Single Sample Tests

Descriptive statistics

Describes the data you collected.

• Shows the center, spread, and shape of the data

• Mean, median, mode, standard deviation, graphs

Inferential Statistics

Uses sample data to make conclusions about a population.

• Helps test hypotheses and estimate population values

• t-tests, chi-square tests, ANOVA, correlation, regression

Hypothesis Testing Cannot Tell Us

• If the design of a study is flawed

• If the data were appropriately collected

• If the sample is representative of the population

• If the theory is true or false

Power

Probability that a study will reject the null hypothesis, when the original (H0) is false. The likelihood of detecting an effect if it exists.

Single Sample Test

Compares one sample to one known or expected value.

• Checks whether the sample is significantly different from that value.

A single-sample test can compare your sample to:

• A hypothetical population mean (e.g., 50)

• A previously known population mean

• A target/expected value

Z-Score

A z-score is a standardized score that shows how far a value is from the mean.

• Data is standardized by converting values into z-scores. mean = 0 std = 1

Hypothesis Testing Steps

1. Identify the population parameter (what you're studying).

2. State the null hypothesis (H₀) (no difference or no effect).

3. State the research hypothesis (H₁) (there is a difference or effect).

4. Choose the correct statistical test (e.g., z-test, one-sample t-test, proportion test).

5. Choose a significance level (usually p = .05) and determine the rejection region.

6. Calculate the test statistic.

7. Decide whether to reject or keep the null hypothesis.

Steps 1–5 are done before collecting data.

Effect Size

Effect size tells you how big the difference is between two groups. Differences between means is Cohen's d.

d = 0.2 small

d = 0.5 medium

d = 0.8 large

P-value: Is there a difference?

Effect size: How big is the difference?

P-Value

• Tells us the p given H0 is true

• Rejecting H0 does not mean that it's false - could still have type I error

• Failing to reject H0 does not mean H0 is true - could still have type II error

Remember: controlling one type of error can increase the other one

Statistical Significance: means that it is unlikely that the results are due to chance alone

Critical Values of Z

Critical values tell you when to reject the null hypothesis

• For a non-directional test, we have both an upper AND a lower boundary (+ - 1.96)

• For a directional test (one tailed), we have an upper OR a lower boundary (-1.645 or +1.645)

One Sample t-Test

• Used to test whether the average of one sample is different from a known population mean.

• The t-value tells you how far your sample mean is from the expected mean.

• The t-distribution depends on the degrees of freedom (df = n − 1).

• As the sample size (and df) increases, the t-distribution becomes more like the normal (z) distribution.

z-test population standard deviation is known.

Randomization Test

Estimates the p-value by repeatedly creating new samples from the original data.

Large sample is when n>30

1. Take a random sample from the original data.

2. Calculate the t-value.

3. Repeat many times.

4. Create a distribution of the t-values.

5. Compare your original t-value to the distribution to get the p-value.

Proportions

A proportion test checks whether a percentage (proportion) is different from an expected percentage or whether two percentages are different.

Example:

You expect 50% of students to prefer online classes. You survey 100 students and find 65% prefer online classes. A proportion test tells you whether 65% is significantly different from 50%.

Z-test and t-Test Comparison

Use a Z-test

• You know the true population standard deviation.

Use a One-Sample t-test

• You don't know the population standard deviation.

• You only have your sample data.