[Math] AP Statistics practice test question: Why is a two-proportion z-test invalid

statistical-inferencestatistics

The question is as follows:

A pollster is interested in comparing the proportions of women and men in a particular town who are in favor of a ban on fireworks within town borders. The pollster plans to test the hypothesis that the proportion of women in favor of the ban is diﬀerent from the proportion of men in favor of the ban. There are 4,673 women and 4,502 men who live in the town. From a simple random sample of 40 women in the town, the pollster finds that 38 favor the ban. From an independent simple random sample of 50 men in the town, the pollster finds that 27 favor the ban. Which of the following statements is true about this situation?

(A) Because the samples are from normal populations, a two-proportion z-test would be valid.

(B) Because the size of each sample is greater than 30, a two-proportion z-test would be valid.

(C) Because the number who favored the ban is greater than 10 in both groups, a two-proportion z-test would be valid

(D) Because of the relative sizes of the populations and samples, a two-proportion z-test would be valid.

(E) A two-proportion z-test would not be valid for these data.

I know the answer is E, but why? I'm assuming the questions thinks a X^2 test of homogeneity should be used instead but can I determine that? What makes a two-proportion z-test invalid? p-bar * n1, q-bar * n1, p-bar * n2, q-bar * n2 are all greater than 5 so the conditions for a two-proportion test seem to be met. What am I missing?

Thanks a lot!

Best Answer

HINT: You are testing between two proportions here. Look at your number of successes for females versus the total number of samples collected for females. Now, look at the requirements for using a z-test for two proportions.

Related Solutions

Two sample z test: Applicability

Here is Minitab output for fake data in such a table. I did not try to match the percentages you give in your problem. The null hypothesis is that recognition of GB and of France are independent abilities. The small p-value indicates the null hypothesis is rejected.

Chi-Square Test for Association: France, GB 

Rows: France   Columns: GB

         Yes     No  All

Yes       43     21   64
       33.58  30.42
       2.640  2.915

No        10     27   37
       19.42  17.58
       4.566  5.042

All       53     48  101

Cell Contents:      Count
                    Expected count
                    Contribution to Chi-square

Pearson Chi-Square = 15.163, 
    DF = 1, P-Value = 0.000

Computations:

The observed count for the upper-left cell is $X_{11} = 43.$

The expected count for the upper-left cell is $E_{11} = 64(53)/101 = 33.58.$

The contribution for that cell is $(X_{11} - E_{11})^2/E_{11} = 2.64.$

The chi-squared statistic $15.163$ is the sum of the 'contributions' from all four cells.

From the row with DF=1 in a printed table of chi-squared distributions, you can see that the value $3.8415$ cuts 5% from the upper tail of the distribution $\mathsf{Chisq}(1),$ so that any value of the chi-squared statistic above 3.8415 would lead you to believe that identification of GB and identification of France are not independent abilities (at the 5% level of significance). The chi-squared statistic here is $15.16 > 3.84.$

Perhaps you can find a more complete discussion of this kind of test later in your text.

Addendum. Suppose my data are real. In these data, the 43 + 27 who got both countries right or neither, you have no info whether GB or France is easier to identify on a map. Of the other 31, who got exactly one country right, there are only 10 who got only GB wrong.

Those 10 are in the lower tail of $\mathsf{Binom}(31, .5).$ That is, assuming both countries are equally easy to identify, there is only probability 0.0354 < 5% that 10 or fewer get only GB wrong. I would hesitate to draw strong conclusions from only 31 useful responses, but there does seem to be evidence more people recognize GB than France on a map. (That wouldn't be surprising, because many people know GB is an island nation, and there aren't many big islands on a map of Europe.)

In R:

pbinom(10, 31, .5)
[1] 0.03537777

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