Calculate the degrees of freedom for a one-sample t test and locate critical values in the t table.

1. Locate sample size 

2. Plug sample size number into the formula n-1

3. locate the total from the formula in Table C.2 in Appendix C. find the critical value under alpha level and df.

1. Normality (assume normal distribution -- meaning that most of the data is close to the mean or median at zero).

2. Random Sampling (the sample was selected randomly minimizing error).

3. Independence (one outcome does not influence the other this is true because of the random sampling factor). 

Compute a one-sample t test and interpret the results.

Step 1: State Hypotheses 

H0 =

H1 not equal

Step 2: Set the criteria 

- compute df=n-1

- locate df in Table C.2 in Appendix C

- if the test is two tailed then include +/- value

- if the test is one tailed then include either + or - value

Step 3: Compute the Tobt = M-μ/√(n)

Step 4: Make a decision 

- compare Tobt to critical value

- For two-tailed, if the Tobt number is higher than the critical value number regardless if positive or negative then the answer is to reject null-hypothesis. 

-For one-tailed, if the direction is expected to go lower then look in positive direction, if it is expected to go higher then look in negative direction for rejection region. 

Compute and interpret effect size and proportion of variance for a one-sample t test.

When we decide to retain the null hypothesis, we conclude that an effect does not exist. However, when we decide to reject the null hypothesis we conclude that an effect does exist. However, we don't know how much of an effect. 

- Cohen's d measures effect size

- Eta squared

- omega-squared 

Compute and interpret confidence intervals for the one-sample t test.

Step 1: Compute the sample mean (Μ) and standard error (SΜ).

Step 2: Choose the level of confidence and find the critical values at that level of confidence

- Level of confidence = alpha value, convert % into alpha by subtracting % from 100. 

- Look up alpha & df at C2

Step 3: Compute the estimation formula to find confidence limits. 

- Compute estimated point t(SM)

- Compute upper confidence limit (M + (SM)

- Compute lower confidence limit (M -t(SM))

Step 1: Read the question.

Step 2: Rephrase the claim in the question with an equation.

1. In sample question #1, Drop out rate = 25%
2. In sample question #2, Drop out rate < 25%
3. In sample question #3, Drop out rate > 25%.

Step 3: If step 2 has an equals sign in it, this is a two-tailed test. If it has > or < it is a one-tailed test.

Describe the between-subjects design, and identify two appropriate sampling methods used to select independent samples.

Between-subjects design is a research design in which different participants are observed one time in each group or at each level of one factor. Example, we could observe learning outcomes during a study session held in a low or well-lit room. In this example lighting is the factor and it has two levels low and well lit. Each level of the factor constitutes a group. 

Selecting independent samples includes: 

1. quasi-experimental method of collecting independent samples: population 1, population 2. 

2. The experimental method of collecting independent samples: Population 1: Group 1, Group 2, Sample. 

Compute a two-independent-sample t test and interpret the results.

Step 1: State Hypothesis

- H0=

or H0 =/><

- H1 does not equal

or H1 </>

Step 2: Set the criteria for a decision

- df-two-independent sample t-test = df1 + df2

- Look up critical value in C.2

Step 3: Compute the test statistic 

- tobt = (M1-M2)-(μ1-μ2)/SM1-M2)

Step 4: Make a decision 

- compare tobt to critical value if the tobt does not exceed the critical value then the statistic does not fall in the rejection region and is therefore retained.

Compute and interpret effect size and proportion of variance for a two-independent-sample t test.

When we reject the null hypothesis, we conclude that an effect does exist in the population. When we retain the null we conclude that an effect does not exist in the population, which was the decision. 

Effect size calculations: 

Formulas provided. 

Compute and interpret confidence intervals for a two-independent-sample t test.

Step 1: compute the sample mean (M1-M2) and standard error (SM1-M2).

Step 2: Choose the level of confidence. 

- subtract % from 100 for alpha value

- Look up alpha value and df in Table C.2 

- determine critical value

Step 3: Compute the estimation formula. 

-compute estimation formula

- compute upper confidence limit

- compute lower confidence limit. 

Describe two types of research designs used when selecting related samples.

Related Sample Designs - aka. dependent sample, participants are related.

1. Repeated-Measures Design -- participants are observed in more than one group, for example, the same participants are observed in different treatments. There are pre-post (dependent variable is measured before and after a treatment) and within-subject designs (researchers observe the same participants across many treatments but not necessarily before and after a treatment).   

2. Matched Pairs Design (participants are matched based on common characteristics or traits).   

State three advantages for selecting related samples.

Most disadvantages pertain specifically to a repeated measures design and not a matched pairs design. 

Advantages: 

1. It is more practical. Example, it can be more practical to observe the behavior of the same participants before and after a treatment or to compare how well participants of similar ability master a task.

2. Selecting related samples reduces standard error. The value for the estimate of standard error for a related samples t-test will be smaller than that for a two-independent-sample t-test. 

