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# One-proportion z-test

last edited by 5 years, 8 months ago

A one-proportion z-test is a test of the null hypothesis that the proportion of a single sample equals a specified value ($H_{0}:&space;p&space;=&space;p_{0}$) by assigning the statistic

to a standard normal model. This is useful because it allows one to determine whether or not a null hypothesis should be rejected or will fail to be rejected. For any z-test, you must identify the population and the sample, establish hypotheses, identify the correct model (normal in this case) and check the appropriate conditions. The conditions that must be checked are for randomness, the 10% condition, and the success/failure condition. Then you must compute the test statistic, sketch the sampling distribution model, and shade appropriately. You will then calculate the P-value, and state a contextually meaningful conclusion based on the P-value.

For example, suppose that we have some data on the effects of acid rain on trees in the Hopkins Forest. 25 of 100 trees sampled showed some sort of damage from acid rain. This rate seemed to be higher than the 15% quoted in a recent article on the average proportion of damaged trees in the Northeast. Does the sample suggest that trees in the Hopkins Forest are more likely to be damaged by acid rain than trees from the rest of the region?

First, you are going to identify the population and sample. The population is all trees in the Hopkins Forest, and the sample is the 100 trees picked in the forest. You are then going to determine the null hypothesis, and the alternate hypothesis. In this case, the null hypothesis is

and the alternate hypothesis is

Third, you are going to check the three conditions. The problem does not specifically state that the trees are random, and we are unsure of how the trees were selected. We will proceed to do the rest of the test with caution. Because the trees are in a forest, it is certain that 100 trees is less than 10% of the total tree population of Hopkins Forest. Lastly, the calculations for the success/failure condition are as follows

$np&space;=&space;100(0.15)&space;=&space;15&space;$

$nq&space;=&space;100(0.85)&space;=&space;85&space;$

Now we will compute the test statistic.

You then must go to to SPSS, go to "Transform", and select "Compute Variable". From here you will choose "CDF.Normal" and enter the numbers so it shows "1-CDF.Normal(2.78,0,1)". This will give you the P-value. Note that you do "1-CDF" because you are shading to the right, and SPSS shades to the left. The P-value given by SPSS is .003. Since the P-value is less than .05, we are going to reject the null hypothesis. There is sufficient evidence to suggest that trees in the Hopkins Forest are more susceptible to acid rain damage.

If you do the binomial test through SPSS, you will receive slightly different values.

Please note that doing this calculation by hand will result in different results because SPSS is using an exact test, and we are using a normal model to approximate the sampling distribution.

Generating a one-proportion z-test in SPSS:

• Go to the “Analyze” menu and select “Nonparametric Tests”.
• From “Nonparametric Tests”, select “One Sample”.
• On the “Objective” tab, select “Customize analysis”.
• On the “Fields” tab, make sure the proper variable is in the “Test Fields” box.
• On the “Settings” tab, select “Customize tests”, and then the “Binomial Test”.
• Then click on the “Options” button under the “Binomial Test”.
• Change the “Hypothesized proportion” to the null hypothesis value, and select “Specify success values”.
• The success value you enter must be what you are trying to measure, and must be entered exactly how it appears in the data sheet.
• Also on the "Settings" tab, on the left hand side, click "Test Options". Here you can change the significance level if necessary.
• Click “Okay”, then “Run”.
• The "Hypothesis Test Summary" box gives you the decision to reject or to fail to reject the null hypothesis, this is located in the output window.
• Double click the “Hypothesis Test Summary” box to open the "Model Viewer" window.
• The "Model Viewer" window contains the graph and information about the problem.
• "Total N" is equal to the sample size. "Test Statistic" is equal to the observed successes, not the z-score. "Standardized Test Statistic" corresponds to the z-score, but is not the same because SPSS is using exact numbers, while computing by hand is using a normal approximation. Lastly, "Asymptotic Sig." is the p-value.
• Keep in mind the p-value for this example is for a 1-sided test. If it is a 2-sided test, the p-value must be doubled.

The following video illustrates these steps: