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Use the normal approximation to the binomial to find the probabilities for the specific value(s) of X.a. n = 30, p = 0.5, X = 18b. n = 50, p = 0.8, X = 44c. n = 100, p = 0.1, X = 12

a. 0.0811 b. 0.0516 c. 0.1052

Intro Stats / AP Statistics

Chapter 6

The Normal Distribution

Section 4

The Normal Approximation to the Binomial Distribution

Probability Topics

Claire A.

June 8, 2021

n=10 , p=0.5 , X?7

Temple University

University of North Carolina at Chapel Hill

University of St. Thomas

Idaho State University

Lectures

0:00

19:36

Use the normal approximati…

04:30

Let x be a binomial random…

08:31

01:02

Compute $P(x)$ using the b…

01:14

02:19

Use a computer to find the…

04:41

Let $x$ be a binomial rand…

01:08

02:53

Check each binomial distri…

Okay, so these are the formulas will be using here on the top. Okay. For part A an equal, sturdy and P equals 0.5. All right, so first we have to solve for I'm you. Okay, That's gonna be our mean, all right? And that's just n times p, which equals 15. Okay. And for our standard deviation sigma, that's gonna be n times P times Q and the square root of the whole thing. But we already did end times p. So that's just 15. Okay. And then one minus p. It's just one minus 10.5. Okay, that's our cue. And the standard deviation is gonna equal 2.74 Okay, You just rounded to two decimal places, all right? Now they want the probability that X equals 18. Okay, but we have to use our continuity correction factor. Okay, so for this one, we're going to subtract 0.5 and add 0.5. Okay, so let's see. So 18 minus 0.5 is gonna be 17.5, okay, and 18 plus 0.5 is gonna be 18.5. All right, So X is just bound by those two numbers All right, so we're gonna have to find out the Z score for each of those. All right, So, Zine One is gonna be X minus mu, which is our mean, whoops. Made a mistake there. All right are mean is actually 15. Okay. And that's gonna be divided by 2.74 That's gonna be divided by sigma. All right, so, Z one, we're going to get 0.91 okay? And you have to look up. You have to look up this Z score on the table. Okay? You have to look at up on the Z score table, and you should get 0.8186 Okay, now for Z two. All right. Now we're going to use the 18.5. That's our X minus 15. That's arm you or mean. And again, we have 2.74 for Sigma. Our standard deviation. Okay, so this is gonna be 1.28 Okay? Just rounded to two decimals, if you need to. All right. And when you look this up on the Ziese on the Z table, the value for 1.2 way is 0.8997 Okay, so Now you just subtract thes two values and take the absolute value, okay? Or just put the larger number first. So this minus this is going to give us point zero a 11 Okay. And converting that 2% that's gonna be 8.11%. Okay, heart be so for part B, you was gonna be n times P. That's 50 times 0.8. Okay. And that's going to give us 40. And then for Sigma, it's gonna be the square root of NP, which we already said was 40 times one minus p. Okay, so we're gonna get eight for our standard deviation. Okay? And we wanted We want to find the probability that X equals 44. So we have to use our continuity correction factor. And that means we have to add and subtract 0.5, and our X value is going to be bound by those two numbers. Okay, So 44 minus 0.5 and 44 plus 0.5 and X is bound by 43.5 and 44.5. Okay, so now we have to do to two values for Z. Okay? It doesn't matter which one is the one for which one is easy to. Okay, so that's gonna be X, which is 43.5, minus the mean, which is 40 divided by our standard deviation, which is eight. Okay. And for Z two, we're going to use this value for X 44.5, minus 40. Invited by A. Okay, so for this, I got 0.44 rounded off the two decimals, if you need to. And for this, I got 20.56 Okay, so you need to use the table, the Z score table, okay. And when you look up 0.44 you should yet 0.6700 When you look up 0.56 you should get points 7123 Okay. And the probability is in between these two. So you just subtract thes two numbers and take the absolute value Or just put the larger number first and subtract. So I got 0.4 to 3. And when you convert that to percentage, just move the decimal over to places, and that is 4.23%. Okay? Finally port. See, we have mu equals. Ah, 100 times 1000.1, which is 10. Okay. And we have Sigma equals the square root of N P. Q. Okay, this is and this is P. So we already have 10 for NP and cue is just one minus p. So that is going to equal three. All right now they want the probability that X equals 11. I'm sorry, X equals 12 okay? And we have to use our continuity correction. So we add and subtract 0.5. All right, and X is bound by those two numbers, so 12 minus 120.5 and 12 plus 120.5. Now we do our ZEEZ OK. It doesn't matter which one is Z one or Z two. All right, we have X, which is 11.5 minus mu, which is 10 divided by three. And for Z two, we have X, which is 12.5 minus 10 divided by three. Okay, so this is gonna be 0.50 So round off to two decimal places if you need to. And this is going to be 0.83 All right now. Yet this is disease score. You have to look up the value on the Z table. Okay, so for 0.50 we get 0.69 15 Okay. And for 0.83 we get 0.7967 Okay, So once again, you just subtract thes two values, all right? And take the absolute value and you should get 0.1052 In order to convert this to decimal, just move the desk point over two places, so that will be 10.52 percent.

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