PSYC FPX 4700 Assessment 3 Hypothesis Effect Size Power and Tests
PSYC FPX 4700 Assessment 3 Hypothesis Effect Size Power and Tests
Name
Capella University
PSYC FPX 4700 Statistics for the Behavioral Sciences
Prof. Name
Date
PSYC FPX 4700 Assessment 3 Hypothesis Effect Size Power and Tests
Hypothesis testing, effect size, statistical power, confidence intervals, and t tests are essential tools for analyzing psychological and behavioral research data. Together, these statistical methods help researchers determine whether observed differences are likely to reflect meaningful population effects rather than random sampling variation. In this assessment, you will apply these concepts to sampling distributions, directional and nondirectional hypotheses, p values, confidence intervals, one-sample t tests, and independent-samples t tests using JASP and Microsoft Excel.
Complete all problems in the designated Word document and do not submit additional files unless specifically instructed. For calculations, show your work clearly and make each final response easy to identify. You may use bold text, highlighting, or another formatting method to distinguish final answers from calculations.
Problem Set 3.1: Sampling Distribution of the Mean
Interpreting the Population Mean and Variance
A sampling distribution describes how a statistic, such as a sample mean, varies across repeated samples drawn from the same population. Understanding the population mean and variance provides an important foundation for interpreting sample statistics and determining how much variability can be expected from one sample to another.
Suppose a researcher wants to study the average attention span of individuals in a hypothetical population. Attention span is measured in minutes and represents the amount of time a person remains focused on a particular task. The population follows a normal distribution, with the population parameters provided in the original problem.
Use the information given in the problem to answer the following questions.
1. What is the population mean (μ)?
Answer: __________________________
The population mean, represented by μ, is the average value of the variable across the entire population.
2. What is the population variance (σ²)?
Answer: __________________________
Population variance, represented by σ², describes how widely individual scores are distributed around the population mean.
3. Sketch the population distribution
Draw a normal distribution curve and label the population mean and the values located one, two, and three standard deviations above and below the mean.
Your diagram should identify:
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μ − 3σ
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μ − 2σ
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μ − 1σ
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μ
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μ + 1σ
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μ + 2σ
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μ + 3σ
A normally distributed population should appear symmetrical around μ. The standard deviation values show the expected distance of observations from the population mean.
Problem Set 3.2: Effect Size and Statistical Power
Understanding Effect Size and Power
Effect size and statistical power provide different but related types of information. Effect size describes the magnitude of a difference or relationship, whereas statistical power describes the probability that a statistical test will detect a genuine effect when one exists.
Power is generally affected by several factors. Larger effect sizes and larger sample sizes tend to increase power, while greater variability can make it more difficult to detect an effect.
Scenario 1: Effect Size
Two researchers examine the effectiveness of a drug-use treatment. Researcher A obtains an effect size of d = 0.36 among males, while Researcher B obtains an effect size of d = 0.20 among females.
Which researcher is expected to have greater statistical power when all other factors are equal? Explain your answer.
Answer: __________________________
When sample size, variability, significance level, and other relevant factors are held constant, Researcher A is expected to have greater statistical power because the effect size of 0.36 is larger than 0.20. Larger effects are generally easier for statistical tests to detect.
Scenario 2: Sample Size
Two researchers investigate marital satisfaction among military families. Researcher A studies 22 married couples (n = 22), while Researcher B studies 40 married couples (n = 40).
Which researcher is expected to have greater statistical power? Explain your answer.
Answer: __________________________
Researcher B is expected to have greater statistical power because the larger sample provides more information and generally produces a more precise estimate of the population effect. When other factors remain equal, increasing sample size generally increases statistical power.
Scenario 3: Population Standard Deviation
Two researchers examine standardized examination performance among senior high school students from two local communities. Researcher A studies a population with a standard deviation of σ = 110, while Researcher B studies a population with a standard deviation of σ = 60.
Which researcher is expected to have greater statistical power? Explain your answer.
Answer: __________________________
Researcher B is generally expected to have greater statistical power because the smaller standard deviation indicates less variability. When other factors are equal, lower variability makes it easier for a statistical test to distinguish a genuine effect from random variation.
Problem Set 3.3: Directional and Nondirectional Hypothesis Testing
Understanding Directional and Nondirectional Hypotheses
Hypothesis testing involves evaluating competing statements about a population. The null hypothesis generally represents the absence of the predicted effect or difference, while the alternative hypothesis represents the research expectation.
A directional hypothesis predicts not only that a difference exists but also the direction of that difference. A nondirectional hypothesis predicts that a difference exists without specifying which group will have the higher or lower value.
