Basic Statistics Cheat Sheet | Cheat Sheet Statistics

Hypothesis Testing: One-Sample T-test

● A hypothesis test evaluates whether the null hypothesis (H₀) is a reasonable explanation for the observed data, using evidence from a sample. The outcome is always to reject H₀ or

fail to reject H₀—you never accept or prove the null hypothesis.

● The H0 is a statement about population parameters or distributions. The alternative hypothesis (Hₐ) includes all values not speciﬁed by H₀ and often reﬂects the research hypothesis.

● There are one-sided tests (left- and right-tailed tests) and two-sided tests. The one-sided test has the same p-value. A two-tailed p-value is twice the one-tailed p-value.

○ Two-tailed: μ ≠ μ₀ | Right-tailed: μ > μ₀ | Left-tailed: μ < μ₀

● The assumptions are normality (the sample observations come from a normal distribution) and independence (the sample observations are independent of one another).

● The alpha level α is the threshold below which a p-value is small enough to reject the null hypothesis. We reject if p-value < 𝛼 and fail to reject if p-value .

≥ 𝛼

● Type I error (α): H₀ is true, but we mistakenly reject it | Type II error (β): H₀ is false, but we fail to reject it.

○ Lowering α reduces Type I error but increases Type II error, and vice versa. The only way to reduce both is to increase the sample size.

● Power is the probability of rejecting a false null hypothesis and is deﬁned as 1 − β. A high-power test is more likely to detect a true eﬀect when one exists.

Dependent (Paired) Samples T-test: Compares means of 2 groups measured on the same continuous variable, where observations can be meaningfully paired (before vs. after treatment).

● The null hypothesis typically states that the mean paired diﬀerence (d) equals zero (H₀: d = 0).

● The assumptions are normality and independence. To decide whether you can use a paired samples t-test.

○ If n1=n2 > 30, you can rely on the Central Limit Theorem, and the test is likely to be robust to normality violations.

○ If n < 30, you must check normality through the skewness statistic or the Shapiro-Wilk test.

■ If normality holds, use the paired t-test; if not, the paired t-test may be invalid.

● When interpreting results, ﬁrst determine the test type.

○ If it is 1-tailed (Ha: 𝜇d > 0 or Ha: 𝜇d < 0), check if the mean diﬀerence goes in the hypothesized direction.

■ If yes, check whether (2-tailed p-value)/2 <0.05

● If yes: Statistically signiﬁcant; report result and calculate eﬀect size

● If no: Not signiﬁcant; report result.

○ If it is 2-tailed (i.e., Ha: 𝜇d ≠ 0), check if the 2-tailed p-value < .05 or if 0 is not in the conﬁdence interval.

■ If yes → Statistically signiﬁcant; report result, determine direction (μd > 0 or < 0), and calculate eﬀect size.

■ If no → Not signiﬁcant; report result.

Independent Samples T-test (Two-sample T-test): Compares the means of two populations whose individuals cannot be meaningfully paired. For example, a drug trial comparing a

treatment and control group uses an independent samples test. The goal is to estimate the diﬀerence between population means (𝜇₁ − 𝜇₂) using the diﬀerence between sample means (ȳ₁ − ȳ₂).

● Assumptions include normality, independence, and independent groups (scores in one group do not depend on the other).

● The null hypothesis is that the diﬀerence between the population means equals a hypothesized value Δ₀: H₀: 𝜇₁ − 𝜇₂ = Δ₀, commonly Δ₀ = 0.

● Before running the test, we decide which formula for degrees of freedom to use by performing Levene’s test for equality of variances.

○ If we fail to reject the null hypothesis (i.e., if p ≥ 0.05), we should use the formula for populations with equal variances.

○ If we reject the null hypothesis (i.e., if p < 0.05), we should use the formula for populations with unequal variances.

● Assumptions and result interpretation follow the same procedures as described for the dependent t-test.

One-way ANOVA: Used when comparing the means of three or more groups deﬁned by a single factor (e.g., treatment type).

● The H0 states that all population means are equal (μ₁ = μ₂ = ... = μ). The HA states that at least one mean diﬀers.

