Explain a test of proportions like I'm 5.
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Okay, imagine you have a bag with candy. Some are red and some are green. You want to know if you have the same amount of red and green candies or not. To find out, you take out a few candies and count the number of red and green ones. If there are the same number of red and green candies, the proportion of red and green candies in the candies you took out should be the same as the proportion of red and green candies in the whole bag.

This is called a test of proportions. It helps you figure out if two things are equal or not. It's just like a test you might take at school to see if you understand a lesson. But instead of testing what you know, it's testing if the red and green candies are equal.
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Show me the equation for it, along with dummy data.
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The formula for a test of proportions is:

Z = (p1 - p2) / sqrt(p * (1 - p) * (1/n1 + 1/n2))

where:
Z is the test statistic, which measures how far the observed proportion is from the expected proportion.
p1 and p2 are the proportions of successes in the two groups.
p is the average of p1 and p2.
n1 and n2 are the sample sizes in the two groups.

Let's say we have a sample of 100 candies, and we count that 40 of them are red and 60 are green. We can use the formula like this:

p1 = 40/100 = 0.4 (proportion of red candies)
p2 = 60/100 = 0.6 (proportion of green candies)
p = (p1 + p2) / 2 = 0.5
n1 = 100 (sample size)
n2 = 100 (sample size)

Z = (0.4 - 0.6) / sqrt(0.5 * (1 - 0.5) * (1/100 + 1/100)) = -2.236

The resulting Z score of -2.236 tells us that the proportion of red candies is significantly lower than the proportion of green candies.
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