Essentials of Statistics Chapter 9
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Essentials of Statistics Chapter 9

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Questions and Answers

What is the formula for calculating the t-statistic?

  • $t = \frac{sM}{M - \mu}$
  • $t = \frac{M - \mu}{sM}$ (correct)
  • $t = \frac{M - \mu}{\sigma}$
  • $t = \frac{n - 1}{M - \mu}$

    • The t distribution has less variability and is narrower than the normal distribution.

    False

    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.

    <p>σ</p> Signup and view all the answers

    Match the following components of the hypothesis testing process with their descriptions:

    <p>State hypotheses = Formulate null and alternative hypotheses, select alpha level Locate critical region = Use t distribution table with df Calculate t statistic = Use the sample mean and the population mean Make a decision = Decide whether to reject or fail to reject H0</p> Signup and view all the answers

    What does an $r^2$ value of 0.09 indicate about the effect size?

    <p>Medium effect</p> Signup and view all the answers

    An $r^2$ value of 0.01 suggests a large effect.

    <p>False</p> Signup and view all the answers

    What is the formula to calculate the estimated population mean ($ u$) using the sample mean (M)?

    <p>μ = M ± tsM</p> Signup and view all the answers

    In a t distribution, values tend to concentrate around the value of ______.

    <p>0</p> Signup and view all the answers

    Match the following $r^2$ values with their corresponding effect sizes:

    <p>0.01 = Small effect 0.09 = Medium effect 0.25 = Large effect 0.00 = No effect</p> Signup and view all the answers

    Under what condition should a t statistic be used instead of a z-score?

    <p>When the sample size is less than 30</p> Signup and view all the answers

    The t statistic requires knowledge of the population standard deviation.

    <p>False</p> Signup and view all the answers

    What does Cohen's d measure?

    <p>Effect size</p> Signup and view all the answers

    The estimated standard error is calculated using the formula __________.

    <p>sM = s / √n</p> Signup and view all the answers

    Which of the following is NOT a tool needed to perform hypothesis testing with t statistics?

    <p>Population mean</p> Signup and view all the answers

    The z-score statistic can be used when the sample size is greater than 30.

    <p>True</p> Signup and view all the answers

    The formula for the t statistic is __________.

    <p>t = (M - μ) / sM</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>t Statistic = An alternative to z-score used when population standard deviation is unknown Cohen's d = A measure of the standardized effect size Standard Error = An estimate of the variability of sample means Hypothesis Testing = A method for making decisions using data</p> Signup and view all the answers

    When n is small (less than 30), the t distribution is primarily characterized as:

    <p>Flatter and more spread out than the normal z distribution</p> Signup and view all the answers

    Two samples from the same population that have the same mean will also have the same t statistic.

    <p>False</p> Signup and view all the answers

    What is one reason why hypothesis test results should be accompanied by a measure of effect size?

    <p>To indicate the size of the treatment effect.</p> Signup and view all the answers

    The estimated Cohen’s d is computed using the formula: d = (M - μ) / ______.

    <p>s</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Cohen's d = A measure of effect size indicating standard deviation differences t distribution = A probability distribution used in hypothesis testing for small sample sizes Effect size = Quantifies the size of a treatment effect Hypothesis test = Determines if a treatment effect is greater than chance</p> Signup and view all the answers

    Compared to the z-score, a hypothesis test with a t statistic requires:

    <p>Less information about the population</p> Signup and view all the answers

    The t statistic is the same as the z-score in terms of requiring population variance.

    <p>False</p> Signup and view all the answers

    What alternative method can be used to measure effect size besides Cohen's d?

    <p>Percentage of variance explained</p> Signup and view all the answers

    What is the t value for df = 9 and α = 0.10?

    <p>1.833</p> Signup and view all the answers

    Increasing the confidence level will decrease the width of the confidence interval.

    <p>False</p> Signup and view all the answers

    What does it mean if a test result is reported as significant?

    <p>The null hypothesis was rejected.</p> Signup and view all the answers

    The confidence interval for the population mean (μ) is calculated using the formula μ = M ± ____ (sM).

    <p>t-value</p> Signup and view all the answers

    Match the following terms with their descriptions:

    <p>t(12) = 3.65 = t statistic with 12 degrees of freedom p &lt; 0.05 = Significance level indicating a 5% threshold two-tailed test = Tests for deviations in both directions one-tailed test = Tests for deviations in one direction</p> Signup and view all the answers

    What is the effect of increasing the sample size on the standard error (SE) of the mean?

    <p>It decreases SE.</p> Signup and view all the answers

    Directional tests are more commonly used than non-directional (two-tailed) tests.

    <p>False</p> Signup and view all the answers

    In a hypothesis test, what indicates that the null hypothesis has not been rejected?

    <p>Not significant result.</p> Signup and view all the answers

    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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    Related Documents

    t-Statistic.pdf

    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.

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