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# Hypothesis test for difference in proportions

Hypothesis test for difference in proportions.

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• I didn't understand why we used combined sample proportion here ? And also what if combined sample proportion was not given? Couldn't we just use both the proportion separately and use the standard error formula
• Yah I'm not exactly sure what he did there, but yes you can just use the standard error formula and use the proportions separately. Both methods give the same exact answer.
• I'm slightly confused why our Z score calculation uses the difference between p^A vs. p^B - when the standard deviation is calculated using p^C (or the combined p^ of A & B samples).

In my mind - we're saying 'is it reasonable that A & B have come from the same sample?'. If they have, the difference between P^A and the estimated overall sample p^ (p^C) will likely not be more than 1 and a bit standard deviations - and that's why we calculate the standard deviation value of a combined sample.

I'm confused why we're then measuring the difference between p^A and p^B and not measuring the difference between p^A and p^C? I understand that my thinking is wrong, I'm just not sure why!
• The problem statement does not state that the sampling was random. Is that another reason to conclude that sufficient evidence does not exist here?
• I have a question. To use the formula sig(A-B) = sqrt (sig(A)^2 + sig(B)^2) it is supposed there is no dependency or relationship between A and B.
Besides the 10% rule or independency for each variable.

So, what is the solution if there is some relationship between A and B.
• I think there might be a problem here... In the video, Sal writes (.55(.45))/100 twice and doesn't add the variance of the other sample... Is this right or wrong?
• It is right; according to H0 p1=p2=approx 0.55
this yields the same Variance, but only because N1=N2=100.
(1 vote)
• Why z = (p_hat_a - p_hat_b)/SE?
(1 vote)
• Because it is a test statistic.
Remember the 𝒛 for any test statistic is =
(Estimator﹣Null) / SE

Let's focus on the numerator (Estimator﹣Null):
∙ The "estimator" in this case is the difference between proportions. This is what we are trying to estimate from the question. Thus,
Estimator = p̂₁﹣ p̂₂

∙ The "null" in this case is zero. Because we are assuming both proportions are the same and equal, meaning there is no difference. Thus,
Null = 0

Now to put them all together, the numerator becomes:
Estimator ﹣ Null
(p̂₁﹣ p̂₂) ﹣ 0

Since it is minus 0 at the end we can always leave it out, which is what Sal passed over.
So the numerator just becomes:
p̂₁﹣ p̂₂

As for the denominator, SE (Standard Error) acts as the Standard Deviation which is always the denominator of 𝒛

So now we get the formula:
𝒛 = (p̂₁﹣ p̂₂) / SE
• Hey, to find out significance of a difference in proportion test, don't we first need to calculate the sample size required at some alpha and power ?
(1 vote)
• I would argue that the Variance of the Difference of 2 Samples with Size 100 each would result in