● The assumptions include normality, independence, independent groups, and homogeneity of variances (populations have the same variance).

● The ANOVA test compares the between-group variance (MSB) and the within-group variance (MSW).

○ The MSB is the spread of the sample means across treatment groups, while the MSW is the spread of scores within each treatment group.

○ ANOVA uses the F-ratio (MSb/MSw), which follows an F-distribution when assumptions are met, and the null hypothesis is true.

■ If MSB ≤ MSW (i.e., F ≤ 1), we fail to reject the null hypothesis because the variation we see is close to what we expect to see if the null is true.

■ If MSB > MSW (i.e., F > 1), we may reject the null hypothesis because the variation we see is unusual if the null is true.

● The eﬀect size η2 (eta-squared) is calculated SSB/SST (variance explained by groups). The Cohen eﬀect size is used to categorize the size of the eﬀect.

○ η2 < 0.01 negligible, 0.01 ≤ η2 < 0.06 small, 0.06 ≤ η2 < 0.14 medium, and η2 ≥ 0.14 large

● Post-hoc multiple comparisons tests identify which pairs of means diﬀer while controlling Type I error. Tukey's Honestly Signiﬁcance Diﬀerence compares all pairs after a

signiﬁcant ANOVA, while the Bonferroni adjustment is used for planned comparisons.

● To decide whether you can use a one-way ANOVA.

○ If n₁, n₂, ....nk > 30, you can rely on the CLT and proceed with checking assumptions.

○ If either sample is < 30 → check normality

■ If the skewness ratio <2 or the Shapiro-Wilk test is p>0.05, then normality is tenable → proceed with checking assumptions

■ If the skewness ratio is not <2 or the Shapiro-Wilk test is not p>0.05 → normality is not tenable → do not use one-way ANOVA, result may not be

valid

○ If n1 = n2 = ... = nk → homogeneity of variance is not an issue → proceed with the one-way ANOVA

○ If n1 ≠ n2 ≠ ... ≠ nk → homogeneity of variance is an issue → perform Levene's test

■ If Levene's test p > .05 → homogeneity of variance is not an issue → proceed with the one-way ANOVA

■ If Levene's test is not p > .05 → homogeneity of variance is not tenable → do not use one-way ANOVA, test result may not be valid

● When interpreting results, ﬁrst determine whether the ANOVA test is signiﬁcant.

○ If the ANOVA p < .05, the results are statistically signiﬁcant.

■ If k > 2 → Conduct post-hoc, interpret the result, and calculate eﬀect size.

■ If k=2 → Use sample means to interpret the result and calculate the eﬀect size

Two-way ANOVA: Examines how two factors aﬀect a dependent variable and whether they interact, by partitioning variability into main and interaction eﬀects.

● The null hypotheses state that there is no interaction and no main eﬀects of Factors 1 (equal row means) and 2 (equal column means), while the alternative hypotheses state that

an interaction exists and that at least one row or column mean diﬀers.

● The assumptions include normality, independence, independent groups, and homogeneity of variances.

● An interaction occurs when the eﬀect of one factor depends on the level of the other. Interaction plots help visualize this:

○ Parallel lines indicate no interaction, even if they are curved or have kinks. Nonparallel or crossing lines suggest a possible interaction, though lines may cross

without a signiﬁcant interaction or not cross despite one. If the interaction is signiﬁcant, interpretation should focus on the interaction rather than the main eﬀects.

● Eﬀect sizes show the proportion of total variance explained by each eﬀect:

○ Interaction Eﬀect: η²INT = SSINT / SST | Factor 1 Eﬀect: η²F1 = SSF₁ / SST | Factor 2 Eﬀect: η²F2 = SSF₂ / SST

● Assumptions follow the same procedures as described for the one-way ANOVA. When interpreting results, ﬁrst determine whether the interaction eﬀect is signiﬁcant.

○ If p < 0.05 for the interaction eﬀect, the eﬀects are statistically signiﬁcant → Conduct a simple

eﬀects analysis, interpret the interaction result, and calculate the eﬀect size (do not interpret

main eﬀects result).

○ If p is not <0.05 → check if p <0.05 for any of the main eﬀects

■ If yes, the results are statistically signiﬁcant → check if k >2.

