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Problem 27 Medium Difficulty

A simple random sample of size 100 is selected from a population with $p=.40$ .
a. What is the expected value of $\overline{p}$ ?
b. What is the standard error of $\overline{p} ?$
c. Show the sampling distribution of $\overline{p}$ .
d. What does the sampling distribution of $\overline{p}$ show?

Answer

a. 0.40
b. 0.049
c. See graph
d. The probability distribution of the sample proportions $\hat{p}$

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Video Transcript

All right. So we have a simple random sample of size 100 were given that the population proportion Is there a point for Oh, so we have some questions. Answer. First off, what's our expected value? A sample proportions. And we know that this is just There we go. This is just the proportion itself. So that gives us 0.40 for part B and wants us to find the standard deviation of the population distribution. And because it doesn't give us a population size, we can assume it is infinite. So that formulas p times one minus p all over our sample size. So that's gonna be 0.60 times 0.40 over 100. Then take the square root of that. When you that that's about 0.49 All right, now it's time to sketch this, So I'm just gonna That's not the most horizontal line in the world, but sure works. So if we're to sketch, this army's gonna be in the middle. And there we go, there we go. I'm gonna scoot just down just so it's a little closer to the and I'm also gonna make this slightly more symmetrical. Their looks kind of rough, but there you have it. Then it's gonna be one standard deviation away. So this is zero point 331 and this is gonna be 0.449 right? No, not 33351 My mistake. And we know that it's gonna be That's 68% of the data right there. Party asks us to. Here we go. Explain what this graph is. What is Ah, the sampling distribution of the sampling sample probabilities. And that is just the probability. Oops. Distribution of the sample proportion. Uh, P hat. I don't think I have a shortcut for the hat on top. So there and there you have it.