3. Selecting related samples increases power because of the reduction in standard error. 

Calculate the degrees of freedom for a related-samples t test and locate critical values in the t table.

Step 1: Determine df = n-1

Step 2: Determine alpha value.

Step 3: Look up df and alpha value in table C.2

Step 4: Locate critical value 

Compute a related-samples t test and interpret the results.

Related Sample t-test

Step 1: State the hypotheses 

- H1 = 0

- H2 does not equal 0 or if two-tailed test <or>

Step 2: Set the criteria 

- Look up df & alpha value in Table C.2.

- Determine critical value 

Step 3: Compute the test statistic

Formula provided. 

- MD = ΣD(differences)/nD(#participants (n))

- μD = null hypothesis # = 0 

- SD = √(nD)

- S2D = SS(ΣD)^2/nD-1

Step 4: Make a decision

Compare tobt to critical value. We reject the null if the obt. value exceeds the critical value. 

Identify the analysis of variance test and when it is used for tests of a single factor.

Identify each source of variation in a one-way between-subjects ANOVA and a one-way within-subjects ANOVA.

If group means are equal, then the variance of group means is equal meaning they do not vary. The larger the difference between group means, the larger the variance of group means. 

Sources of variation one-way between-subjects ANOVA

-- variance is due to differences between group means = MSBG -- numerator.

Sources of variation one-way within-subjects ANOVA

-- variance that has nothing to do with having different groups, but the error that happens by chance = MSE -- denominator. 

Calculate the degrees of freedom and locate critical values for a one-way between-subjects ANOVA and a one-way within-subjects ANOVA.

Between-subjects 

1. Degrees of freedom

- dfBG=k-1

- goes in the numerator

2. Critical Value

- locate dfBG in table C.3

Within-subjects 

1. Degrees of freedom

-dfE = N-k

- goes in the denominator

2. Critical Value

-locate dfE in table C.3

Identify the assumptions for a one-way between-subjects ANOVA and a one-way within-subjects ANOVA.

One-way between subjects test Assumptions

1. Normality -- normal distribution

2. Random Sampling -- the data is obtained from a sample that was selected using a random sampling procedure

3. Independence -- the outcomes of the study are independent 

4. Homogeneity -- we assume that the variance in each population is equal to that of the others. 

One-way within-subjects test assumptions

1. Normality

2. Independence

3. Homogeneity of variance -- variance in each population is equal to that in the others. 

4. Homogeneity of covariance -- participants scores in each group are related 

Follow the steps to compute a one-way between-subjects ANOVA and a one-way within-subjects ANOVA, and interpret the results.

Steps for Between-Subjects Test

Step 1: State Hypotheses

H0 = 0

H1 > 0

Step 2: Set the criteria for a decision 

dfBG provided -- numerator

dfE provided -- denominator

dfT provided

Locate dfBG & dfE in table C.3 for critical value.

Step 3: Compute test statistic 

Stage 1: 

k = groups number of groups or levels of factors

n = number of participants per group

N = number of participants overall 

ΣxT = sum of all scores in a study

Σx(2/T) sum of all scores individually squared in a study

Stage 2: 

[1] (ΣxT)²/N

[2] Σ(x²/n)

[3] Σx(2/T)

Stage 3:

SSBG = [2]-[1]

SST = [3]-[1] 

SSE = SST-SSBG

Stage 4 provided

Step 4: Make a decision

- When Fobt < critical value retain the null hypothesis

- When Fobt > critical value reject null hypothesis 

Steps for Within-Subjects Test

Step 1: State Hypotheses

H0

H1

Step 2: Set the criteria for a decision 

dfBG provided -- numerator

dfBP provided

dfE provided -- denominator

dfT provided

Locate dfBG & dfE in table C.3 for critical value.

Step 3: Compute test statistic 

Stage 1: 

k = groups number of groups or levels of factors

n = number of participants per group

ΣxT = sum of all scores in a study

Σ p sum of scores for each person 

Σx(2/T) sum of all scores individually squared in a study

Stage 2: 

[1] (ΣxT)²/k*n

[2] Σ(x²/n)

[3] Σx(2/T)

[4] Σ p²/k

Stage 3:

SSBG = [2]-[1]

SSBP = [4]-[1]

SST = [3]-[1] 

SSE = SST-SSBG-SSBP

Stage 4 provided

Step 4: Make a decision

- When Fobt < critical value retain the null hypothesis

- When Fobt > critical value reject null hypothesis 

Compute and interpret Tukey’s HSD post hoc test and identify the most powerful post hoc test alternatives.

Step 1: compute the test statistic for each pairwise comparison

- formula provided.

Step 2: compute the critical value for each pairwise comparison

- compute Tukey'sHSD: qα√(MSe/n)

- q is the studentized range statistic -- find in table C.4

- to find q we need to know dfE and real range r (equal to the number of groups).

-dfE provided formula. 

