Employee Wellness A survey of employed U.S. adults found that ...

Question

Employee Wellness A survey of employed U.S. adults found that only 35% believe their employer cares about their well-being. You randomly select a sample of U.S. employees. Find the probability that fewer than 100 believe their employer cares about their well-being. (Source: Gallup)

c. You select 400 U.S. employees.

c. You select 400 U.S. employees.

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. According to a survey, 28% of employees report having flexible work hours. If a company randomly selects 350 employees, what is the probability that fewer than 80 of them report having flexible work hours? Awesome. So it appears for this particular prom we're asked to determine. What the probability will be that fewer than 80 of the 350 total employees surveyed will report having flexible work hours, so fewer than 80 of them will report having flexible work hours. So now that we know that we're ultimately trying to determine this probability value, that is our final answer we're ultimately trying to solve for, let's take a moment to read off our multiple choice answers to see what our final answer might be. A is 0.0138. B is 0.2338. C is 0.7662, and finally D is 0.9862. Awesome. So first off, let us define the random variable and distribution. So let us state that X will be the number of employees with flexible work hours, and let us also note that X will follow a binomial distribution, as we should recall. So, X, which is a binomial, which I'll call B for short, Has a population that is equal to 350 and a percentage of 28%, which is 0.28 when it's written as a decimal. So now, our next step is we need to check if the normal approximation is appropriate for the case of this problem. So we need to calculate. N multiplied by P, so N multiplied by P is equal to 350 multiplied by 0.28, which is equal to Drum roll 98. And we also need to note that N multiplied by 1 minus P is equal to 350, and then we need to take 350 multiplied by parentheses 1 minus 0.28, which will be equal to when we plug that expression into our calculator, you will find that it's equal to. 252, and bolt values are greater than 5, so normal approximation is appropriate for this case. So now our next step is we need to determine the mean and standard deviation by recalling and using the equations to solve for the mean and standard deviation. So let us recall that the mean, which will denote as mu, is equal to N multiplied by P, which is equal to 98, and our standard deviation, which we will denote as sigma is equal to the square root of N multiplied by P. Multiplied by 1 minus P, which will be equal to the square root of 350 multiplied by 0.28, multiplied by 1 minus 0.28. So that will mean when we plug that expression into our calculator, that sigma, or standard deviation when we round to two decimal places, will be equal to the square root of 70.56. Fantastic. So moving right along here, we need to recall and apply the continuity correction for fewer than 80. So we need to use the condition that X is less than 80. So the continuity correction will be X is less than or equal to 79.5. So, our next step now is we need to recall and use the Z score, since we need to convert our value to a Z score. So the Z score Z is equal to 79.5 minus 98. Divided by the square root of 70.56. So once again, 79.5 minus 98, all divided by the square root of 70.56, because that's our value for the standard deviation. So when we plug that into our calculator, we will find that it's approximately equal to -2.2024. Let me round to 4 decimal places. Awesome. So now our next step is that we need to find the probability using the standard normal distribution table. So we need to note that Z equals -2.2024 is in between -2.20 and -2.21. So when we use our standard normal distribution table, which you should have one of those in your textbook, we will determine that P of Z is less than -2.20 is approximately. Equal to 0.0139, and that P of Z is less than -2.21, is approximately equal to 0.0136. So by interpolation, we will determine, as we should recall, that P of Z is less than -2.2024. Is going to be equal to. -2.2024. Minus -2.20. All divided by -2.21 minus -2.20. All multiplied by parentheses 0.0136 minus 0.0139. And we need to Add. We also need to add the value of 0.0139. So when we plug in that long expression into our calculator, we will find that it's approximately equal to when we round to four decimal places like all of our multiple choice answers, we will find that it's equal to 0.0138, and that is our final answer. Hooray, we did it. So therefore. Without a doubt, our final answer when we look at our multiple choice answers, the correct answer has to be multiple choice answer letter A, which once again is 0.0138. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Probability Distribution

Binomial Probability Formula

Normal Approximation to the Binomial

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