Independent T Test

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Questions and Answers

What is the null hypothesis (H0) for an Independent-Samples t-Test?

The means of two populations are equal. (correct)

The samples are dependent.

The variances of two populations are equal.

The means of two populations are not equal.

What is required to create the sampling distribution of the difference between the means?

Select two random samples from one population. (correct)

Select samples of different sizes.

Select one random sample and compute its variance.

Select two random samples from different populations.

What does the standard error of the difference between the means represent?

The maximum possible difference between the sample means.

The average distance between the sampling distribution mean and all differences of the means. (correct)

The margin of error in measurements.

The actual difference between the sample means.

For which test is the standard deviation of the population known?

Single-Sample z-Test. Signup and view all the answers

Which condition must be satisfied for conducting an Independent-Samples t-Test regarding the variances?

Homogeneity of variance must be established. Signup and view all the answers

What decision should be made if the calculated t-value (tobt) is greater than the critical t-value (tcrit)?

Reject the null hypothesis (Ho) Signup and view all the answers

In the results, what does t(8) = 2.77, p < .05 imply about students who calculate statistics by hand compared with those who use software for calculations?

They are significantly better at selecting statistical tests Signup and view all the answers

What is the estimated d calculated between the two groups of students?

1.75 Signup and view all the answers

How is the value of r² computed based on the provided t-value and degrees of freedom?

r² = t² / (t² + df) Signup and view all the answers

What conclusion can be drawn from the computed p-value of p < .05 in an independent t-test result?

There is a significant difference between groups Signup and view all the answers

The pooled variance formula is used to estimate what in the context of hypothesis testing?

The variance within two samples Signup and view all the answers

What does the formula for a single-sample t-test primarily compare?

Sample mean and population mean Signup and view all the answers

In a t-test for independent samples, what is the estimated standard error of the difference between the means based on?

Pooled variance of the samples Signup and view all the answers

Which formula correctly represents the calculation of pooled variance for unequal sample sizes?

(SS1 + SS2) / (n1 + n2 - 2) Signup and view all the answers

What is the main purpose of calculating the estimated standard error?

To estimate the reliability of a statistic Signup and view all the answers

In the formula for the independent-samples t-test, what does the term (X1 - X2) represent?

Difference between the sample means Signup and view all the answers

How is the t-value calculated in a single-sample t-test?

By dividing the difference between sample mean and population mean by the estimated standard error Signup and view all the answers

What is one of the first steps in performing a t-test?

State the null hypothesis Signup and view all the answers

What is a key requirement for using the t-test with independent samples?

Samples should be randomly selected Signup and view all the answers

When calculating the t statistic in hypothesis testing, what does a higher t value indicate?

A larger difference between sample means Signup and view all the answers

What do degrees of freedom in the context of t-tests refer to?

Number of scores that can vary while estimating a parameter Signup and view all the answers

Study Notes

Independent-Samples t-Test

Used to compare means of two independent groups
Assesses if there's a statistically significant difference between the groups
Assumes homogeneity of variance (variances of the groups are similar)
Randomly selected samples
DV measured using ratio or interval scale
Group means are known

Single-Sample t-Test

Compares a sample mean to a known population mean
Assesses if the sample mean is significantly different from the population mean
Assumes DV is normally distributed
Randomly selected sample
Ratio or interval scale DV
Population mean is known

z-Test

Tests the difference between a sample mean and a known population mean when population standard deviation is known
Assumes DV is measured using a ratio or interval scale, normally distributed
Randomly selected sample
Population mean and standard deviation are known

General Model

Independent-Samples t-Test: Model shows relationship between sample A, sample B, population A, and population B. The null hypothesis (H₀) states that there is no difference between population A and population B (μ₁ - μ₂ = 0). The alternative hypothesis (H_A) states that there's a difference between population A and population B (μ₁ - μ₂ ≠ 0).
Single-Sample t-Test and z-Test: Model shows relationship between sample, original population, and treated population. The null hypothesis (H₀) states that there is no difference between samples and populations. The alternative hypothesis (H_A) states that there's a difference.

Hypotheses

Null hypothesis (H₀): States no difference between groups or no relationship
Alternative hypothesis (H_A): States a difference between groups or a relationship

Steps

Step 1: State the hypotheses
Step 1a: Decide if it's a one-tailed or two-tailed test.
Step 2: Set the significance level (alpha), usually .05
Step 3: Calculate the appropriate degrees of freedom

-For Independent t-Test:df=(n₁ - 1) + (n₂ - 1)= n₁ + n₂ - 2

-For Single-Sample t-Test: df= n-1
Step 3 (continued): Select and calculate the appropriate statistical test .Calculate Sx, Sx1-X2 etc
Step 4: Make a decision: Determine if the test statistic is in the critical region. Draw a picture.

-If +tobt > tcrit or -tobt < -tcrit → Reject H₀

-If -tcrit < tobt < +tcrit → Retain H₀
Step 5: Report the statistical results (p-value)
Step 6: Write a conclusion: Summarize findings in words, including statistical results
Step 7: Calculate the estimated effect size(d)
Step 8: Calculate the percentage of variance explained (r²)

Formulas

Independent-Samples t-Test: t_obt = (X̄₁ - X̄₂) - (μ₁ - μ₂) / √S²_X₁-X₂
Single-Sample t-Test: t_obt = (X̄ – μ)/S_X
Pooled Variance: S_pool² = (SS₁ + SS₂) / (df₁ + df₂)
Estimated Standard Error: S_X₁-X₂ = √((S_pool²/n₁)+(S_pool²/n₂))

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Description

This quiz covers the fundamentals of Independent-Samples t-Test, Single-Sample t-Test, and z-Test. It assesses your understanding of when and how to use these statistical methods, including their assumptions and applications. Test your knowledge and improve your statistical analysis skills!