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 (a) at most 40.

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 at most, 45 exercise weekly? Awesome. So it appears for this particular problem we're asked to determine if we were to select randomly 60 adults, what would be the probability that at most, 45 out of the 60 adults exercise weekly? Awesome. So with that in mind, 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 0.3617. B is 0.2099, C is 0.4106, and D is 0.8126. Awesome. So first off, let us write down all of our given information, all of our given variables. We're told that the probability of success, which is an adult exercising weekly, we, we'll call this value P and P will be equal to 0.80, which is 80% written as a decimal. We're told that the sample size, which we're gonna denote as N, is equal to 60. And we will define the random variable as X, and this is roughly the binomial, which I'll call B, and once again for our binomial, we have an N equals 60 and P equals 0.80. And the binomial in this, since we're dealing with a binomial in this case, that means we could use normal approximation if certain conditions are met. So to approximate a binomial distribution, as we should recall, Using a normal distribution, the following conditions must be met first. Firstly, a fixed number of trials, the number of trialsn must be fixed, and then for the case of our problem, N equals 60, which is a fixed value. Our second condition is only 2 outcomes per trial, so each trial must result in either a success, exercises weekly, or a failure does not exercise at all. So, moving right along here, in that case, in this condition, this second condition is also met. So for independent trials is our next step, so we need to recall a note that the outcome of one trial must not affect the outcome of another. So assuming a random selection, independence is reasonable. We also need to consider the constant probability of success, meaning, as we should recall, each trial must have the same probability P of success. Given that P equals 0.80, this is constant for all trials and Next, we need to consider a large enough sample is present, meaning the normal approximation condition, meaning, as we should recall, to safely use the normal approximation, we need to check that, we need to check the following. We need to check that N multiplied by P is greater than or equal to 5 and that 1 minus p is greater than or equal to 5. So when we plug in our known variables, we'll find that N multiplied by P is equal to 60 multiplied by 0.80, which is equal to. 48 and 1 minus. P, so we have 1 minus P, but when we, we missed a variable on accident, so we need to note that I'll go back for a moment that N multiplied by 1 minus P is greater than or equal to 5. So, We need to note that when we plug in our known variables for N multiplied by 1 minus P, So we will find when we plug in our known variables that this will be equal to 60 multiplied by 1 minus our value of P, which is 0.80, which will mean that N multiplied by 1 minus P will be equal to simply 12. And both 48 and 12 are greater than 5, so that means that both conditions are satisfied, so I'll put a check mark. Next to both values. So, therefore, because both conditions are satisfied, and since all 5 conditions are met, it is appropriate to use the normal approximation to the binomial distribution. So with that in mind, Our first step now is we need to compute the mean and the standard deviation. So the mean, which we're going to denote as mu is equal to N multiplied by P, which is equal to 60 multiplied by 0.80, which gives us an answer of 48, fantastic. So now when we solve for sigma, our standard deviation, we should recall that this is equal to the square root of N multiplied by P, multiplied by 1 minus P, fantastic. And when we plug in our known variables, we will find that. It will be equal to the square root of 60 multiplied by 0.8, multiplied by 1 minus 0.80. And when we plug that into our calculator, we will find that sigma will be equal to the square root of 9.6. So now, At this point, we need to determine the probability that at most 45 exercise weekly, which you could write this as P of X is less than or equal to 45, where P represents our probability. So, when we recall and use the normal approximation with continuity correction, we can calculate the following, meaning we could write the following, we could write that P of X is less than or equal to 45, is approximately equal to P of Z is less than or equal to. 45 + 0.5 minus 48 divided by the square root of 9.6. And we will find that this is equal to P of Z is less than or equal to -2.5 divided by the square root of 9.6. Fantastic, which will mean that it's approximately equal to P of Z is less than or equal to 0.807. Fantastic. So now we need to recall that we need to interpolate the value of P of Z is less than or equal to 0.807. More precisely, we can use the standard normal table, which you should have a standard normal table in your textbook. We will use linear interpolation between the closest Z scores. So the value that we're interested in is where Z is equal to negative 0. 807. So considering this, when we use our standard normal table, we will find that P of Z is less than or equal to 0.80. We will find that this is equal to 0.2119, and for P of Z is less than or equal to negative 0.81. We will find that this value is equal to 0.2090, and once again we get both of these values by using a standard normal table. So now, At this point, we need to interpolate between these values, so we need to find the decimal position of negative 0.807 between 0.80 and 0.81. So the distance from -0.80 is equal to 0.007, and the total interval, which I'll call TI is equal to 0.01. And the fraction interval, which I'll call FI, and the fraction interval is equal to 0.007 divided by 0.01, which is equal to 0.7. So now, moving right along here, we need to interpolate between the two probabilities, and when we do that, we will find that P of Z is less than or equal to negative 0.807. is approximately equal to P of Z is less than or equal to 0.80. Minus. 0.7 multiplied by Parenthesis P. Of Z is less than or equal to 0.80. Fantastic. And then we need to Therefore, after this, we need to subtract. So we're subtracting key of Z is less than or equal to -0.81. Fantastic. And don't forget your other set of parentheses. So now, with that in mind, we can go ahead and write that P of Z is less than or equal to 0.807 is approximately equal to 0.2119 minus 0.7 multiplied by 0.2119 minus 0.202, wait. So it's gonna be 2090, so 0.2090. And when we plug this expression into our calculator, we will find that P of Z is less than or equal to 0.807 will be approximately equal to when we round to 4 decimal places, 0.2099, and that is our final answer. Hooray, we did it. So looking at our multiple choice answers, our final answer without a doubt will have to be multiple choice answer letter B, which once again is 0.2099, 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.

Key Concepts

Binomial Distribution

Normal Approximation to the Binomial

Probability Calculation

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