In Exercises 5 and 6, determine whether you can use a normal d...

Question

In Exercises 5 and 6, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities.

A survey of U.S. undergraduates found that 37% of those attending in-state colleges would prefer to take a job in a different state after graduation. You randomly select 18 U.S. undergraduates attending in-state colleges. Find the probability that the number who would prefer to take a job in a different state after graduation is (c) at least 10. Identify any unusual events. Explain.

Accepted Answer

Welcome back, everyone. A study finds that 29% of university freshmen own a car. If you randomly select 15 freshmen, what is the probability that at least 5 of them own a car? Decide whether the normal approximation is suitable and use it if possible. Is this event unusual? Now there are 3 things worth trying to figure out in this problem. First, we want to know if it is to solve it, we can use a normal approximation. If so, then we're going to solve and find that probability and then determine if it's unusual. So let's first start by checking if the normal approximation is suitable. So from this problem, given that we know the proportion of freshmen P who own a car equals 29% or 0.29%. The sample size N is equal to 15. Then let's see if we can tell if it's suitable by using the test for normal approximations to the binomial. What does that mean? Well, if NP is greater than or equal to 5, and NQ is greater than or equal to 5, where Q is equal to 1 minus P. Then the normal approximation, OK, the normal approximation. Is suitable So is that the case for our problem? Well here Notice that NP. is equal to 15 multiplied by 0.29, which equals 4.35. OK. And NQ would be equal to 15 multiplied by 1 minus 0.29, which is 0.71, and that is 10.65. OK. Since NP is less than 5, OK. And, well, Let me write it another way. Since NP is not greater than our equal to 5, OK. Then the normal approximation. In a normal approximation. Is not suitable. Because now, remember, the condition is that both this and NQ has to be greater than 5. So even though NQ is greater than 5, because NP is not greater than or equal to 5, that means the approximation would not be suitable. So for us to find the probability, we'll need to use something else. We'll need to use the binomial probability formula. So let's go ahead and try this out, OK? Now, here, We're solving for as our problem said, the probability that at least 5 of them own a car. So we're solving for the probability that X is greater than or equal to 5, and we know that that will be equal to 1 minus the probability that X is less than 5. And that probability The probability that X is less than 5 is going to be equal to the sum of all the probabilities before X equals 5. In other words, the probability X equals 0, plus the probability X equals 1 plus the probability X equals 2. That x equals 3. And that X equals 4. OK, so we need to find these 5 probabilities. And then sum them to find a probability X is less than 5. Now what do we know about binomial distributions? Well, recall. Now, the probability X equals X in a binomial distribution is equal to the combination of N and X or N choose X multiplied by P to the pore of X multiplied by Q or 1 minus P to the power of N minus X, OK? So what this means then. Is that the probability X is less than 5, is going to be equal to probability X is equals 0, that's 15 to 0, multiplied by 0.29 to the power of 0, multiplied by 0.71 to the power of 15 minus 0, which is 15. Plus the probability X equals 1, so that's 15 to 1, 0.29 to the power of 1, multiplied by 0.71 to the power of 1415 minus 1. Plus 15 choose 2 supplied by 0.29 squared multiplied by 0.71 to the power of 13. Next, the probability X equals 3, that's 15 to 3. Uh, multiplied by 0.29 cubed. But by 0.71 to the 12th. Plus 15 choose 4 multiplied by 0.29 to the 4th, multiplied by 0.71 to the 11th, OK? So now, this is how we can find the probability that X is less than 5. Now when we go ahead and uh do our calculations here and add all of these probabilities together, then we should find to three significant figures the probability X is less than 5 is equal to 0.550. Therefore, that means the probability that we were searching for that X is greater than or equal to 5. It's going to be equal to 1 minus, the probability X is less than 5, which is 0.55, and that equals 0.45. Therefore, there is a 45% chance that out of our freshmen that we choose, out of the uh a random sample from the university freshmen that we choose, at least 5 of them will own a car. No Is this event unusual? That's the 3rd part of this question. How can we figure that out? We'll recall that an event is unusual if its probability is less than 0.05. Now, since the probability that X is greater than or equal to 5 is equal to 0.45 and 0.45 is greater than 0.05, that means then that this event. Is not. Unusual. Based on what we know about unusual probabilities. So in summary, A normal approximation was not suitable. The probability that at least 5 of them own a car is 0.45, and this event is not unusual. Thanks a lot for watching everyone. I hope this video helped.

��app

Key Concepts

Binomial Distribution

Normal Approximation to the Binomial

Probability Calculation

Watch next