Dependent-Samples t-Test Overview

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Questions and Answers

What defines dependent samples in hypothesis testing?

  • Samples must consist of more than 100 observations.
  • The samples are assumed to have the same variance.
  • Samples are collected independently from different populations.
  • Each participant's data in one sample is paired with a specific participant's data in another sample. (correct)

How is the mean difference calculated in dependent samples?

  • By averaging the differences between each pair of observations. (correct)
  • By multiplying the means of both samples together.
  • By using only the largest and smallest differences.
  • By subtracting the mean of sample A from the mean of sample B.

Which of the following is correct regarding hypothesis testing in t-tests for dependent samples?

  • Dependent t-tests cannot have a two-tailed hypothesis.
  • The null hypothesis states that the two sample means are equal. (correct)
  • The alternative hypothesis proposes no effect or difference.
  • Significance levels for one-tailed tests must be set at 0.01.

Which formula is used to calculate the t-statistic in dependent-samples t-tests?

<p>$t = \frac{M_D - 0}{s_D / \sqrt{n}}$ (B)</p> Signup and view all the answers

What is the degrees of freedom (df) for a dependent-samples t-test with 8 pairs of observations?

<p>7 (B)</p> Signup and view all the answers

What does the null hypothesis (H0) state in a dependent-samples t-test?

<p>The mean difference is equal to 0. (A)</p> Signup and view all the answers

Which formula is used to calculate the t-obtained for a dependent-samples t-test?

<p>$D - ar{D}$ (A)</p> Signup and view all the answers

What does a t-obtained value indicate in hypothesis testing?

<p>The standardized difference between the sample mean difference and the population mean difference. (B)</p> Signup and view all the answers

Which step is NOT included in calculating the estimated standard error of the mean difference in a dependent-samples t-test?

<p>Calculate the population mean difference. (C)</p> Signup and view all the answers

What is measured by the symbol $s_D^2$ in the context of a dependent-samples t-test?

<p>The population variance of the difference scores. (D)</p> Signup and view all the answers

In hypothesis testing, when do you reject the null hypothesis in a dependent-samples t-test?

<p>When the t-obtained is greater than t-critical. (A)</p> Signup and view all the answers

Which calculation is necessary to obtain the sample mean difference in a dependent-samples t-test?

<p>Add together all difference scores and divide by n. (D)</p> Signup and view all the answers

What characterizes a dependent-samples t-test?

<p>It requires dependent samples. (D)</p> Signup and view all the answers

What is the first step when calculating the mean difference in a dependent-samples t-test?

<p>Find the difference for each pair of scores. (D)</p> Signup and view all the answers

In hypothesis testing using a dependent-samples t-test, what is the null hypothesis (H0)?

<p>The means of the two populations are equal. (B)</p> Signup and view all the answers

What does the formula D = X1 - X2 represent in the context of a dependent-samples t-test?

<p>The difference between each pair of scores. (B)</p> Signup and view all the answers

What is required for the application of a dependent-samples t-test concerning sample size?

<p>Equal sample size is necessary. (D)</p> Signup and view all the answers

For both dependent and independent-samples t-tests, what is one common requirement for the dependent variable?

<p>It must be normally distributed. (C)</p> Signup and view all the answers

When conducting a dependent-samples t-test, what is the requirement regarding the variance?

<p>Homogeneity of variance is necessary. (D)</p> Signup and view all the answers

Which of the following tests would show the best fit if comparing the means of two related groups?

<p>Dependent-samples t-test (B)</p> Signup and view all the answers

Flashcards

Hypothesis

A statement about a population parameter that is tested using the data of a sample.

Significance Level (alpha)

The probability of rejecting a true null hypothesis. Commonly set at .05.

Degrees of Freedom (df)

The number of independent pieces of information used to estimate a population parameter.

t-critical (tcrit)

The value from a t-table that is used to determine if a calculated t-statistic is significant to reject the null hypothesis.

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Dependent-Samples t-Test

A statistical procedure used to compare the means of two related groups.

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Difference Scores (D)

The difference between scores from two related groups (e.g., before and after a treatment).

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Null Hypothesis (H0) for Dependent-Samples t-test

The assumption that the population mean difference (μD) is zero, meaning no difference between the groups.

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Alternative Hypothesis (H1) for Dependent-Samples t-test

The claim that the population mean difference (μD) is not zero, implying a significant difference between the groups.

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Estimated Variance of Difference Scores (sD^2)

A measure of the spread of difference scores, calculated from the sample data.