Cho and Abe (2013) examined issues surrounding one-tailed and two-tailed tests in research involving directional hypotheses.
Consider the following hypotheses for a study examining whether males self-disclose more than females:
H₀: μmales − μfemales ≤ 0
H₁: μmales − μfemales > 0
Is this a directional or nondirectional test?
Answer: __________________________
These hypotheses represent a directional test because the alternative hypothesis specifically predicts that the mean self-disclosure score for males will be greater than the mean for females.
Do these hypotheses account for all possible values of the population mean difference?
Answer: __________________________
Yes. The hypotheses divide the possible values of the population mean difference into two regions. The null hypothesis covers values at or below zero, while the alternative hypothesis covers values greater than zero.
This division reflects the directional nature of the research question. The researcher is specifically testing whether the male population mean exceeds the female population mean rather than simply asking whether the two means are different.
Problem Set 3.4: Understanding p Values and Statistical Decisions
How p Values Guide Statistical Decisions
A p value indicates how compatible the observed sample results are with the null hypothesis under the statistical model being used. Researchers compare the p value with a predetermined significance level, commonly α = .05, to make a statistical decision.
Lambdin (2012) emphasized the importance of applying significance criteria consistently rather than treating results just above the chosen threshold as if they were statistically significant.
At α = .05, researchers generally make one of two decisions:
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Reject the null hypothesis when the p value is less than .05.
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Fail to reject the null hypothesis when the p value is greater than or equal to .05.
For example, a result of p = .03 meets the conventional α = .05 criterion, whereas p = .067 does not.
What does this mean for the statement about p = .067?
Answer: __________________________
If the predetermined significance level is α = .05, a p value of .067 does not meet the criterion for statistical significance. Therefore, the researcher would fail to reject the null hypothesis. Describing the result as “marginally significant” simply because it is close to .05 can create an inconsistent interpretation of the predetermined statistical criterion.
Failing to reject the null hypothesis does not prove that the null hypothesis is true. It means that the sample does not provide sufficient statistical evidence to reject it at the selected significance level.
Problem Set 3.5: One-Sample t Test in JASP
What Is a One-Sample t Test?
A one-sample t test evaluates whether the mean of a sample differs significantly from a specified population or hypothesized mean. It is particularly useful when researchers have one sample and want to compare its average score with a known or theoretical reference value.
For this problem, use the minutesreading.jasp dataset. The dataset contains the weekly reading times, measured in minutes, of Riverbend City online news readers.
Riverbend City Online News claims that its readers spend more time reading its content than people spend reading national news. The population mean for national-news reading time is 8 minutes per week.
JASP procedure
Open minutesreading.jasp in JASP and follow these steps:
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Select T-Tests from the toolbar.
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Under Classical, select One-Sample T-Test.
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Select Time and move it into the Variables box.
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Make sure Student is selected.
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Enter 8 in the Test Value field.
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Run the analysis.
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Review the JASP output.
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Copy and paste the relevant JASP output into the designated Word document.
1. State the nondirectional hypothesis
Answer: __________________________
For a nondirectional test, the hypotheses should evaluate whether the population mean differs from 8 minutes in either direction.
2. What is the critical t value for α = .05 using a two-tailed test?
Answer: __________________________
The exact critical t value depends on the sample’s degrees of freedom. Determine the degrees of freedom from the JASP output and use the appropriate two-tailed critical value.
3. Is the average reading time significantly different from 8 minutes?
Answer: __________________________
Use the p value from the JASP output to make the decision. Compare the p value with α = .05. If the p value is below .05, reject the null hypothesis. If the p value is .05 or greater, fail to reject the null hypothesis.
Explain the statistical decision in relation to the research question and the population mean of 8 minutes.
Important: Keep the minutesreading.jasp dataset available because it is also required for the confidence interval analysis in Problem Set 3.6.
Problem Set 3.6: Calculating a 95% Confidence Interval
What Is a 95% Confidence Interval?
A confidence interval provides an estimated range of plausible values for a population parameter based on sample data. A 95% confidence interval communicates the uncertainty associated with estimating a population mean from a sample.
Continue using the minutesreading.jasp dataset from Problem Set 3.5.
JASP procedure for the confidence interval
In JASP:
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Select Location Estimate.
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Select Confidence Interval.
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Enter 95.0% as the confidence level.
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Review the resulting output.
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Copy the relevant output into the Word document.
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Report the lower and upper limits of the confidence interval.
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Interpret the interval in relation to the population mean of 8 minutes.
How should the confidence interval be interpreted?
Answer: __________________________
Report the exact confidence interval produced by JASP and explain whether the hypothesized population mean of 8 minutes falls inside or outside the interval.
