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Questions and Answers
What is the formula for calculating the t-statistic?
What is the formula for calculating the t-statistic?
The t distribution has less variability and is narrower than the normal distribution.
The t distribution has less variability and is narrower than the normal distribution.
False
What does 'df' represent in the context of t-tests?
What does 'df' represent in the context of t-tests?
degrees of freedom
The t-statistic is used to test hypotheses about an unknown population mean when the value of ______ is also unknown.
The t-statistic is used to test hypotheses about an unknown population mean when the value of ______ is also unknown.
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Match the following components of the hypothesis testing process with their descriptions:
Match the following components of the hypothesis testing process with their descriptions:
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What does an $r^2$ value of 0.09 indicate about the effect size?
What does an $r^2$ value of 0.09 indicate about the effect size?
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An $r^2$ value of 0.01 suggests a large effect.
An $r^2$ value of 0.01 suggests a large effect.
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What is the formula to calculate the estimated population mean ($
u$) using the sample mean (M)?
What is the formula to calculate the estimated population mean ($ u$) using the sample mean (M)?
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In a t distribution, values tend to concentrate around the value of ______.
In a t distribution, values tend to concentrate around the value of ______.
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Match the following $r^2$ values with their corresponding effect sizes:
Match the following $r^2$ values with their corresponding effect sizes:
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Under what condition should a t statistic be used instead of a z-score?
Under what condition should a t statistic be used instead of a z-score?
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The t statistic requires knowledge of the population standard deviation.
The t statistic requires knowledge of the population standard deviation.
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What does Cohen's d measure?
What does Cohen's d measure?
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The estimated standard error is calculated using the formula __________.
The estimated standard error is calculated using the formula __________.
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Which of the following is NOT a tool needed to perform hypothesis testing with t statistics?
Which of the following is NOT a tool needed to perform hypothesis testing with t statistics?
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The z-score statistic can be used when the sample size is greater than 30.
The z-score statistic can be used when the sample size is greater than 30.
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The formula for the t statistic is __________.
The formula for the t statistic is __________.
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Match the following terms with their definitions:
Match the following terms with their definitions:
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When n is small (less than 30), the t distribution is primarily characterized as:
When n is small (less than 30), the t distribution is primarily characterized as:
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Two samples from the same population that have the same mean will also have the same t statistic.
Two samples from the same population that have the same mean will also have the same t statistic.
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What is one reason why hypothesis test results should be accompanied by a measure of effect size?
What is one reason why hypothesis test results should be accompanied by a measure of effect size?
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The estimated Cohen’s d is computed using the formula: d = (M - μ) / ______.
The estimated Cohen’s d is computed using the formula: d = (M - μ) / ______.
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Match the following terms with their definitions:
Match the following terms with their definitions:
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Compared to the z-score, a hypothesis test with a t statistic requires:
Compared to the z-score, a hypothesis test with a t statistic requires:
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The t statistic is the same as the z-score in terms of requiring population variance.
The t statistic is the same as the z-score in terms of requiring population variance.
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What alternative method can be used to measure effect size besides Cohen's d?
What alternative method can be used to measure effect size besides Cohen's d?
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What is the t value for df = 9 and α = 0.10?
What is the t value for df = 9 and α = 0.10?
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Increasing the confidence level will decrease the width of the confidence interval.
Increasing the confidence level will decrease the width of the confidence interval.
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What does it mean if a test result is reported as significant?
What does it mean if a test result is reported as significant?
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The confidence interval for the population mean (μ) is calculated using the formula μ = M ± ____ (sM).
The confidence interval for the population mean (μ) is calculated using the formula μ = M ± ____ (sM).
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Match the following terms with their descriptions:
Match the following terms with their descriptions:
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What is the effect of increasing the sample size on the standard error (SE) of the mean?
What is the effect of increasing the sample size on the standard error (SE) of the mean?
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Directional tests are more commonly used than non-directional (two-tailed) tests.
Directional tests are more commonly used than non-directional (two-tailed) tests.
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In a hypothesis test, what indicates that the null hypothesis has not been rejected?
In a hypothesis test, what indicates that the null hypothesis has not been rejected?
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Study Notes
Introduction to the t Statistic
- Use t statistic when the population standard deviation is unknown, rather than the z-score.
- Requires knowledge of sample standard deviation and estimated standard error.
- Hypothesis tests with t-statistics follow similar principles as z-tests.
Hypothesis Testing Review
- Sample Mean (M) estimates the population mean (μ).
- Standard error quantifies the variability between sample mean and population mean.
- Z-score is used when population parameters are known; critical regions depend on normal distribution.
Working with z-Scores
- Z-scores quantify how far sample means deviate from population means.
- Requires population standard deviation (σ), often unknown in practice, which limits research.
Transition to the t Statistic
- t statistic is suitable for estimating population parameters when using sample data.
- Identifies "estimated standard error" (sM) computed from sample standard deviation (s) and sample size (n).
Degrees of Freedom (df)
- Degrees of freedom are calculated as n - 1, reflecting the independence of sample scores.
- Important for accurate statistical analysis with t distribution.
Characteristics of the t Distribution
- Flatter and more spread out compared to the normal distribution; has "fatter tails."
- Family of t distributions corresponds to different degrees of freedom.
Performing Hypothesis Tests with t Statistic
- Conduct four-step hypothesis tests:
- State null and alternative hypotheses, select alpha level.
- Identify critical regions using t distribution tables based on df.
- Calculate t-test statistic.
- Decide on the null hypothesis based on the calculated statistic.
Assumptions of the t Test
- Assumes independence among sample observations.
- The population should ideally be normally distributed, particularly with small sample sizes.
Measuring Effect Size
- Hypothesis testing identifies if a treatment effect exists but not its magnitude.
- Effect size measures provide context for significance:
- Cohen’s d estimates the magnitude of treatment effects.
- Variability accounted for (r²) differentiates between small, medium, and large effects.
Confidence Intervals for Estimating μ
- Construct intervals around sample means to estimate population means.
- Confidence level affects width:
- Higher confidence requires wider intervals.
- Larger samples yield smaller intervals due to reduced standard error.
Reporting in Research
- Results must clearly present significance levels (e.g., p < .05).
- Report t statistic value with degrees of freedom (e.g., t(12) = 3.65) in research findings.
Directional Hypotheses and One-Tailed Tests
- Non-directional (two-tailed) tests are standard, while directional tests may focus on one tail of the t distribution.
- Careful design reflects specific research objectives.
Learning Check Insights
- Understanding distribution shapes and variance in t-tests strengthens statistical reasoning.
- Recognizing the implications of sample sizes impacts the reliability of test results.
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Description
This quiz covers key concepts from Chapter 9 of 'Essentials of Statistics for the Behavioral Sciences.' You will learn when to use the t statistic instead of the z-score, perform hypothesis testing using t statistics, and compute effect sizes such as Cohen's d and percentage of variance. Test your understanding of these statistical fundamentals.