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Estimated Standard Error of the Mean Difference (sD)

The standard deviation of the sampling distribution of the mean difference, used to standardize t-value.

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t-obtained (t-obt)

The calculated value of the t-statistic used to assess if the mean difference is statistically significant in a dependent-samples t-test.

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Within-Subjects Design

An experimental design where each participant is measured under multiple conditions (e.g., before and after a treatment).

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Independent Samples Design

An experimental design where participants are randomly assigned to different groups.

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Matched-Groups Design

An experimental design where participants are matched based on similar characteristics before being randomly assigned to groups.

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Null Hypothesis (H0)

The assumption that there's no significant difference between groups or conditions.

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Alternative Hypothesis (HA)

The statement that there is a significant difference between groups or conditions.

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Homogeneity of Variance

The assumption that the variability (variance) of the outcome measure is similar across different groups.

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Ratio or Interval Scale

Types of measurements used for the dependent variable in t-tests (e.g., height, weight, time).

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Dependent Variable (DV)

The variable being measured and potentially affected by the independent variable(s).

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Study Notes

Dependent-Samples t-Test Overview

  • A statistical test used to compare means of two related groups
  • Used for matched-groups designs and within-subjects designs
  • Assesses if there is a significant difference between means of related groups

Comparison of Between-Subjects and Within-Subjects Designs

  • Between-Subjects Design:
    • Different participants are exposed to different levels of the independent variable (IV)
    • Each person serves in only one condition
    • Independent or matched samples are used in each condition
  • Within-Subjects Design:
    • One group of participants experiences all levels of the IV
    • Each participant serves in all conditions of the IV
    • The same sample is used in all conditions

Appropriate Statistics

  • Independent-Samples Design: independent-samples t-test
  • Matched-Groups Design: dependent-samples t-test
  • Within-Subjects Design: dependent-samples t-test

Dependent-Samples t-Test Requirements

  • Randomly selected samples
  • Dependent variable (DV) measured using ratio or interval scale
  • DV is normally distributed
  • Homogeneity of variance
  • Requires dependent samples (and equal sample sizes)

General Model for Z-Test and Single-Sample t-Test

  • Compares a sample to a population
  • Hypotheses are tested against an original population and a treated population

General Model for Independent-Samples t-Test

  • Compares two independent samples to see if they significantly differ
  • Two populations are compared
  • Null hypothesis (HO): μ₁ - μ₂ = 0
  • Alternative hypothesis (HA): μ₁ ≠ μ₂

General Model for Dependent-Samples t-Test

  • Compares two related samples to see if there is a significant difference
  • Compares pairs of scores from the same individuals
  • Null hypothesis (HO): μD = 0
  • Alternative hypothesis (HA): μD ≠ 0

Definitional Formulas

  • Single-Sample t-Test: tobt = (X - μ) / sX
  • Dependent-Samples t-Test: tobt = (D - μD) / sD

t-Tests Formulas

  • Single-Sample t-Test: tobt = (sample mean - population mean) / estimated standard error
  • Dependent-Samples t-Test: tobt = (sample mean difference - population mean difference) / estimated standard error of the mean difference

Steps for a Dependent-Samples t-Test

  • Step 1: Calculate the estimated variance (sD2) of the population using ∑(D - D̄)2 / (n - 1)
  • Step 2: Calculate the estimated standard error (sD) of the mean difference using √sD2 / n
  • Step 3: Calculate the t-obtained (tobt) using (D̄ - μD) / sD

Hypothesis Testing

  • Step 1: State the hypotheses (research and statistical)
  • Step 2: Set the significance level (α = .05) and determine the critical t-value (tcrit)
  • Step 3: Select and compute the appropriate statistic (dependent-samples t-test)
  • Step 4: Make a decision (reject or fail to reject null hypothesis based on comparing tobt and tcrit)
  • Step 5: Report statistical results (t(df) = tobt, p < .05)
  • Step 6: Write a conclusion (IV and DV relationship in words)
  • Step 7: Compute estimated effect size (d)
  • Step 8: Compute r2: Measures the proportion of variance in the DV explained by the IV

Degrees of Freedom

  • Single-Sample t-Test: df = n - 1, where n is the number of scores
  • Dependent-Samples t-Test: df = n - 1, where n is the number of pairs of scores

Effect Size (d) Interpretation

  • 0.2: Small effect
  • 0.5: Medium effect
  • 0.8: Large effect

Percentage of Variance Explained (r2) Interpretation

  • 0.01: Small effect
  • 0.09: Medium effect
  • 0.25: Large effect

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