Introduction to the t Statistic

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Questions and Answers

What is the critical value for a one-sample t-test at α = 0.05 with 8 degrees of freedom?

± 2.262

± 2.306 (correct)

± 3.107

± 1.645

What action is taken when the obtained t statistic falls into the critical region?

Fail to reject the null hypothesis

Reject the null hypothesis (correct)

Accept the alternative hypothesis

Increase the sample size

Which of the following statements about sample size is true?

Smaller samples always yield more reliable results.

Larger samples provide smaller standard errors. (correct)

Larger sample sizes reduce the likelihood of obtaining a significant treatment effect.

Larger sample sizes generally increase measurement error.

What is a key reason for calculating effect size after a hypothesis test?

To understand the magnitude of the treatment effect Signup and view all the answers

Which measure is NOT typically used as a form of effect size in statistical analysis?

Alpha error Signup and view all the answers

What does a high variance indicate in the context of hypothesis testing?

Decreased reliability of test results Signup and view all the answers

How is the effect size related to the null hypothesis mean in the one-sample t-test?

It shows the difference between the treatment effect and the null hypothesis mean. Signup and view all the answers

In a hypothesis test, what does a significant t statistic indicate?

The treatment effect is significantly greater than chance. Signup and view all the answers

What is the first step in conducting a t test for hypothesis testing?

State the null and alternative hypotheses and select an alpha level Signup and view all the answers

In Example 9.2, what was the average score of the sample of research participants?

46 Signup and view all the answers

What is the null hypothesis in the study by Chang et al. regarding light-emitting eReaders?

Late-night reading from a light-emitting screen has no effect on alertness Signup and view all the answers

What is the level of significance used in the study for the hypothesis test?

0.05 Signup and view all the answers

How many degrees of freedom (df) are used when the sample size (n) is 9?

8 Signup and view all the answers

Which of the following is not a step in the t testing procedure?

Calculate the effect size Signup and view all the answers

What method is used to find the critical values for the t distribution?

Utilizing a t table for specified degrees of freedom Signup and view all the answers

What was the sample size (n) of volunteers in the study examining the effect of reading from an eReader?

9 Signup and view all the answers

What happens to the width of the confidence interval when more confidence is desired?

The width increases Signup and view all the answers

What is the meaning of a directional hypothesis?

It specifies which direction the effect will occur Signup and view all the answers

In hypothesis testing, what does a t statistic of -2.67 indicate if it falls into the critical region?

Reject the null hypothesis Signup and view all the answers

How does sample size affect the standard error (SE) and subsequently the confidence interval?

Larger samples lead to smaller SE Signup and view all the answers

What is the null hypothesis in the case presented regarding alertness scores?

Reading from a screen will have no effect on alertness scores. Signup and view all the answers

What does the critical region represent in a hypothesis test?

The area where the t statistic must fall to reject the null hypothesis Signup and view all the answers

If the p-value is less than .05 in a hypothesis test, what conclusion can be drawn?

Reject the null hypothesis Signup and view all the answers

Which of the following statements about confidence intervals is true?

A smaller sample size will result in a wider confidence interval. Signup and view all the answers

What does the t statistic estimate in hypothesis testing?

The deviation of sample means from the population mean Signup and view all the answers

When is the t statistic typically used?

When the population variance is unknown Signup and view all the answers

What is the formula for the t statistic?

$t = \frac{M - \mu}{s_M}$ Signup and view all the answers

In the context of the t distribution, what does 'degrees of freedom' refer to?

The number of independent scores in a sample Signup and view all the answers

What is a characteristic of the t distribution compared to the normal z distribution?

It is flatter and more spread out Signup and view all the answers

Which assumption must be satisfied to effectively use a one-sample t test?

The population data must be normally distributed Signup and view all the answers

What does a 'fatter tails' characteristic in the t distribution indicate?

Increased probability of extreme values Signup and view all the answers

What does the term 'estimated standard error' refer to in the t statistic formula?

The standard deviation of the sample mean across samples Signup and view all the answers

What does a larger value of d indicate regarding its effect?

It indicates a larger effect size. Signup and view all the answers

What does r^2^ = 0.25 represent in terms of effect size?

Large effect Signup and view all the answers

Which step is NOT part of estimating the value of a population mean using confidence intervals?

Determine the sample size needed. Signup and view all the answers

In the context of one-sample t tests, what is the role of treatment?

It refers to any unique characteristic or manipulation of the sample. Signup and view all the answers

What percentage of variability in cognitive alertness was explained by using an e-reader before bed?

46% Signup and view all the answers

Which of the following describes a medium effect size according to r^2^?

0.09 Signup and view all the answers

How many steps are involved in estimating the value of a population mean using confidence intervals?

Three Signup and view all the answers

What is the purpose of calculating confidence intervals in research?

To provide a range of values likely containing the population mean. Signup and view all the answers

Study Notes

The t Statistic: An Introduction

The t statistic estimates how far a sample mean deviates from the population mean stated in the null hypothesis.
It's used when the population standard deviation (σ) is unknown.
The t distribution is a family of distributions used when the population standard deviation is unknown, with a specific distribution for each degree of freedom.
The t distribution is flatter and more spread out than the normal distribution.
Degrees of freedom (df) represent the number of independent scores in a sample and are calculated as df = n - 1.

Hypothesis Testing with the t Statistic

The one-sample t test is used to test hypotheses about a single group mean when the population variance is unknown.
It relies on assumptions of normality, random sampling, and independence of data.
The four steps of hypothesis testing using the t statistic include: stating hypotheses, setting criteria, calculating the t statistic, and making a decision.
The critical region is defined using the t distribution table and the alpha level (α).

Measuring Effect Size for the t Statistic

Effect size measures the magnitude of the treatment effect.
Cohen's d is a common measure of effect size, representing the difference between the sample mean and the population mean in terms of standard deviations.
Measures of proportion of variance, such as eta-squared (η2), represent the percentage of variability in the dependent variable explained by the treatment effect.
Effect size is typically calculated only after rejecting the null hypothesis.

Confidence Intervals

Confidence intervals provide an estimate of the population mean by constructing an interval around the sample mean.
The wider the interval, the less precise the estimate of the population mean.
The width of the confidence interval is influenced by the level of confidence and the sample size.

Directional Hypotheses and One-Tailed Tests

One-tailed tests are used for directional hypotheses, where the researcher predicts a specific direction for the effect (e.g., treatment will increase scores).
The critical region is defined in only one tail of the t distribution for a one-tailed test.
Directional hypotheses are appropriate for exploratory investigations or pilot studies.

Factors Affecting Hypothesis Testing Results

Larger sample sizes reduce the standard error, increasing the likelihood of rejecting the null hypothesis.
Larger sample variances increase the standard error, making it less likely to reject the null hypothesis.

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Description

This quiz covers the fundamentals of the t statistic, including its use in estimating deviations from the population mean when the population standard deviation is unknown. It also addresses hypothesis testing with one-sample t tests and the importance of degrees of freedom. Test your understanding of these statistical concepts!