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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?
• Could someone please explain how we came to the conclusion that p-value is just p(z>1.83)?
• 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.
• 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)
• 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.
• Sal used a simple shortcut.
A z table indicates the proportion of the area of the distribution TO THE LEFT of a given z score. Given that normal distributions are by definition symmetric around their means, if we're looking for the area of just one tail in the positives, we can either subtract the proportion given by the z table from 1, or simply look at the corresponding negative z-score. To put it more formally:

P(z ≤ -a) = P(z ≥ +a)

Hope that helps!
• 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?