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 (b) exactly 50.

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 health survey finds that 80% of adults exercise at least once a week. If you randomly select 60 adults, what is the probability that exactly 50 exercise weekly? Awesome. So it appears for this particular problem we're asked to determine what the probability will be that exactly 50 out of these 60 adults Exercise weekly. So what is the probability that exactly 50 individuals out of the 60 total being surveyed exercise weekly? That's our final answer we're ultimately trying to solve for. What is this probability value? So with that in mind, let's read off our multiple choice answers to see what our final answer might be. A is 0.1043. B is 0.3017. C is 0.7201, and D is 0.1750. Awesome. So note first off that we need to write down all the variables or all the information that we are given. So we are told that the probability of success, which is an adult exercising weekly, which we'll call P is equal to 0.80 because we have to write 80% as a decimal. We're told that the sample size N is equal to 60. And we're also told to define the random variable, which will define the random variable as X, which is roughly a binomial, which I'll call B for short. So it's a binomial. And once again, N is equal to 60 and P is equal to 0.80. So in order to apply the approximation to a binomial using normal distribution, the following conditions have to be met first, as we should recall, we can use normal approximation to binomial distribution only if these specific conditions are satisfied. So firstly, we need to satisfy the condition that there is a fixed number of trials, meaning that the value of N has to be a fixed value, and the number of trials in this particular problem is 60 and N equals 60 is indeed a fixed value. So that condition is met. Our second condition is we need to determine that each trial results in success or failure. So each adult either exercises, which is considered a success, or does not exercise, which is considered a failure. So therefore, the second condition is met. Our third condition, as we should recall, is the constant probability of success. So as we should recall, each selected adult has the same probability of exercising weekly since P equals 0.80. So that means the third condition is met. Our fourth condition is we need our focus is on independent trials, and as we should recall, the behavior of one adult does not influence another, assuming random selection, so this is reasonable. So therefore condition 4 is met. And our fifth condition focuses on sample size is large enough, meaning this is what N is large means. So in order to use the normal approximation, we need to meet the following two conditions and multiplied by P must be greater than or equal to 5 and 1. So now we also need to note that N multiplied by P has to be greater than 5, and N multiplied by 1 minus P has to be greater than or equal to 5. So, when we plug in our known variables, we will find that N multiplied by P is equal to 60 multiplied by 0.80. Which is equal to 48 and N. And we also need to do our second equation, so N multiplied by 1 minus P. And when we plug in our known variables to solved for this expression, we will find that it's equal to 60, multiplied by 1 minus 0.80, which will give us an answer of. 12, and since 48. Is greater. Then 5 and 12 is greater than 5. That will mean that without a doubt, all the conditions are satisfied, so therefore, as a result, It is appropriate to use the normal approximation. So now our first major step to solve this problem is we need to compute the mean and the standard deviation. As you should recall, we can solve for mu, which denotes the mean as, so the mean mu is equal to N multiplied by P, which is equal to 48 in this case as we discovered. And sigma, which is the standard deviation, as you should recall, is equal to the square root of N multiplied by P, multiplied by 1 minus P. Which when we plug in our known variables, we will find that it's equal to the square root of 60 multiplied by 0.8, multiplied by 1 minus 0.80, which I put 0.8, but to be precise, let's say it's 0.80. So when we plug this expression into our calculator, we will find that sigma, or our value for the standard deviation, will be equal to the square root of 9.6. Fantastic. So now, it's important to note that we're trying to determine the probability that exactly 50 individuals exercise weekly from this survey. So we're trying to find P of X is equal to 50. So we need to recall and use the continuity correction for a single value, and when we do that, we will determine that P of X is equal to 50. will be approximately equal to P of 49. 0.5. is less than X. And X is less than 50.5. And we can say that this is equal to P 49.5 minus 48 divided by the square root of 9.6. is less than Z and Z is less than 50.5 minus 48 divided by the square root of 9.6. And we could safely say that this is equal to P of 0.484 is less than Z and Z is less than 0.807. Fantastic. So, moving right along here. Our next step now, and don't forget your parentheses. Our next step is we need to interpolate between the values. So we want to calculate P of 49.5 is less than X and X is less than 50.5, which is approximately equal to P of 0.4. 84 is less than Z, which is less than 0.807. Which means that this is all equal to P of Z is less than 0.807. Minus P of Z is less than 0.484. Fantastic. So once again we need to interpolate both. So we need to interpolate P of Z is less than 0.807. Fantastic. And when we use our Z table, which you should have a Z table in your textbook. We will find that P of Z is less than 0.80 is equal to 0.78. 81, and we can also determine that P of Z. is less than 0.81 will be equal to, so once again using our Z table, we will find that it's equal to a value of 0.7910. So therefore, moving right along here. Since 0.807 is 70% of the way from 0.80 to 0.81. Fantastic. So that will mean that we could go ahead and write that P of Z is less than 0.807. is equal to 0.78. 81 + 0.7 multiplied by parentheses 0.7910 minus 0.7881, which when we plug that into our calculator, we will find that it will be equal to 0.79013. So now our next step is we need to interpolate P of Z is less than 0.484. So once again using our Z table we can state that P of Z is less than 0.48. Is gonna be equal to a value of 0.6844 and the value of P of Z is less than 0.49 will be equal to 0. Once again it's gonna be equal to 0.68. 79. Awesome. So once again, 0.6879. So, With that in mind, since 0.484 is 40% of the way from 0.48 to 0.49. So therefore we could go ahead and write that P of Z is less than 0.484. will be equal to 0.6844 plus 0.4 multiplied by parentheses 0.6879 minus 0.6844. And when we plug in that expression into our calculator, we will find that it's equal to 0.6858. Let me round to 4 decimal places. So now we can perform our final approximation. So we go ahead and write that P of X is equal to 50, is approximately equal to P 49.5 is less than X and X is less than 50.5, which will be equal to 0.79013 minus 0.68. 58, which when we plug that into our calculator, we will find that it will be equal to, when we round to 4 decimal places, 0.1043, and that's it. That is our final answer. Hooray, we did it. So going back to look at our multiple choice answers, our final answer without a doubt will have to be what appears to be multiple choice answer letter A, which once again is 0.1043. 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

Probability Calculation

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