The confidence interval should be interpreted as an estimate of the range of plausible population mean values based on the sample and the statistical model. It should not be interpreted as meaning that there is a 95% probability that a fixed population parameter changes from sample to sample.
Problem Set 3.7: Independent-Samples t Test in JASP
What Is an Independent-Samples t Test?
An independent-samples t test compares the means of two separate groups. It can be used to determine whether the observed difference between two group means is statistically significant.
For this problem, use the scores.jasp dataset. Dr. Z wants to determine whether depression scores differ between clients who temporarily avoid watching or reading the news and clients who continue therapy as usual.
Participants are assigned to two groups. Group 1 is instructed not to watch or read the news for two weeks while continuing therapy. Group 2 continues therapy as usual. Depression scores are measured after the two-week period.
JASP procedure
Open scores.jasp in JASP and complete the following steps:
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Select T-Tests from the toolbar.
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Under Classical, select Independent-Samples T-Test.
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Select Score and move it into the Dependent Variables box.
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Select Group and move it into the Grouping Variable box.
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Make sure Student is selected.
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Select Descriptives.
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Deselect options that are not required for the assignment.
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Review the statistical output.
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Copy the appropriate JASP results into the designated Word document.
Review the group means, standard deviations, t statistic, degrees of freedom, and p value when interpreting the findings.
Problem Set 3.8: Interpreting an Independent-Samples t Test
Identify the Independent and Dependent Variables
Use the information from Problem Set 3.7 to identify the variables.
Independent variable (IV): __________________________
The independent variable is the condition assigned to the participants: temporarily avoiding news while continuing therapy versus continuing therapy as usual.
Dependent variable (DV): __________________________
The dependent variable is the depression score measured after the two-week period.
State the hypotheses
Null hypothesis (H₀): __________________________
The null hypothesis should state that there is no difference in mean depression scores between the two groups.
Directional alternative hypothesis (H₁): __________________________
The directional alternative hypothesis should reflect the expected direction of the difference between the two groups based on the research question or study prediction.
Evaluate the null hypothesis
Can the null hypothesis be rejected at α = .05?
Answer: __________________________
Use the appropriate JASP test result and p value to make the decision.
If p < .05, reject the null hypothesis. If p ≥ .05, fail to reject the null hypothesis.
The conclusion should be written in terms of the research question rather than simply stating whether the p value is significant.
Problem Set 3.9: Independent-Samples t Test Using Excel
Performing an Independent-Samples t Test in Excel
Microsoft Excel can perform an independent-samples t test for two separate groups. In this exercise, the data will be analyzed twice: first under the assumption of equal population variances and then without assuming equal variances.
The two approaches use different assumptions and can produce different degrees of freedom and p values.
Depression Score Data
Group 1: 34, 25, 4, 64, 14, 49, 54
Group 2: 24, 78, 59, 68, 84, 79, 57
Conducting the Equal-Variance t Test
Open Microsoft Excel and enter the data into a blank worksheet.
Use the following labels:
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Cell A1: 1
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Cell B1: 2
Enter the Group 1 scores below the first label and the Group 2 scores below the second label.
Then:
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Select Data Analysis.
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Choose t-Test: Two-Sample Assuming Equal Variances.
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Select OK.
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For Variable 1 Range, enter $A$2:$A$8.
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For Variable 2 Range, enter $B$2:$B$8.
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Run the analysis.
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Review the output generated on the new worksheet.
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Copy the complete output into the Word document.
The equal-variance procedure assumes that the two populations have the same variance.
Conducting the Unequal-Variance t Test
Return to the original data worksheet and select Data Analysis again.
Choose t-Test: Two-Sample Assuming Unequal Variances, then select OK.
Enter the following ranges:
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Variable 1 Range: $A$2:$A$8
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Variable 2 Range: $B$2:$B$8
Run the analysis and review the output generated on the new worksheet.
Copy the results into the designated Word document.
The unequal-variance procedure does not require the assumption that both populations have identical variances and is commonly associated with Welch’s independent-samples t test.
Compare the Equal- and Unequal-Variance Results
Compare the outputs from the two Excel analyses. Pay particular attention to:
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The calculated t statistic
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Degrees of freedom
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p value
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Whether the result meets the selected α = .05 significance criterion
Use these statistics to determine whether there is sufficient evidence of a difference between the two group means.
Key Concepts to Remember
Hypothesis testing begins with a null hypothesis and an alternative hypothesis. Researchers use sample data to determine whether there is sufficient statistical evidence to reject the null hypothesis.
Effect size and statistical power answer different questions. Effect size describes how large a difference or relationship is, while statistical power describes the probability of detecting a genuine effect when one exists.
