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# Calculating a P-value given a z statistic

In a significance test about a population proportion, we first calculate a test statistic based on our sample results. We then calculate a p-value based on that test statistic using a normal distribution.

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• If you calculate the z-score, it will actually be 1.75, not 1.83.
• It is not mention in video, but in practice: Calculating the P-value in a z test for a proportion. How do I know it is 2 tail or 1 tail?
• The P-value equation is misleading here. Whether Ha: is p>26 or p<26, P-value = P(z <= -|1.83|). That is the only way you can validly argue that p≠26 is P-value = 2 * P(z <= -|1.83|).
• Realize P(z ≤ -1.83) = P(z ≥ 1.83) since a normal curve is symmetric about the mean. The distribution for z is the standard normal distribution; it has a mean of 0 and a standard deviation of 1. For Ha: p ≠ 26, the P-value would be P(z ≤ -1.83) + P(z ≥ 1.83) = 2 * P(z ≤ -1.83). Regardless of Ha, z = (p̂ - p0) / sqrt(p0 * (1 - p0) / n), where z gives the number of standard deviations p̂ is from p0.
• Could someone please explain how we came to the conclusion that p-value is just p(z>1.83)?
• At why do we not divide by n - 1 to get an unbiased estimate?
• Because the sampling distribution of the sample proportion, whose standard deviation we're calculating, is itself a population and not a sample. We're not trying to estimate anything there, this is a "true" standard deviation.

Think of it this way: while a single sample is part of a population, several samples are collectively a separate thing, a population of samples.

And because of the central limit theorem, the mean of the sampling distribution will be the mean of the parent distribution:
µ[p̂] = p
µ[x̄] = µ
(1 vote)
• for this question, am i right in saying that the p value is also known as the probability of getting a sample proportion of 1/3, given that the null hypothesis is true?
• hm. Let me think "loud". For sure the test statistic here is z, and so we run the p-value calculation on our test statistic, namely the probability of z being at least as big as in the sample. Now as we got the reference z value from a sample showing 1/3 sample proportion, yes, I would say this is true what you are saying that
P(z at least as this extreme | H0 is true) = P(sample proportion is at least 1/3 | H0 is true)
or at least I can not imagine a different situation how else we could have an at least this large z value from a population of the same size.

any mistake in my logic?
• Why do we decide what kind of p value we're using based on the alternative hypothesis?
e.g. If our Ha was p > 10, then we would have a one tailed p-value of the probability of getting a sample proportion at least as deviant as our actual sample proportion, given that Ho is true.

What's the logic behind this?
• At , Sal says, "...all of that over the standard deviation of the sampling distribution of the sample proportions." Why doesn't he just say "over the standard error"?
(1 vote)
• He might be talking about a different equation, I'm taking this course right now and I hate it.
• In Z-Score Table. P value for -1.83 is 0.0336 but for +1.83 is .96638.. Could you please tell me which one to chose.. but Sal told .0336 for both + and - 1.83.