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College students often make up a substantial portion of the population of college cities and towns. State College, Pennsylvania, ranks first with 71.1% of its population made up of college students. What is the probability that in a random sample of 150 people from State College, more than 50 are not college students?

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Let n = 150, x = 50 p = 0.711, q = 0.289.

First, we must verify that np and nq are at least 5 in order to use the Normal approximation:

nq = 106.65 and np = 43.35.

Now, we compute the mean and standard deviation for the Normal Approximation to the Binomial distribution:

m = np = 43.35s = $\sqrt(npq)$ = 5.55

With this, we may calculate the z-score, keeping in mind that we must add the continuity correction of 0.5 to our fixed sample value x = 50:

z = (50.5 - 43.35)/(5.55) ~ 1.29

We are looking for P(z > 1.29) = 1 - P(z < 1.29).

Using a z-score table, we find that:

P(z < 1.29) ~ 0.9032.

Thus:

P(z > 1.29) ~ 1 - 0.9032 = 0.0968.

\end{document}

Intro Stats / AP Statistics

Chapter 6

The Normal Distribution

Section 4

The Normal Approximation to the Binomial Distribution

Probability Topics

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Okay, Number 16. We have an equals 150. Okay. And we're given that 71.1% or college students. But we want to find the probability that are not college students the percentage that are not college students. So we're going to have to take the compliment. We do one minus 0.711 Okay, that 71.1%. And we get 0.289 Okay, that's how that's the percentage that are not college students. 28.9%. So you have to do that step at the beginning. Okay, So our mean is just n times p. That's gonna be 1 50 times 0.289 And that equals 43.35 Okay, standard deviation. It's a square root of NP. Just 43.35 times one minus p. Okay. And really, instead of one minus p, we could have just said 10.711 cause we already calculated that Anyhow, that is going to equal 5.55 Okay, round to two decimal places. Okay. And we want the probability that X is greater than 50. And when we go to our continuity conversion short when X is greater than a number, you add 0.5 to that number, so 50 plus 0.5 is going to be 50.5. And that is our new X value. And now we can calculate our Z score X. It's 50.5 minus the mean divided by the standard deviation, and we get a Z score of 1.29 When you look that up on the Z score table, you should get 0.9015 Now that's the percentage that is less than 50.5. We want to solve it for greater than 50.5, so we have to subtract that from one. So one minus 0.9015 is gonna give us point 0985 and moving the decimal over to places we get 9.85%. So that's the probability that X is greater than 50

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