In a survey of U.S. adults, 81% feel they have little or no co...

Question

In a survey of U.S. adults, 81% feel they have little or no control over data collected about them by companies. You randomly select 250 U.S. adults and ask them whether they feel they have control over data collected about them by companies. Use this information in Exercises 11 and 12. (Source: Pew Research Center)

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.

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.

Accepted Answer

Hello there. Today we want 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 survey reports that 73% of drivers use GPS navigation regularly. If you randomly select 120 drivers, determine whether the normal distribution can be used to approximate the binomial distribution. If so, compute the mean and standard deviation. Awesome. So it appears for this particular problem, we're asked to solve for three separate answers, or technically one answer if it turns out we cannot use the normal distribution in this particular case. However, though, if we can use the normal distribution, then we're asked to solve for the mean and standard deviation for this particular problem. So now that we know what we're ultimately trying to solve for, Our first step that we need to take in order to solve this problem is we need to check if we can even use normal approximation. So is normal approximation appropriate in this case? So that's our first step that we need to take to approach this problem. Let us also quickly write down all of our given variables, so we're told that N, the number of trials, is equal to 120. And we're also told that P, which is the probability of success, is equal to 0.73, which is 73% written as a decimal. And let us also recall that Q is equal to 1 minus P, which will be equal to 0.27. So now, at this point, we need to recall that the normal approximation to the binomial distribution is appropriate if both N multiplied by P is greater than or equal to 5, and if N multiplied by Q is greater than or equal to 5. So with that in mind, we need to calculate and multiplied by P, which is equal to 120 multiplied by 0.73 when we plug in our known variables, which will give us an answer of 87.6. Fantastic, and N multiplied by Q will be equal to 120 multiplied by 0.27, which will give us an answer of 32.4. Awesome. So, In conclusion, as you can see, based on the results we just got together, the normal approximation can be used. So I'll put a green check mark. So yes, the normal approximation can be used. So now, moving right along here, since we know that the normal distribution can be used to approximate the binomial distribution in this case, that means we now need to solve for the mean and standard deviation. So let us solve for the mean first. So let us denote the mean as mu, and mu is equal to N multiplied by P, which is equal to 120 multiplied by 0.73, which is equal to 87.6, and that is our final answer for the mean. Hooray, we did it, and that's when we round to one decimal place. So once again, 87.6 is our mean. Awesome. So now let us solve for our standard deviation. So the standard deviation we can denote this value as sigma, and sigma, our standard deviation as we should recall, is equal to the square root of N multiplied by P multiplied by Q, which will be equal to the square root of 120 multiplied by 0.73 multiplied by 0.27. You could also put your values in parentheses so you don't. So you, so it doesn't get confused in your calculator, so you get a more accurate result. You sometimes calculators want everything to be in parentheses. So when you plug in this square root expression into your calculator and you round to two decimal places, you will find that it's equal to 4.86. And that's it. That is our final answer for our standard deviation. Hooray, we did it. So to quickly recap our final answers, our final answers without a doubt have to be the following. The normal distribution can be used. Our mean mu is equal to 87. 60, because you could add a 0 to the end, even though we wrote originally 87.6, you can say 87.60. And I said we rounded to 4.86, but you could also round up to 4.87 because I didn't round technically, but if you wanted to, you could just easily round up to the nearest. Value, so we're gonna since we're rounding the two values. So when you do, you'll get 4.87, 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

Mean and Standard Deviation of a Binomial Distribution

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