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Question 1142670: You are testing the claim that the proportion of men who own cats is larger than the proportion of women who own cats.
You sample 90 men, and 85% own cats.
You sample 50 women, and 40% own cats.
Find the pooled value of p, as a decimal, rounded to two decimal places.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the proportion of men who own cats is 85%.
you get:
p = .85
q = .15
s = sqrt(.85 * .15 * 90) = 3.87476937.
the proportion of women who own cats is 40%.
you get:
p = .4
q = .6
s = sqrt(.4 * .6 * 50) = .0692820323.
in solving your problem, this reference comes in handy.
https://stattrek.com/hypothesis-test/difference-in-proportions.aspx
the pooled value of p is equal to (p1 * n1 + p2 * n2) / (n1 + n2).
this becomes (.85 * 90 + .40 * 50) / (90 + 50) = .6893 rounded to 4 decimal places.
the pooled value of p is used to provide the standard error of the test.
the standard error of the test is equal to sqrt(p * q * (1/n1 + 1/n2)).
p is the pooled proportion.
q is equal to 1 - p
n1 is the sample size of the first sample.
n2 is the sample size of the second sample.
SE = standard error.
in this problem, you get SE = sqrt(.6893 * .3107 * (1/90 + 1/50)) = .0816 rounded to 4 decimal places.
your test statistic is z = (p1 - p2) / SE
in this problem, that becomes z = (.85 - .40) / .0816 = 5.51 rounded to 2 decimal places.
at a .05 one tailed alpha on the right, your critical z-score would be 1.65.
a z-score of 5.51 is well beyond this critical z-score.
you can therefore conclude that the proportion of men who own cats is larger than the proportion of women who own cats.
the results of the test are statistically significant.
this means that the test z-score is greater then the critical z-score.
you were not, however, asked to do all this.
your solution, based on what you were asked, is that the pooled p is equal to .69 rounded to 2 decimal places.
there is a z-score calculator that does all the dirty work for you.
this calculator can be found at https://www.socscistatistics.com/tests/ztest/default2.aspx
this calculator doesn't tell you how to calculate the pooled p.
it does, however, use the pooled value of p to find the standard error.
this, in turn, contributes to the z-score of the test.
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