● If k>2, conduct post-hoc, interpret the result, and calculate the eﬀect

size.

● If k=2, use sample means to interpret the result and calculate eﬀect size.

■ If not, the results are not statistically signiﬁcant and report result

Correlation and Simple Regression

● A scatterplot is a graphical representation of paired data values (x,y) as individual points on a grid, with the x-axis (horizontal) as the IV and the y-axis (vertical) as the DV.

● The direction of the association between the two variables can be positive (bottom left → upper right) or negative (upper left → bottom right). The form of the association can be

linear (points appear stretched in a consistent straight form) and non-linear (points do not follow a straight form–curved, bending, etc). The strength of association can be strong

(points are tightly clustered in a clear pattern–either linear or nonlinear) or weak (points form a vague cloud with barely discernible pattern).

● Pearson’s Correlation Coeﬃcient (r) measures the strength and direction of a linear relationship between two variables. The range is 1 ≤ r ≤ 1. The direction of r indicates a

negative or positive direction, while the absolute value |r| indicates the strength, with |r| close to 1 → strong and |r| close to 0 → weak. There is no universal cutoﬀ for “strong” r.

● Regression analysis is used when a moderate or strong linear association exists and can be used for prediction.

○ A linear model equation is: where is the predicted value of y, a is the y-intercept, and b is the slope.

𝑦 = 𝑎 + 𝑏𝑥 𝑦

○ The slope b represents the change in the predicted value of y for a one-unit increase in x. The intercept represents the predicted value of y when x equals zero.

If zero is not a meaningful value for x, the intercept is simply a starting point and not a meaningful prediction.

● Residual (e): diﬀerence between observed and predicted value, where .

𝑒 = 𝑦 − 𝑦

● The Coeﬃcient of Determination (R2) is the proportion of variation in y explained by the model, and ranges from 0 to 1. It is used as a measure of eﬀect size.

● The regression assumption includes linearity, independence (error terms must be independent), equal variance (variance of the error terms should be the same), and normal

population (the error terms along the regression line should follow a normal distribution).

● The regression inference focuses on testing whether the population slope equals zero. The null hypothesis states that there is no relationship between x and y. A t-statistic is used

to test this hypothesis and to construct conﬁdence intervals for the slope. A regression model is statistically signiﬁcant if the F-test rejects the null hypothesis.