Step 3: Make a decision to retain or reject the null

hypothesis for each pairwise comparison

- if the test statistic is larger than the critical value we computed then the two groups are significantly different.  

Compute and interpret proportion of variance for the one-way between-subjects ANOVA and the one-way within-subjects ANOVA

Proportion of variance means effect size.

Eta squared provided

Omega squared provided

Summarize the results of the one-way between-subjects ANOVA and the one-way within-subjects ANOVA in APA format.

Summarize results of a one-way ANOVA test

Report the test statistic, degrees of freedom, and the p-value, additionally, the effect size for significant analyses. Summarize the means, standard error, or standard deviations measured in a study in a figure or a table or in the main text of the article. To report the results of a post hoc test, identify which post hoc test you computed and the p-value for significant results. 

Define and explain the following terms: cell, main effect, and interaction.

Main effect -- is a source of variation associated with mean differences across the levels of a single factor. In the two way ANOVA, there are two factors and therefore two main effects; one for Factor A and for B. 

Interaction -- is a source of variation associated with the variance of group means across the combination of levels of two factors. It is a measure of how cell means at each level of one-factor change across the levels of a second factor. 

Cell -- is the combination of one level from each factor, as represented in a cross-tabulation. Each cell is a group in a research study. 

Calculate the degrees of freedom for the two-way between-subjects ANOVA and locate critical values in the F table.

Critical values two-way between-subjects ANOVA

1. Numerator Factor A degrees of freedom = dfA = p-1 (p is levels of factor A)

2. Factor B degrees of freedom = dfB = q-1 (q is level of factor B).

3. A x B degrees of freedom = dfA x B = (p-1)(q-1)

4. Denominator degrees of freedom error = dfE = pq(n-1)

5. total degrees of freedom = dfT= (npq)-1

Locate critical value in table C.3

Compute and interpret the Pearson correlation coefficient and the coefficient of determination, and test for significance.

Pearson correlation coefficient = r provided

Coefficient of determination = r² = η²

Test for significance = hypothesis testing

Step 1: state the hypotheses

H0: p = 0

H1: p≠ 0

Step 2: Set the criteria for a decision

- determine alpha level

- df = n-2

- locate critical value in table C.5

Step 3: Compute the test statistic

- compute r (formula provided).

Step 4: Make a decision

- if r is greater than critical value reject the null hypothesis 

Identify and explain three assumptions and three limitations for evaluating a correlation coefficient.

1. Homoscedasticity - constant variance among data points, assume equal variance of data points dispersed along the regression line. 

2. Linearity -- assumption that the best way to describe a pattern of data is using a straight line. 

3. Normality-- 

4. Causality -- correlation doesn't equal causation it merely shows the direction and the strength of the relationship between two factors. Other possibilities: changes could be due to a third variable, reverse causality, systematic cause an effect. 

5. Outliers -- can obscure the relationship between two factors by altering the direction and the strength of an observed correlation.

6. Restriction of Range -- is a problem that arises when the range of data for one or both correlated factors in a sample is limited or restricted, compared to the range of data in the population from which the sample was selected. 

1. Spearman -- rs, is a measure of the direction and strength of the linear relationship of two ranked factors on an ordinal scale of measurement. 

2. Point-biserial -- rpb, is a measure of the direction and strength of the linear relationship of one factor that is continuous on an interval or ratio scale of measurement and a second factor that is dichotomous on a nominal scale of measurement. 

3. Phi correlation Coefficient -- rω, is a measure of the direction and strength of the linear relationship of two dichotomous factors on a nominal scale of measurement. 

Distinguish between a predictor variable and a criterion variable.

Predictor variable (X) is the variable with values that are known and can be used to predict the values of another variable. 

Criterion variable -- (Y) is the variable with unknown values that can be predicted or estimated, given known values of the predictor variable. 

Compute and interpret the method of least squares.

The method of least squares is a statistical procedure used to compute the slope and y-intercept (a) of the best fitting straight line to a set of data points. 

Equation provided. 

Step 1: Compute preliminary calculations

Step 2: Calculate the slope (b)

Step 3: Calculate the y-intercept (a).

Identify each source of variation in an analysis of regression, and compute an analysis of regression and interpret the results.

Sources of variation:

regression variation -- is the variance in Y that is related to or associated with changes in X. The closer data points fall to the regression line, the larger the value of regression variation.

residual variation -- is the variance in Y that is not related to changes in X. This is the variance in Y that is left over or remaining. The farther data points fall from the regression line, the larger the value of residual variation. 

Analysis of Regression

Step 1: State Hypothesis: 

H0 variance in the Y is not related to change in X

H1 variance in the Y is related to change in X. 

Step 2: Set the criteria

- set alpha level

- dfreg= # of predictor variables - numerator

- dfres = n-2 -denominator

- look up critical value in table C.3 

Step 3: Compute test statistic 

- formula provided

Step 4: Make a decision 

- compare fobt to critical value if fobt exceeds critical value then reject the null hypothesis.