Approximating a Binomial Distribution In Exercises 17 and 18, ...

Question

Approximating a Binomial Distribution In Exercises 17 and 18, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.

Bachelor’s Degrees Twenty-two percent of adults over 18 years of age have a bachelor’s degree. You randomly select 20 adults over 18 years of age and ask whether they have a bachelor’s degree.

Accepted Answer

Hello there. Today we're going to 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 poll finds that 35% of voters support a new policy. If you randomly select 25 voters, can you use the normal distribution to approximate the binomial distribution for the number of supporters? If so, find the mean and standard deviation. If not, explain why. Awesome. So it appears for this particular problem we're ultimately trying to solve for three separate answers, ideally. So our first answers are trying to determine whether or not, yes or no, we can use the normal distribution to approximate the binomial distribution for the number of supporters. That's our first answer we're trying to solve for. And if we can use the normal distribution. We need to determine our 2nd and 3rd answers, which we need to figure out the mean and the standard deviation. Awesome. And if not, we, if we cannot use the normal distribution, you need to explain why. Awesome. So now that we know what we're ultimately trying to solve for, our first step that we need to take is we need to check whether or not we can use the normal approximation. We need to determine if the normal approximation is appropriate for this particular problem. So with that in mind, We need to use the normal approximation to so once again we're trying to use the normal distribution to approximate the binomial distribution, and we could do that if both N multiplied by P is greater than or equal to 5, and if N multiplied by parentheses 1 minus P is greater than or equal to 5. Awesome. So, with that in mind, we need to note that in this particular case, N is equal to 25, because that is the number of the population, the number of people that we're sampling. So in this case, once again, N is equal to 25, and P, in this case is our percentage, and we're told that it's 35%, but we need to write 35% as a decimal. So P in this case will be equal to 0.35. Fantastic. So now we need to solve for N multiplied by P. So N multiplied by P is equal to 25 multiplied by 0.35, which is equal to, let me plug that into our calculator, 8.75. And we need to note that N multiplied by parentheses 1 minus P is equal to 25 multiplied by 1 minus 0.35, which will give us an answer of when we plug it into our calculator. 16.25. So, looking at our two answers, we will find that both values are indeed greater than 5, so the normal approximation is appropriate, so we'll put a green check mark and say, yes, it is appropriate. So, moving right along here, so since we can use the normal approximation, since it's appropriate, we now need to determine the values for the mean and the standard deviation. These those are our last two answers we need to solve for. So we need to find the mean of the binomial distribution first. So as we should recall, the mean mu is equal to N multiplied by P, which is equal to 25 multiplied by 0.35, which is equal to, once again, 8.75. So that's one of our final answers. Hooray, we did it. So now we need to solve for the standard deviation of the binomial distribution, and as we should recall, the standard deviation, sigma. is equal to the square root of N multiplied by P multiplied by 1 minus P, which is equal to the square root of 25 multiplied by 0.35 multiplied by 1 minus 0.35. And when we plug that into our calculator and we round to two decimal places, we will find that it's approximately equal to 2.38, and that is our third and final answer. Hooray, we did it. So therefore, without a doubt, to quickly recap our answers, our final three answers without a doubt will have to be once again. Yes. And mu, our mean, is equal to 8.75, and our standard deviation sigma is equal to 2.38. 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

Mean and Standard Deviation of a Binomial Distribution

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