A critical value is a predetermined threshold on a statistical distribution used to make a decision about the null hypothesis. Derived from the specified alpha level (such as $$\alpha = 0.05$$) and the degrees of freedom, it marks the boundary of the most extreme expected scores. If a calculated test statistic falls beyond this critical value, the result is deemed statistically significant, leading to the rejection of the null hypothesis.

Critical Value

How are the degrees of freedom ($$df$$) calculated when conducting a one-sample $$t$$-test?

When conducting a one-sample $$t$$-test, the degrees of freedom ($$df$$) are calculated by subtracting one from the total sample size ($$N - 1$$). This specific calculation dictates the exact shape of the $$t$$ distribution, enabling researchers to accurately identify the $$p$$ value associated with their test statistic.

Google

The $$t$$ distribution illustrates the spread of possible $$t$$ scores that researchers would expect to observe if the null hypothesis were entirely true. This distribution forms a unimodal and symmetrical curve centered precisely around a mean of $$0$$. By utilizing this known distribution curve, researchers can evaluate any computed $$t$$ score to find its exact $$p$$-value, which represents the proportion of expected scores that are at least as extreme as the observed sample result.

Distribution of t Scores

The one-sample $$t$$-test is a statistical procedure used to compare a single sample mean ($$M$$) against a specific, hypothetical population mean ($$\mu_0$$) that serves as an interesting standard of comparison. By defining a null hypothesis ($$\mu = \mu_0$$) and an alternative hypothesis ($$\mu \neq \mu_0$$), researchers use this test to determine if the sample data provide enough statistical evidence to conclude that the true population mean differs from the hypothesized value.

One-Sample t-Test

Degrees of freedom is a core statistical concept that dictates the precise mathematical shape of a test statistic's distribution when the null hypothesis is true. By determining this exact structural shape, the degrees of freedom make it possible for researchers to accurately calculate the probability, or $$p$$ value, associated with any given test score.

Degrees of Freedom

https://kpu.pressbooks.pub/psychmethods4e/

KPU Research Methods in Psychology - 4th American Edition

Degrees of Freedom (One-Sample t-Test)

To illustrate finding a $$p$$-value, consider a sample of $$25$$ individuals (which provides $$24$$ degrees of freedom) resulting in a computed $$t$$ score of $$1.50$$. The probability of obtaining a $$t$$ score at least this extreme depends on the proportion of expected scores in the corresponding distribution that are far from zero. Defining extreme in both directions, the $$p$$-value is the proportion of expected $$t$$ scores that are $$1.50$$ or greater, combined with those that are $$-1.50$$ or lower. In this scenario, that combined proportion is $$.14$$, meaning the final $$p$$-value is $$.14$$.

Example of Finding a p-Value for a t Score

What does the t distribution specifically illustrate?

Statement that says there is no relationship between the two measured items

Null Hypothesis

Statement that says something is HAPPENING! New theory is true instead of an old one

Alternative Hypothesis

1. The dependent variable must be continuous (interval/ratio).
2. The observations are independent of one another.
3. The dependent variable should be approximately normally distributed.
4. The dependent variable should not contain any outliers.

Assumptions

To conduct a one-sample $$t$$-test, researchers compute the $$t$$ statistic using the exact formula: $$t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$$. In this mathematical equation, $$M$$ represents the sample mean, $$\mu_0$$ is the hypothetical population mean serving as the standard of comparison, $$SD$$ is the standard deviation of the sample, and $$N$$ indicates the total sample size.

Formula for the One-Sample t-Test

The dependent-samples $$t$$-test, sometimes called the paired-samples $$t$$-test, is used to compare two means obtained from the same sample tested at two different times or under two distinct conditions. This statistical comparison is particularly appropriate for analyzing data from pretest-posttest designs or within-subjects experiments. Conceptually, it functions as a special case of the one-sample $$t$$-test.

Dependent-Samples t-Test

In a one-sample $$t$$-test, the null hypothesis posits that the true population mean ($$\mu$$) is identical to the hypothetical population mean ($$\mu_0$$), expressed formally as $$\mu = \mu_0$$. Conversely, the alternative hypothesis argues that the actual population mean significantly deviates from this expected standard of comparison, formalized as $$\mu 
eq \mu_0$$.

Null and Alternative Hypotheses for the One-Sample t-Test

To demonstrate a one-sample $$t$$-test, imagine a health psychologist investigating whether university students accurately estimate the $$250$$ calories in a chocolate chip cookie. The null hypothesis ($$\mu = 250$$) assumes accurate estimation. The researcher asks a sample of $$10$$ students, resulting in a sample mean ($$M$$) of $$212.00$$ calories and a standard deviation ($$SD$$) of $$39.17$$. Plugging these statistics into the one-sample $$t$$-test formula yields a $$t$$ score of $$-3.07$$. For a two-tailed test with $$9$$ degrees of freedom, the critical value is $$\pm 2.262$$. Because the calculated $$t$$ score is more extreme than this critical value, the researcher rejects the null hypothesis and concludes that students significantly underestimate the cookie's calories. If the researcher had a strong theoretical reason to expect an underestimation, a one-tailed test with a less severe critical value of $$-1.833$$ could have been used, making it easier to reject the null hypothesis if the results trended in the expected direction.

Example of a One-Sample t-Test

What is the primary purpose of a one-sample t-test?

Which core statistical concept dictates the precise mathematical shape of a distribution when the null hypothesis is true, thereby allowing researchers to calculate the probability, or $$p$$ value, associated with a given score?

Learn Before

Related

Learn After