Hypothesis Testing and t-Distribution: Key Concepts and Formulas

What is the purpose of using a z-score in hypothesis testing?

To determine whether a sample is part of a population with known mean (µ) and standard deviation (s).

What does the sample mean (M) approximate?

The population mean (µ).

What is the standard error in hypothesis testing?

A measure of how much difference to expect by chance.

Why do we use the t-statistic instead of the z-score?

Because the population standard deviation (s) is rarely known.

What does the t-statistic use instead of the population standard error?

The estimated standard error from the sample.

What is the formula for the t-statistic based on?

The sample variance and the estimated standard error.

What does the degrees of freedom (df) indicate?

How well the sample standard deviation (s) approximates the population standard deviation (s).

What is the formula for degrees of freedom in a sample?

df = n - 1.

How does the t-distribution change with increasing degrees of freedom?

It approaches the z-distribution.

What is the critical region in hypothesis testing?

The area determined by the alpha level where we reject the null hypothesis (Ho).

What is the first step in hypothesis testing?

State the hypotheses (Ho and H1).

What does Ho represent in hypothesis testing?

The null hypothesis, stating that the treatment has no effect.

What is the significance of the p-value?

It is the probability of observing a sample statistic as extreme or more extreme than the one observed.

What does it mean if the p-value is less than alpha (0.05)?

We reject the null hypothesis (Ho).

What is the formula for calculating the t-statistic?

t = (M - µ) / sM.

What does Cohen's d measure?

The effect size, indicating the magnitude of the difference between groups.

What is the formula for calculating r²?

r² = (Ho variance - actual variance) / Ho variance.

What is the confidence interval (CI) in hypothesis testing?

A range of values that is likely to contain the population mean.

What is the critical value for a two-tailed test with df = 10 and alpha = 0.05?

± 2.179.

What does it mean if the t-obtained value is greater than the t-critical value?

We reject the null hypothesis (Ho).

How is the sample variance (s²) calculated?

s² = SS / (n - 1).

What is the estimated standard error (sM)?

It is based on the sample statistic rather than the population parameter.

What is the significance of a two-tailed test?

It tests for the possibility of an effect in two directions (greater than or less than).

What does a t-test assess?

Hypotheses about the population mean when the population standard deviation is unknown.

What is the null hypothesis in the Pretty Students example?

Ho: µ = 4.

What is the alternative hypothesis in the Pretty Students example?

H1: µ ≠ 4.

What is the critical value for a two-tailed test with df = 20?

± 2.086.

What is the relationship between sample size and the approximation of the t-statistic to a normal curve?

Larger sample sizes provide a better approximation.