Overview

This applet simulates a test of the null hypothesis that the population mean is equal or less than a specified value (e.g., μ ≤ 100). The alternative hypothesis is that the population mean is greater than this value (e.g., μ > 100). We assume a normal population distribution with a known standard deviation (e.g., σ = 15) so we can conduct a one-tailed z test.

The user can enter desired values into the text boxes or use a mouse to ‘drag’ the mean of the Alternate Population, the sample size, or power to see how a change of one value affects others

Place the cursor over different parts on the picture above for a description of applet features.

A: Population distributions and a sample

The population distribution assumed by the null hypothesis is shown in blue (Null Population) while the actual population from which our sample is drawn (Alternative Population) is red. When we click the ‘Sample’ button (C) a sample of n cases represented by yellow boxes is drawn from the Alternative Population. The sample mean is indicated by the red arrow. The mean is also shown by an arrow in B and numeric value in F.

The mean of the Alternative Population can be increased or decreased by ‘dragging’ the distribution with the mouse. Place the cursor near the population distributions, hold the left mouse button, and move left or right. The value of the Alternative Population mean can also be changed by entering a new value in the text box, μ₁, in D.

B: Sampling distributions and a sample mean

The blue sampling distribution is the theoretical distribution of all possible sample means for samples of size n drawn from the Null Population. The comparable distribution for samples drawn from the Alternative Population is shown in pink (and red). The red arrow and black box indicate the mean of the last sample. If the sample mean is greater than a “critical value” (indicated by the vertical red dashed line) we conclude that it is unlikely that the sample was drawn from the Null Population and we reject H₀.

The dark blue area to the right of the critical value represents the ‘alpha’ error set by the data analyst, the probability of rejecting a true null hypothesis. The pink area to the right of the critical value represents statistical power, the probability of rejecting the null hypothesis when we actually are sampling from the Alternative Population.

C: Draw a sample

Click this button to draw a sample of size n from the Alternative Population. The n observed scores are shown as yellow boxes. The mean is indicated with a red arrow in A and B. The numeric value of the mean, the z-score on the Null Population, the p-value, and the decision (reject or fail to reject H₀) are shown in F.

D: Population values, sample size, and power

Values for the population means (μ₀ and μ₁), the standard deviation (σ), the effect size (d), alpha (α) and beta error rates (β), sample size (n), and power may be entered by the user. Highlight the value in a box, replace it with a new value, and press Enter to activate.

In the ‘live’ applet you can click on each term for a brief description.

E: ‘Thermometers’ for sample size n and power

Click on a desired height of the n thermometer to change the sample size from 1 to 100.

Hold down the left mouse button on the Power thermometer and ‘drag’ up or down to change power. A left-click above the level of power will increase sample size by 1, while a click below the power level will decrease the sample size by 1.

F: Sample information

After a sample is drawn this box shows the sample mean, the z and p values associated with the null distribution, the critical value, and the decision (reject or fail to reject H₀).