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One-proportion z-interval

Page history last edited by Shelby Chaffos 6 years ago

A one-proportion z-interval is a confidence interval for the true value of a proportion. Since a proportion only deals with categorical data the one-proportion z-interval is only used when you have categorical data. One-proportion z-interval is used in the Inference rubric in which once the conditions; the random condition, the 10% condition and the success/failure condition, then the confidence interval is calculated. The confidence interval is  where  is a critical value from the standard normal model corresponding to the specified confidence level,  is the sample proportion. q(hat) is 1-, and  is the standard error. 



There are two methods of computing a one-proportion z-interval; one by hand and the other using SPSS. Although both are correct, SPSS gives us a more exact answer.


For example, a study of the effects of acid rain on trees in the Hopkins Forest shows that 25 of 100 tress sampled exhibited some sort of damage from acid rain. We want to find the 95% confidence interval of the true value of this proportion. By hand, we start by checking the conditions which are random, 10% and Success/Failure. The question states that the survey was indeed random, 100 trees is not more than 10% of all the trees in the Hopkins Forest and =25=>10 and q-hat=75=>10. Then we proceed and use the formula,

 =(0.25-1.96(squareroot(0.25)(0.75)/100), 0.25+1.96(squareroot(0.25)(0.75)/100))=(0.1651, 0.3349). We use  we want a 95% confidence interval. If we wanted a 90% confidence interval we would use =1.645 and if we want 99% we would use =2.576. In SPSS, the interval is expressed in the decimal format as portrayed under the "95% confidence area in the Confidence Interval Summary chart on the left side of the photo below. 


One can see that the outcome of SPSS, and concluded that we are 95% confident that the true population of trees damaged by acid rain in captured in the interval 16.9% and 34.7%. The outcomes from both methods. By-hand we are 95% confident that the true population of trees damaged by acid rain is captured in the interval 16.5% and 33.5%. The outcomes from both methods (SPSS and by-hand) are slightly different, because the process that SPSS uses to calculate a Z-Interval is more accurate, which gives us a more precise answer. 



Generating a one-proportion z-interval in SPSS

  • Go to the "Analyze" menu, select "Non-Parametric Test" and click on "One Sample". 
  • In the One-Sample Nonparametric Test window, under "Objective" tab and select "Customize Analysis". 
  • Click "Fields" tab, in the “Test Fields” window make sure the variable of interest is being tested. 
  • Under the "Settings" tab, within “Select an item”, click “Choose Tests” then select “Customize Tests”. Select the option called “Compare observed binary probability to hypothesized (Binomial Test)". 
  • Then go to “Options” underneath "Compare observed binary probability to hypothesized (Binomial Test)". 
  • In the "Binomial Options" window, under the "Confidence Interval", select "Clopper-Pearson (Exact)". Under "Define Success for Categorical Fields" and click "Specify Success values". Under "Value", type in the success in your problem. For this specific example we are using "Damaged" because the interval of trees damaged by acid rain is what our successes are. Be sure to include all capitalization and proper spelling as SPSS is searching for the true value of successes.
  • Click “OK”.
  • Under “Select an item”, click “Test Options” and here you can change the confidence level. For this example, we are computing a 95% confidence interval therefore we are going to leave it.  
  • Click "Run".
  • In the Output window, "Hypothesis Test Summary" Chart will appear, double click on this chart. It opens a new window called "Model Viewer". Go to "View" option (at the bottom of the page) and change from "Hypothesis Summary View" to "Confidence Interval Summary View". The Confidence Interval Summary Chart now appears. 


 The following video illustrates these steps: 


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