In Exercises 69 and 70, determine whether you can use a normal...

Question

In Exercises 69 and 70, 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. adults found that 72% used a mobile device to manage their bank account at least once in the previous month. You randomly select 70 U.S. adults and ask whether they used a mobile device to manage their bank account at least once in the previous month. Find the probability that the number who have done so is (c) greater than 60.

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 health survey finds that 80% of adults exercise at least once a week. If you randomly select 60 adults, what is the probability that more than 55 exercise weekly? Awesome. So it appears for this particular prompt, we're asked to determine if we were to select 60 adults, what is the probability that more than 55 out of these 60 adults survey surveyed exercise weekly. So, what is the probability that more than 55 of these 60 adults exercise weekly and that is our final value we're ultimately trying to solve for? So now that we know that we're trying to determine this probability value, let's read off our multiple choice answers to see what our final answer might be. A is 0.27078. B is 0.78007. C is 0.07810, and D is 0.00778. Awesome. So first off, let us take a moment to write down all of our given variables. We're told that the probability of success, which means an adult exercising weekly, which we'll call P, is equal to 0.80. Once again, this is 80% written as a decimal, and our sample size N is equal to 60. And once again, this is because we are sampling 60 adults total. We also need to define the random variable as X and X is roughly a binomial, which I'll call B, and once again for this case, N is equal to 60 and P is equal to 0.80. So now at this point, we need to recall and use the conditions for approximating the binomial using normal distribution. So as we should recall, we can use the normal approximation to the binomial distribution only if the following conditions are satisfied. Our first condition that we need to focus on is Does it have a fixed number of trials? Is N a fixed number? For this particular prong, the number of trials N is equal to 60, which is a fixed value. So this condition is met. Our second condition is that each trial results in a success or failure, which for this particular problem, as we should recall, each adult either exercises, which is a success, or does not exercise, which is a failure. So therefore, the second condition is met. The third condition that we need to focus on is the constant probability of success, which, as we should recall, each selected adult has the same probability of exercising weekly since P is equal to 0.80. So that will mean that the third condition is met. The 4th condition focuses on independent trials, which, as we should recall, the behavior of one adult does not influence another, assuming random selection. This is a reasonable statement, so that means that the 4th condition is met. The 5th condition focuses on sample size. Is sample size large enough, meaning this is what N is large means. So in order to prove or disprove this condition is being met, we need to Recall and use the normal approximation, which means we need to use two separate equations, or rather expressions. So we need to recall that N multiplied by P is greater than or equal to 5, and N multiplied by 1 minus p is greater than or equal to 5. So that means when we plug in our known variables, and multiplied by P is equal to 60 multiplied by 0.80, which is equal to 48 and N multiplied by 1 minus P is equal to 60 multiplied by 0. So it's gonna be 60, that's our value for N multiplied by 1 minus 0.80, which when we plug that into our calculator, we will find that it's simply equal to 12. And since 48, And 12 are greater than 5, that means that this particular condition is met, because once again, 48 and 12 are indeed greater than the value of 5, which is fantastic. So therefore, without a doubt, all the conditions are satisfied. So as a result, the normal approximation is appropriate for this particular problem. So now our first step to solve for this problem is we need to compute the mean and standard deviation. Let us denote the mean as mu, and the mean is equal to N multiplied by P, which, as we determine N multiplied by P is equal to 48. And to solve for the standard deviation, let us recall that sigma, the standard deviation, is equal to the square root of N multiplied by P multiplied by 1 minus P, which is equal to the square root of 60 multiplied by 0.80 multiplied by 1 minus 0.80. So when we plug that into our calculator, rather the square root expression into our calculator. We will find that sigma is equal to the square root of 9.6. Fantastic. So now, moving right along here, we are trying once again we're trying to determine the probability that more than 55 individuals exercise weekly, which you can write this as P of X is greater than 55. So, in order to figure out this probability, we need to recall and use the continuity correction. Which as we should recall, we could write this as P of X is greater than 55 is approximately equal to P of Z is greater than 55 plus 0.5 minus 48 divided by the square root of 9.6, which will be equal to P. Of Z is greater than 7.5 divided by the square root of 9.6, which will mean that this is approximately equal to P of Z is greater than 2.421 will be round to three decimal places. Awesome. So now our next step is we need to perform and recall interpolation from the Z table. So to do interpolation using our Z table, which once again you should note that there should be a Z table in your textbook, and when we use our standard Z table values, we will find that P of Z is less than 2.42. Is going to be equal to Drum roll 0.9922. Fantastic. So once again, in order to solve for this particular value, you need to use your standard Z table from your textbook or you can just quickly look it up online. Our second value, which is P of Z is Less than 2.43 will be equal to 0.9924. Fantastic. So it's important to note that we are at Z equals, so once again Z equals 2.421, which is at 10% of the way from 2.42 to 2.43. Fantastic. So that means we can therefore, as a result, simply write P of Z is less than 2.421 is equal to 0. 9922. Plus 0.10 multiplied by parentheses 0.9924 minus 0.9922, which when we plug this into our calculator, we will find that it's equal to 0.99222. So therefore, we can go ahead and write that P of Z is greater than 2.421 is equal to 1 minus P of Z is less than 2.421, which will be equal to when we plug in our known variables, 1 minus 0.99222. Which will be equal to, let me round to 12345 decimal places. So we're rounding the 5 decimal places. We will get 0.00778, and that is our final answer. Hooray, we did it. So looking at our final answer, which is 0.00778, let us compare it to our multiple choice answers, and when we compare it, we will determine without a doubt that our final answer has to be multiple choice answer letter D. Which once again is 0.00778. 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

Probability Calculation

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