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Hypothesis Testing: One-Sample T-test ● A hypothesis test evaluates whether the null hypothesis (H₀) is a reasonable explanation for the observed data, using evidence from a sample. The outcome is always to reject H₀ or fail to reject H₀—you never accept or prove the null hypothesis. ● The H 0 is a statement about population parameters or distributions. The alternative hypothesis (Hₐ) includes all values not specified by H₀ and often reflects the research hypothesis. ● There are one-sided tests (left- and right-tailed tests) and two-sided tests. The one-sided test has the same p-value. A two-tailed p-value is twice the one-tailed p-value. ○ Two-tailed : μ ≠ μ₀ | Right-tailed : μ > μ₀ | Left-tailed : μ < μ₀ ● The assumptions are normality (the sample observations come from a normal distribution) and independence (the sample observations are independent of one another). ● The alpha level α is the threshold below which a p-value is small enough to reject the null hypothesis. We reject if p-value < 𝛼 and fail to reject if p-value ≥ 𝛼. ● Type I error (α) : H₀ is true, but we mistakenly reject it | Type II error (β) : H₀ is false, but we fail to reject it. ○ Lowering α reduces Type I error but increases Type II error, and vice versa. The only way to reduce both is to increase the sample size. ● Power is the probability of rejecting a false null hypothesis and is defined as 1 − β. A high-power test is more likely to detect a true effect when one exists. Dependent (Paired) Samples T-test: Compares means of 2 groups measured on the same continuous variable, where observations can be meaningfully paired (before vs. after treatment). ● The null hypothesis typically states that the mean paired difference (d̄) equals zero (H₀: d = 0). ● The assumptions are normality and independence. To decide whether you can use a paired samples t-test. ○ If n 1 =n 2 > 30, you can rely on the Central Limit Theorem, and the test is likely to be robust to normality violations. ○ If n < 30, you must check normality through the skewness statistic or the Shapiro-Wilk test. ■ If normality holds, use the paired t-test; if not, the paired t-test may be invalid. ● When interpreting results, first determine the test type. ○ If it is 1-tailed (Ha: 𝜇 d > 0 or Ha: 𝜇 d < 0), check if the mean difference goes in the hypothesized direction. ■ If yes, check whether (2-tailed p-value)/2 <0. ● If yes: Statistically significant; report result and calculate effect size ● If no: Not significant; report result. ○ If it is 2-tailed (i.e., Ha: 𝜇 d ≠ 0), check if the 2-tailed p-value < .05 or if 0 is not in the confidence interval. ■ If yes → Statistically significant; report result, determine direction (μd > 0 or < 0), and calculate effect size. ■ If no → Not significant; report result. I ndependent Samples T-test (Two-sample T-test): Compares the means of two populations whose individuals cannot be meaningfully paired. For example, a drug trial comparing a treatment and control group uses an independent samples test. The goal is to estimate the difference between population means (𝜇₁ − 𝜇₂) using the difference between sample means (ȳ₁ − ȳ₂). ● Assumptions include normality, independence, and independent groups (scores in one group do not depend on the other). ● The null hypothesis is that the difference between the population means equals a hypothesized value Δ₀: H₀: 𝜇₁ − 𝜇₂ = Δ₀, commonly Δ₀ = 0. ● Before running the test, we decide which formula for degrees of freedom to use by performing Levene’s test for equality of variances. ○ If we fail to reject the null hypothesis (i.e., if p ≥ 0.05) , we should use the formula for populations with equal variance s. ○ If we reject the null hypothesis (i.e., if p < 0.05 ), we should use the formula for populations with unequal variances. ● Assumptions and result interpretation follow the same procedures as described for the dependent t-test. One-way ANOVA: Used when comparing the means of three or more groups defined by a single factor (e.g., treatment type). ● The H 0 states that all population means are equal (μ₁ = μ₂ = ... = μ). The HA states that at least one mean differs. ● The assumptions include normality, independence, independent groups, and homogeneity of variances (populations have the same variance). ● The ANOVA test compares the between-group variance (MSB) and the within-group variance (MSW). ○ The MSB is the spread of the sample means across treatment groups, while the MSW is the spread of scores within each treatment group. ○ ANOVA uses the F-ratio (MSb/MSw), which follows an F-distribution when assumptions are met, and the null hypothesis is true. ■ If MSB ≤ MSW (i.e., F ≤ 1), we fail to reject the null hypothesis because the variation we see is close to what we expect to see if the null is true. ■ If MSB > MSW (i.e., F > 1), we may reject the null hypothesis because the variation we see is unusual if the null is true. ● The effect size η^2 (eta-squared) is calculated SSB/SST (variance explained by groups). The Cohen effect size is used to categorize the size of the effect. ○ η 2 < 0.01 negligible, 0.01 ≤ η 2 < 0.06 small, 0.06 ≤ η 2 < 0.14 medium, and η 2 ≥ 0.14 large ● Post-hoc multiple comparisons tests identify which pairs of means differ while controlling Type I error. Tukey's Honestly Significance Difference compares all pairs after a significant ANOVA, while the Bonferroni adjustment is used for planned comparisons. ● To decide whether you can use a one-way ANOVA. ○ If n ₁ , n ₂ , ....nk > 30, you can rely on the CLT and proceed with checking assumptions. ○ If either sample is < 30 → check normality ■ If the skewness ratio <2 or the Shapiro-Wilk test is p>0.05, then normality is tenable → proceed with checking assumptions ■ If the skewness ratio is not <2 or the Shapiro-Wilk test is not p>0.05 → normality is not tenable → do not use one-way ANOVA, result may not be valid ○ If n1 = n2 = ... = nk → homogeneity of variance is not an issue → proceed with the one-way ANOVA ○ If n1 ≠ n2 ≠ ... ≠ nk → homogeneity of variance is an issue → perform Levene's test ■ If Levene's test p > .05 → homogeneity of variance is not an issue → proceed with the one-way ANOVA ■ If Levene's test is not p > .05 → homogeneity of variance is not tenable → do not use one-way ANOVA, test result may not be valid ● When interpreting results, first determine whether the ANOVA test is significant. ○ If the ANOVA p < .05, the results are statistically significant. ■ If k > 2 → Conduct post-hoc, interpret the result, and calculate effect size. ■ If k=2 → Use sample means to interpret the result and calculate the effect size Two-way ANOVA : Examines how two factors affect a dependent variable and whether they interact, by partitioning variability into main and interaction effects. ● The null hypotheses state that there is no interaction and no main effects of Factors 1 (equal row means) and 2 (equal column means), while the alternative hypotheses s tate that an interaction exists and that at least one row or column mean differs. ● The assumptions include normality, independence, independent groups, and homogeneity of variances. ● An interaction occurs when the effect of one factor depends on the level of the other. Interaction plots help visualize this: ○ Parallel lines indicate no interaction, even if they are curved or have kinks. Nonparallel or crossing lines s uggest a possible interaction, though lines may cross without a significant interaction or not cross despite one. If the interaction is significant, interpretation should focus on the interaction rather than the main effects. ● Effect sizes show the proportion of total variance explained by each effect: ○ Interaction Effect : η²INT = SSINT / SST | Factor 1 Effect : η²F1 = SSF₁ / SST | Factor 2 Effect: η²F2 = SSF₂ / SST ● Assumptions follow the same procedures as described for the one-way ANOVA. When interpreting results, first determine whether the interaction effect is significant. ○ If p < 0.05 for the interaction effect, the effects are statistically significant → Conduct a simple effects analysis, interpret the interaction result, and calculate the effect size (do not interpret main effects result). ○ If p is not <0.05 → check if p <0.05 for any of the main effects ■ If yes, the results are statistically significant → check if k >2. ● If k>2, conduct post-hoc, interpret the result, and calculate the effect size. ● If k=2, use sample means to interpret the result and calculate effect size. ■ If not, the results are not statistically significant and report result Correlation and Simple Regression ● A scatterplot is a graphical representation of paired data values (x,y) as individual points on a grid, with the x-axis (horizontal) as the IV and the y-axis (vertical) as the DV. ● The direction of the association between the two variables can be positive (bottom left → upper right) or negative (upper left → bottom right). The form of the association can be linear (points appear stretched in a consistent straight form) and non-linear (points do not follow a straight form–curved, bending, etc). The strength of association can be strong (points are tightly clustered in a clear pattern–either linear or nonlinear) or weak (points form a vague cloud with barely discernible pattern). ● Pearson’s Correlation Coefficient (r) measures the strength and direction of a linear relationship between two variables. The range is 1 ≤ r ≤ 1. The direction of r indicates a negative or positive direction, while the absolute value |r| indicates the strength, with |r| close to 1 → strong and |r| close to 0 → weak. There is no universal cutoff for “strong” r. ● Regression analysis is used when a moderate or strong linear association exists and can be used for prediction. ○ A linear model equation is: 𝑦 = 𝑎 + 𝑏𝑥 where 𝑦 is the predicted value of y, a is the y-intercept, and b is the slope. ○ The slope b represents the change in the predicted value of y for a one-unit increase in x. The intercept represents the predicted value of y when x equals zero. If zero is not a meaningful value for x, the intercept is simply a starting point and not a meaningful prediction. ● Residual (e) : difference between observed and predicted value, where 𝑒 = 𝑦 − 𝑦. ● The Coefficient of Determination (R^2 ) is the proportion of variation in y explained by the model, and ranges from 0 to 1. It is used as a measure of effect size. ● The regression assumption includes linearity, independence (error terms must be independent), equal variance (variance of the error terms should be the same), and normal population (the error terms along the regression line should follow a normal distribution). ● The regression inference focuses on testing whether the population slope equals zero. The null hypothesis states that there is no relationship between x and y. A t-statistic is used to test this hypothesis and to construct confidence intervals for the slope. A regression model is statistically significant if the F-test rejects the null hypothesis.

