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Use the normal approximation to find the probabilities for the specific value(s) of X.a. $n=10, p=0.5, X \geq 7$b. $n=20, p=0.7, X \leq 12$c. $n=50, p=0.6, X \leq 40$

Intro Stats / AP Statistics

Chapter 6

The Normal Distribution

Section 4

The Normal Approximation to the Binomial Distribution

Probability Topics

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Lectures

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Use the normal approximati…

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Check each binomial distri…

Okay, so I outlined the formulas will be using for number six. This is for the mean. This is for the standard deviation, and this is for the Z score. So for part A, the main is gonna be 10 times 100.5, which is just five. And the standard deviation is gonna be the square root of NP, which we said was five times Q, which is one minus p. Okay. And this will give us 1.58 Okay, around to two decimals if you need to. And we want to find the probability that X is greater than or equal to seven, and we have to do the conversion factor. And that converts to the probability that X is greater to 6.5. Okay, you just subtract you just subtract 0.5 from seven. In this case, so 6.5 is gonna be our X value when we find the Z score. So now Z is gonna be 6.5, minus the mean, which we said was five divided by the standard deviation 1.58 Okay. And when you do this in the calculator, you should get 0.95 Okay, just round it off to two decimals, if you need to. Then you look this. You look this value up on the Z score table, you look up 0.95 and the value for 0.95 on the Z score table is 0.8 to 89 All right, Now, this is the probability that X is less than 6.5. We want to know the probability for greater than 6.5. So we have to subtract that value from one. Okay? And we get 0.17 11 Okay, And a convert this to percentage. Just move the desk, fall over to places, and you should get 17.11%. Moving on the part b are mean is gonna be n times p, which is 20 times 200.7. And our standard deviation is gonna be the square root of N p, which we said was 14 times Q, which is one minus p. And when you do this in the calculator, you should get 4.2. All right Now we want to find the probability that X is less than or equal to 12. Okay? And we have to do the correction factor. So for this one, we're going to add 0.5 and remove the equal sign. So X is less than 12.5. Okay, so 12.5 is going to be our X value when we find the Ziese score. Okay. So Zia's X, which we said was 12.5, minus the mean, which is 14 invited by the standard deviation. And that is going to equal negative 0.36 Just round to two decimals, if you need to. All right, so we have to look up negative 0.36 on the Z score table and we get a value off 0.3594 Okay, now, that gives us the probability that X is less than 12.5. And that's exactly what we want. Okay, so just to convert this to percentage, move the decimal over to places and you should get 35.94 percent now for port. See, mule is going to be 50 times 500.6, and that gives us 30. Okay, that's end times p. And for the standard deviation, it's a square root of end times P, which we said was 30 times Q, which is one minus p or one minus 10.6. Okay, Using the calculator, we get 3.46 All right, rounded two decimals. If you need Teoh. And we want to find the probability that X is less than or equal to 40. Okay? And when we do our conversion, we have to add 0.5 and remove the equal sign. So X is going to be less than 40.5. Okay, we did 40 plus 0.5 to get 40.5. Okay, So 40.5 is going to be our X value when we find the Z score. So Z equals 40.5, minus the mean, which we said was 30. Divided by the standard deviation, which we said was 3.46 Okay, we get a value of 3.3 round 22 decimals, if you need to. And when you look up 3.3 on the Z score table, we get a value of 0.9 nine a eight. Okay, Now, this gives us the probability that X is less than 40.5. Okay, so that's exactly what we want. And just converting this to decimal. Move the decimal over to places and you should get 99.88%. Okay, so that is problem six. All right, let me just write this year. This is problem six, or it's a be and see.

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