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 (a) exactly 7. Identify any unusual events. Explain.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A study finds that 29% of university freshmen own a car. If you randomly select 15 freshmen, what is the probability that exactly 5 of them own a car? Decide whether or not applying the normal approximation is suitable for this case and if using it is possible. Is this event considered unusual? Awesome. So it appears for this particular problem, we're asked to solve for three separate answers. Our first answer is we're asked to determine yes or no, whether or not that applying the normal. Approximation, so can we, yes or no, apply the normal approximation for this particular prom based on the conditions set to us by the prom itself? That's our first prompt. Can we, or can we not apply the normal approximation? That's our first answer. Our second answer. And we need to determine if using it. Is possible. And We're trying to, in order to determine whether or not we can use this and if it's possible, we need to determine the probability. So that's our second answer. So is the normal approximation suitable for this problem? Yes or no? What is the probability? That's our second answer. And finally, our last and final answer is we're asked to determine whether or not this event is considered unusual, yes or no. So now that we know what we're ultimately trying to solve for, let's take a moment to write down all of our given information. We're told that the population proportion P is equal to 0.29 in this particular case because we're told that The population 29%, which we need to write 29% as a decimal. We're also told that the sample size N is equal to 15, as we're randomly selecting 15 freshmen. And finally, our last value that we know is the value for X, which is the number of successes of interest, which in this case is equal to 5. You are trying to find the probability that exactly 5 of them own a car. So with that in mind, we're ultimately trying to determine that we want to find the probability P, so we're trying to find the value of P of X is equal to 5. And specifically, this is capital X. So P of capital X is equal to 5. So, as we should recall, this is a binomial probability problem because we are counting how many individuals out of 15 satisfy a yes or no condition. Do they yes own a car or no own a car? So with that in mind, our first step that we need to take is we need to determine whether or not is normal approximation suitable. So as you should recall, the normal approximation to the binomial distribution is suitable when And multiplied by P is greater than or equal to 5 and and N multiplied by 1 minus p is greater than or equal to 5. Fantastic. So let us check these conditions. So N multiplied by P, when we plug in our known variables will be equal to 15 multiplied by 0.29, which will give us an answer of 4.35. And for N multiplied by 1 minus P, you will find that it's equal to 15 multiplied by 1 minus our value for P, which is 0.29, which will give us an answer of 10.65. Fantastic. And since N multiplied by P is equal to 4.35, it is less than 5, so the normal approximation is not appropriate here. So I'm gonna put a red X, so it's not appropriate because 4.35 is less than 5. So, with that in mind, now that we know that the normal approximation is not appropriate, We now need to move along to we need to recall and use the binomial formula now because right now we're trying to solve for our second answer, so we need to recall and use the binomial formula. So as you should recall, the binomial. Formula states that when we use the binomial probability formula, we could state that P X is equal to lowercase x, which is equal to And Over X. Multiplied by P to the power of X, multiplied by 1 minus P. All to the power of N minus X. Fantastic. So with that in mind, we need to substitute in our known variables. So in this case, it's gonna be the probability P of capital X is equal to 5, which will be equal to parentheses 15/5. Multiplied by 0.29 to the power of 5, multiplied by 1 minus 0.29, all to the power of drumroll or value of N. Which is 15 minus our value of X, which is 5. So 15 minus 5 will give us a value of 10. So when we plug in this big old long expression into our calculator and we round to three decimal places, we will find that it's approximately equal to 0.201, and that is one of our final answers. Hooray, we did it. So, we can now start to solve for our last answer whether or not the event is considered unusual. So, as we should recall, a common rule of thumb is that if an event is considered unusual, so an event will be considered unusual, if its probability is less than 0.05%, so 0.05. So 0.05 as a decimal or 5%. Fantastic. So now we need to note that for this particular problem, we're told that P of Capital X, so P of capital X is equal to 5. Fantastic. And as we're told, this value is approximately equal to 0, so it's going to be approximately equal to 0.201, and we need to note that since 0.201 is greater than 0.05. This is not an unusual event, so I'll put a a big X. So because 0.201 is greater than 0.05, this is not considered an unusual event. So, looking back at the prom itself to quickly recap what we've learned, our our final answers without a doubt, have to be the following. The normal approximation is not suitable. P X equals 5 is equal to 0.201. And the event is not unusual, and that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Binomial Distribution

Normal Approximation to the Binomial

Unusual Events

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