Multiple Choice Question

In the 1800s, Dr.Wunderlich estimated the mean human body temperature to be 98.60 °F. You want to find out if today’s mean °F is lower. ANSWER : A
From 2019-2021, the mean number of calories consumed per person per day in the U.S. was 3,864, according to a study. You want to know if, on average, Americans today consume fewer than 3,864 calories per person per day. ANSWER : A
You want to see if there are significant differences in mean BP between males and females. ANSWER : B
You want to see whether students who usually do homework with music perform the same with music as without music. ANSWER: C
Many people believe that students gain weight during their first year in college. A random sample of 64 first-year college students was weighted during the first week of the semester and then again during the twelfth week. ANSWER : C
A researcher wants to compare the mean heart rates of adults following three different exercise programs: yoga, cycling, and strength training. ANSWER : D
A psychologist wants to study whether college students’ test anxiety scores are affected by the type of study method (group vs. individual) and by year in school (freshman, sophomore). ANSWER : E

Hypothesis Testing One-sample T-test Test whether the mean gestation time of 70 pregnancies (x̄ = 260.31 days) differs from the known population mean (μ = 266 days)

The normality assumption is not an issue for this analysis since we can rely on the Central Limit Theorem because the sample size (n=70) is greater than 30. Our results are likely to be robust to violations of the normality assumption.
H 0 : The population mean gestation time for human pregnancies in the Nashville hospital is equal to 266 days (i.e., H 0 : 𝜇 = 266)
Since this is a one-sample t-test, there is no second group and therefore no variance comparison to make.
According to the results of the one-sample t-test, there is a statistically significant difference between the mean gestation time at the Nashville hospital and the established population mean of 266 days. The p-value (0.0027) is less than α = 0.05, so we reject the null hypothesis. The sample mean gestation time (260.31 days) is significantly shorter than the population mean, indicating that pregnancies at this hospital have a shorter average gestation period.

Dependent Samples T-test A researcher wants to know how large the average difference in dexterity is between the dominant and non-dominant hands in children.

The normality assumption is not an issue for this analysis since we can rely on the Central Limit Theorem because the sample size (n=93) is greater than 30. Our results are likely to be robust to violations of the normality assumption.
H 0 : The population mean dexterity difference between dominant and non-dominant hands is equal to 0 (i.e., H 0 : 𝜇d=0).
Since this is a dependent t-test, we are analyzing a single set of differences from matched pairs. There is no second, independent group whose variance needs to be compared.
According to the results of the dependent samples t-test, there is a statistically significant difference in mean dexterity between the dominant and non-dominant hands. The p-value (<0.0001) is less than α = 0.05, so we reject the null hypothesis. The mean dexterity score for the dominant hand (higher) is significantly greater than that of the non-dominant hand, with a mean difference of 0.054.

Independent Samples T-test Use NHANES data to conduct an independent sample t-test comparing mean systolic blood pressure between males and females.

