square test?

ANOVA generally analyzes differences between means of a continuous outcome across groups. Chi-square tests are primarily used with categorical frequency data. A goodness-of-fit test compares observed and expected frequencies, whereas a chi-square test of independence examines whether two categorical variables are associated.

What is a Tukey HSD test?

Tukey’s Honestly Significant Difference test is a post hoc procedure used to conduct pairwise comparisons after ANOVA. It helps identify which specific group means differ when an overall ANOVA is statistically significant.

What is a chi-square goodness-of-fit test?

A chi-square goodness-of-fit test evaluates whether observed categorical frequencies differ significantly from expected frequencies or proportions.

What is linear regression used for?

Linear regression examines the relationship between a predictor variable and a continuous outcome. It can determine whether a predictor is statistically associated with an outcome and quantify the direction and amount of explained variation.

What is the difference between parametric and nonparametric tests?

Parametric tests generally rely on assumptions concerning population distributions and are commonly used with interval- or ratio-level data when those assumptions are reasonably satisfied. Nonparametric tests are useful for ordinal or ranked data and for situations where important assumptions of a corresponding parametric test are not met.

Why are JASP and Excel used in PSYC FPX 4700 Assessment 4?

JASP provides a user-friendly environment for conducting statistical procedures such as ANOVA, chi-square analysis, post hoc tests, and regression. Microsoft Excel also provides statistical analysis tools that can be used for procedures including one-way ANOVA and linear regression.

What is the null hypothesis in ANOVA?

The null hypothesis states that the population means are equal across the groups being compared. A statistically significant ANOVA provides evidence against this hypothesis, indicating that at least one group mean differs.

When should a Tukey post hoc test be used?

Tukey HSD is generally used after a statistically significant ANOVA when researchers need to determine which specific pairs of groups differ. It helps control the familywise error rate during multiple pairwise comparisons.

How do you interpret a p value in PSYC FPX 4700 Assessment 4?

The p value indicates how compatible the observed result is with the null hypothesis under the statistical model. When the p value is below the predetermined alpha level, such as .05, the result is typically described as statistically significant and the null hypothesis is rejected.

Which nonparametric test compares three independent groups?

The Kruskal-Wallis H test is commonly used to compare three or more independent groups when the outcome is ordinal, ranked, or otherwise unsuitable for a one-way ANOVA.

Which nonparametric test compares three related measurements?

The Friedman test is commonly used when the same participants are measured at three or more related time points or conditions using ordinal or ranked data.

Key Statistical Concepts to Remember

PSYC FPX 4700 Assessment 4 becomes easier to approach when the purpose of each statistical procedure is clearly understood. ANOVA focuses on differences among means, chi-square procedures focus on categorical frequencies, and regression focuses on relationships and prediction.

For quick reference:

Statistical Method	Primary Purpose	Typical Data/Design
One-way ANOVA	Compare means across groups	One categorical predictor, continuous outcome
Tukey HSD	Identify specific group differences after ANOVA	Multiple group comparisons
Chi-square goodness-of-fit	Compare observed and expected frequencies	One categorical variable
Chi-square independence	Examine association between categorical variables	Two categorical variables
Simple linear regression	Examine prediction/relationship	Continuous predictor and outcome
Mann-Whitney U	Compare two independent groups	Ordinal/non-normal data
Wilcoxon signed-rank	Compare two related measurements	Paired/related ordinal data
Kruskal-Wallis	Compare three or more independent groups	Ordinal/non-normal data
Friedman test	Compare three or more related measurements	Repeated ordinal/ranked data

Conclusion

PSYC FPX 4700 Assessment 4 provides practice with several statistical techniques commonly used in psychological and behavioral research. Students apply one-way ANOVA, Tukey HSD post hoc testing, chi-square analysis, and linear regression while also learning how to distinguish parametric from nonparametric methods.

The most important part of the assessment is not simply producing statistical output. Students should be able to select an appropriate test, identify the relevant assumptions and variables, interpret test statistics and p values, and communicate findings clearly in APA style. JASP and Excel can simplify the computational process, but the resulting output still needs to be interpreted in the context of the research question.

References

Drakou, A., Tzetzis, G., & Mamassis, G. (2006). Leisure constraints experienced by university students in Greece. Sport Management International Journal, 2(1), 1–13.

JASP Team. (n.d.). JASP: A fresh way to do statistics. https://jasp-stats.org/

National Institute of Standards and Technology. (n.d.). NIST/SEMATECH e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook/

PSYC FPX 4700 Assessment 4 Anova Chi Square Tests and Regression

OpenStax. (2023). Introductory statistics 2e. Rice University. https://openstax.org/details/books/introductory-statistics-2e

Penn State Eberly College of Science. (n.d.). STAT 200: Elementary statistics. https://online.stat.psu.edu/stat200/

The jamovi project. (n.d.). jamovi. https://www.jamovi.org/