A one-sample t test compares one sample mean with a specified population or hypothesized mean. An independent-samples t test compares the means of two separate groups.
A confidence interval provides an estimated range of plausible values for a population parameter and communicates the precision and uncertainty associated with a sample estimate.
When interpreting p values, researchers should use the significance level established before conducting the analysis. If α = .05, a p value slightly above .05 should not automatically be described as statistically significant or “marginally significant.”
Frequently Asked Questions About Hypothesis Testing, Effect Size, Power, and t Tests
What is hypothesis testing?
Hypothesis testing is a statistical process used to evaluate whether sample data provide sufficient evidence to reject a null hypothesis about a population. The researcher establishes hypotheses, selects a significance level, conducts an appropriate statistical test, and interprets the resulting evidence.
What is the difference between a null hypothesis and an alternative hypothesis?
The null hypothesis (H₀) generally states that there is no statistically meaningful difference, relationship, or effect. The alternative hypothesis (H₁ or Hₐ) represents the research claim that a difference, relationship, or effect exists. An alternative hypothesis can be directional or nondirectional.
What is statistical power?
Statistical power is the probability that a statistical test will detect a genuine effect when that effect actually exists. Power generally increases with larger samples and larger effect sizes. Lower variability can also make genuine effects easier to detect.
What does effect size mean in statistics?
Effect size describes the magnitude of an observed difference or relationship. Unlike a p value, which addresses the statistical evidence against a null hypothesis, effect size helps researchers understand how substantial the observed effect is.
For example, Cohen’s d is commonly used to describe the standardized difference between two means.
What is the difference between a one-tailed and two-tailed test?
A one-tailed test evaluates an alternative hypothesis that predicts a specific direction, such as one population mean being greater than another. A two-tailed test evaluates whether a difference exists without predicting whether one mean will be higher or lower.
What does a p value below .05 mean?
When α = .05, a p value below .05 generally provides sufficient statistical evidence to reject the null hypothesis under the assumptions of the test. It does not mean that the alternative hypothesis has a 95% probability of being true, nor does it indicate the size or practical importance of an effect.
What does it mean to fail to reject the null hypothesis?
Failing to reject the null hypothesis means that the available sample evidence is not sufficiently strong to reject the null hypothesis at the selected significance level. It does not establish that the null hypothesis is definitively true.
What is a 95% confidence interval?
A 95% confidence interval is an interval-estimation method that provides a range of values based on sample data. Under repeated sampling and the assumptions of the method, approximately 95% of intervals constructed using the same procedure would contain the true population parameter.
What is a one-sample t test used for?
A one-sample t test evaluates whether a sample mean differs from a specified population or hypothesized mean. It is appropriate when there is one sample and a reference mean against which that sample is compared.
What is an independent-samples t test?
An independent-samples t test evaluates whether the means of two independent groups differ. It is commonly used when participants belong to separate groups and the researcher wants to compare their average scores on a dependent variable.
Why are equal-variance and unequal-variance t tests compared in Excel?
The two procedures make different assumptions about population variances. The equal-variance test assumes that the two populations have the same variance, whereas the unequal-variance procedure does not require equal variances and is commonly associated with Welch’s t test.
How do sample size and effect size affect statistical power?
Larger sample sizes generally increase statistical power because they provide more information and reduce sampling uncertainty. Larger effects are also easier to detect, so increasing effect size generally increases power when other factors remain constant.
Why is statistical significance different from practical significance?
Statistical significance addresses whether the observed evidence is sufficiently inconsistent with the null hypothesis at a selected significance level. Practical significance considers whether the size of the effect is meaningful in a real-world or applied context. A statistically significant result can have a very small effect, while a potentially meaningful effect may fail to reach statistical significance in a small study.
References
American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). American Psychological Association. https://apastyle.apa.org/products/publication-manual-7th-edition
Cho, H., & Abe, S. (2013). Is two-tailed testing for directional research hypotheses justified? Journal of Business Research. https://doi.org/10.1016/j.jbusres.2013.08.006
JASP Team. (n.d.). JASP: A fresh way to do statistics. https://jasp-stats.org/
PSYC FPX 4700 Assessment 3 Hypothesis Effect Size Power and Tests
Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4, 863. https://doi.org/10.3389/fpsyg.2013.00863
Lambdin, D. D. (2012). Significance tests as sorcery: Science is empirical—Significance tests are not. Theory Into Practice, 51(2), 75–81. https://doi.org/10.1080/00405841.2012.662865
OpenStax. (2021). Introductory statistics 2e. Rice University. https://openstax.org/details/books/introductory-statistics-2e