Both sample sizes are greater than 30 (i.e., male=4,915 and female=5,436). Thus, the CLT is likely to hold for both samples, and our results are likely to be robust to violations of the normality assumption. Ultimately, the normality assumption is not an issue for this analysis.
H₀: There is no difference in mean systolic blood pressure between males and females in the population.
Since the p-value for each version of Levene’s test is p < 0.001, we should reject the null hypothesis of equal variances and use an independent sample t-test for populations with unequal variances.
The results of the independent samples t-test indicate a statistically significant difference in mean systolic blood pressure between males and females. The mean difference of 3.82 shows that males have higher systolic blood pressure than females. The p-value (<0.001) is less than α = 0.05, so we reject the null hypothesis, indicating that the difference in systolic blood pressure between males and females is statistically significant. One-way ANOVA Suppose you plan to use a one-way ANOVA to answer the following question: Does self-concept in twelfth grade (SLFCNC12) of college-bound students vary as a function of the degree of urbanity (URBAN)?
The sample sizes for urban, suburban, and rural twelfth-grade, college-bound students are all greater than 30 (urban n=123; suburban n=215; rural n=162). Therefore, the Central Limit Theorem is likely to hold across all three groups, and our results are likely to be robust to violations of normality. Thus, the normality assumption is not an issue for this analysis. -The p-values for all versions of Levene’s test are greater than 0.05, leading us to fail to reject the null hypothesis that the group variances are equal. Therefore, we can conclude that the homogeneity of variances assumption is tenable. -The one-way ANOVA result is statistically significant, with a p-value below 0.05 ( p = 0.0045). This allows us to reject the null hypothesis of equal group means at the 5% significance level and conclude that at least one population mean differs from the others.
The effect size (𝜂^2 = 0.022) falls within the range classified as a small effect under Cohen’s conventions, indicating that only a small proportion of the variability in self-concept scores is explained by students’ urbanicity. -A post-hoc test is necessary because the one-way ANOVA for urbanicity was statistically significant (p = 0.0045) and the factor consists of more than two groups, so further analysis is needed to determine which specific groups differ. Two-way ANOVA Does self-concept in 12th-grade college-bound students (SLFCNC12) differ by urbanicity (URBAN), biological sex (GENDER), or their interaction?
The sample sizes for urban, suburban, and rural male and female twelfth-grade, college-bound students are all greater than 30. Therefore, the Central Limit Theorem is likely to hold across all six groups, and our results are likely to be robust to violations of normality. Thus, the normality assumption is not an issue for this analysis.
The two-way ANOVA shows a significant main effect of urbanicity (p = 0.0094) and a significant main effect of biological sex (p = 0.0010). However, the interaction between urbanicity and biological sex is not statistically significant (p = 0.1833). -The effect size for urbanicity (η² = 0.019) and the effect size for biological sex (η² = 0.022) both fall within the range classified as small under Cohen’s conventions, indicating that each factor explains only a small proportion of the variability in students’ self-concept scores.
Since there is no interaction effect, there is no need for a simple effects analysis. Moreover, even though the main effect of biological sex is significant, there is no need for a post-hoc test because biological sex only has two categories. However, we do need to conduct a post hoc test for the highest level of education expected because it has a significant main effect and more than two categories. Interaction _This illustration shows population means (no sampling error) from a fictional 2 × 3 study on biological sex, teaching method, and reading achievement, with equal group sizes.
Main Effect of Sex_ : There is no main effect of biological sex on mean reading achievement. When averaging across teaching methods, both males and females have the same overall mean score of 20. Thus, biological sex does not influence reading performance, and males and females perform at the same level when the teaching method is ignored. - Main Effect of Teaching : There is a main effect of teaching method, as the average reading achievement scores differ across the three methods when biological sex is ignored. Specifically, the averages are 25, 20, and 15 for the respective teaching methods, indicating that the choice of teaching method influences reading performance. - Interaction Effect : There is an interaction between biological sex and teaching method, as shown by the non-parallel lines connecting the group means. The impact of teaching methods varies by sex, with females’ scores decreasing across the methods, while males’ scores remain unchanged. - Effectiveness of Teaching : For females, the most effective method is Whole Language, followed by Synthetic Phonics, with Analytic Phonics being the least effective. For males, all teaching methods result in the same performance, indicating that method choice does not influence their reading scores. This demonstrates that the teaching method has a differential effect depending on the student’s sex. Correlation and Regression Construct a regression model to predict the percent body
A simple linear regression model was calculated to predict the percent body fat based on hip size. The overall regression was statistically significant (F(1, 248)=165.52, p < 0.001, R^2 = 0.40). It was found that hip size significantly predicted body fat (b = 0.81, p < 0.001).
Regression equation: 𝑃𝑐𝑡𝐵𝐹 = 𝑎 + 𝑏 𝑥 𝐻𝑖𝑝 →𝑃𝑐𝑡𝐵𝐹 = − 62. 12 + 0. 814 𝑥 𝐻𝐼𝑃
The slope indicates that for each additional inch of hip circumference, an adult male’s percent body fat is expected to increase by about 0.814 percentage points, on average.
The y-intercept (-62.12) represents the predicted body fat when hip size is zero inches. Since a hip measurement of zero is not realistic, the intercept has no meaningful interpretation in this context. -The coefficient of determination for the model is R² = 0.4003, meaning that hip size predicts about 40% of the variation in percent body fat.

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