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 (b) less than 5. Identify any unusual events. Explain.

Accepted Answer

Hello there. Today we're gonna solve the following practice from 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 fewer than 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 ultimately trying to solve for three separate answers. Our first answers we're trying to determine whether or not, yes or no, the normal approximation is suitable for this particular problem. That's our first answer. Our second answer is we're trying to determine the probability for this particular problem. That's our second answer. And our third and final answer we're trying to solve for is we're trying to determine whether or not, yes or no, if this event is considered unusual. So now that we know what we're ultimately trying to solve for, let us take a moment to write down all of our known variables. So we're told that the proportion of freshmen who own a car, which we'll call this value P, is equal to 0.29 because we're told that it's 29% and we need to write 29% as a decimal. We're also told the sample size N is equal to 15, means we're selecting 15 freshmen for this study. And our interest is the probability P, so P of capital X is less than 5. So the probability of X is less than 5, where X in this case is our binomial and this is for where N is equal to 5 and P is equal to 0.29 as we've stated. But let's make a note that. I'll put it in parenthesis here and write it in blue. So let's make a note that where X is our binomial value, which I'll call B for short. Fantastic. So now, our first step is we need to check if the normal approximation is suitable for this particular problem. So can we even use the normal approximation? And as we should recall, in order to use the normal approximation in applying it to a binomial, both of the following conditions must be met and multiplied by P must be greater than or equal to 5. And N multiplied by 1 minus P must be greater than or equal to 5. Fantastic. So now we need to check if these conditions are met. So N multiplied by P, when we plug in our non variables, we will take 15 multiplied by 0.29, and when we plug that into our calculator, we will find that it's equal to 4.35. And if we take N multiplied by 1 minus P, we will find that it's equal to 15 multiplied by 1 minus 0.29, which will give us an answer of 10.65. Fantastic. So let me check, we will find that N multiplied by 1 minus P. is greater than or equal to 5, however, and multiplied by P. is less than 5, and because N multiplied by P. So once again, N multiplied by P is less than 5, so that means in conclusion the normal approximation is not suitable for this case. So I'll put an X. So it's not suitable for this case. So now, moving right along here, our second step now is we need to recall and use the binomial probability formula, which states that the probability P of capital X. is equal to x, which is equal to parenthesis N 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 now, moving right along here. So we now need to note that we're ultimately trying to determine that the probability, so P of capital X is equal to, so once again, We need to be very clear. So P of capital X is less than 5. is equal to P of Capital X. is equal to 0. Plus, P X is equal to 1. Plus P X is equal to 2. Plus capital P X is equal to 3, and finally plus of X is equal to 4. Fantastic. So moving right along here. We need to substitute in our known variables. So we're gonna have a little bit of a long equation here. So let's write it out slowly together. So we have P of capital. X is is gonna be less than 5. Awesome. So this is the probability that X is less than 5. will be equal to parentheses 15 over 0. Multiplied by 0.29 to the power of 0, multiplied by 0.71 to the power of 15. Plus parentheses 15/1 multiplied by 0.29. To the power of 1, multiplied by 0.71 to the power of 14. Plus parentheses 15/2. Multiplied by 0.29. So the power of 2. And we need to multiply this by. 0.71. To the power of 13. Plus parentheses 15/3 multiplied by 0.29 to the power of 3, multiplied by 0.71 to the power of 12 plus 15/4 multiplied by 0.29 to the power of 4 multiplied by 0.71 to the power of 11. And when we perform that big long calculation in our calculator, which you may need to use like an online calculator such as Sinlaw to perform this calculation, or you could break it into parts, but when you plug in this big old long expression, you should find that it's ultimately approximately equal to, let me round to three decimal places, 0.550. Hooray, we did it, and that is one of our final answers. Hooray, we did it. So, moving right along here. Our 3rd step now is we need to determine is the event considered unusual. So, as we should recall, a result is typically considered unusual if the probability is less than 0.05 or 5%. So considering this, we need to note that P X is less than 5. is equal to 0. 550. And this means that 0.550 is greater than 0.05, and because it's greater than 0.05, that means that this event is not considered unusual because our value is greater than 0.05. So once again, this is not unusual. So going back to look at the problem itself, to quickly recap what we've learned, that will mean that our final set of answers without a doubt will have to be the following. So the normal approximation is not suitable, P X is less than 5 is equal to 0.55. 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.

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Key Concepts

Binomial Distribution

Normal Approximation to the Binomial

Unusual Events

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