Z test for Proportions

 

This statistic is a useful and simple measure.  Many times in MR, we want to know if a proportion is equal to one expected, or if proportions from different samples are equal. 

 

For example, suppose that a client says that he will only manufacture a new product if more than 60% of respondents say they will buy it.  A sample of 500 people is selected.  When the data is analyzed, it is found that 62% said they would buy the product.  Is the client safe in making a decision to build the product?

 

The statistic looks like this:

 

 

Where p = the hypothesis tested, in this case p = .60,  po= .62, and q = 1 – p, and n = 500.

 

Z then is equal to 0.913.  Go to the statistical appendix of a MR or statistical text and find the first table, then look up the value of Z = 0.91 (or look up the value of Z on the net).  It is equal to 0.8186.  Subtracting this value from 1.0 equal “sig.”  So “sig” is equal to 1 - .8186 = 0.1814.  We are still looking for “sig” less than .05.  Therefore we would tell our client that there is no evidence that “more than” 60% will buy his product.

 

Z can also be used to compare two groups.  Suppose that a researcher took a sample of 200 students from both ISU and one from UNI.  At ISU, 70% of the students said they thought they got a good education, at UNI the result was 75%.  We assume that the percent is the same at both universities.  The equation is:

 

 

Where p1 = .70 and p2 = .75, and p = (f1+f2)/(n1+n2) or (140 + 150)/(200 + 200) =  .725, and q = 1 – p.

 

Z in this case is equal to  –1.12.  Sig is 1 - .8686  =  0.1314.  We do not have enough evidence to say that ISU and UNI students are different.

 

 

 

What to do?

 

This test does not exist on SPSS.

 

Do the following:

 

Show all work (handwritten).  Show all answers (typed).

 

1.  A coin is thrown 200 times, is it a fair coin if it comes ups heads 58% of the time?

 

2. A sample of 1000 people found that 38% supported a new tax. The supporters said they would pursue the tax only if 35% supported it.  Should they proceed?

 

3. A die is thrown 72 times, the face with a two showed up 19 times, is the die fair?

 

4. In California a sample of 200 people found that 60% liked a new wine, a sample of 300 people in Iowa found that 50% liked it.  Are the two samples different?

 

5.  A random sample in Waterloo found that 70% supported a new sales tax for education.  Only 65% of the people in Cedar Falls supported it.  Are the two populations different?  The Waterloo sample was of 300 people, the Cedar Falls sample was of 100 people.