In Exercises 25–28, use these parameters (based on Data Set 1 ...

Question

In Exercises 25–28, use these parameters (based on Data Set 1 “Body Data” in Appendix B):

Men’s heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in.

Women’s heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in.

If the Navy changes the height requirements so that all women are eligible except the shortest 3% and the tallest 3%, what are the new height requirements for women?

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 college entrance exam scores are normally distributed with a mean score of 500 and a standard deviation of 100. The college decides to offer scholarships to the students who score within the top 10% of the overall exam score range. What is the minimum score value that is required to qualify for the scholarship? Awesome. So it appears for this particular prompt, we're ultimately trying to figure out what is the minimum score value that is required in order for a student to qualify for the scholarship. So now that we know what 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 580, B is 700. C is 645 and D is 629. Awesome. So first off, in order to solve for this particular problem, we need to we need to ultimately, once again, we're ultimately trying to figure out what the minimum score value that is required to qualify for the scholarship. So in order to do that, we need to identify the percentile corresponding to the top 10%. So we need to note that the top 10% corresponds to the 90th. Percentile of the normal distribution or where where denotes the probability P of Z is less than lowercase c. So P of capital Z is less than lowercase z, which is going to be equal to 0.90. So now, our next step is we need to find the Z score for the 90th percentile. Which is parenthesis of of Z is less than lowercase z is equal to 0.90 by interpolization. So that means that from our standard normal tables, that would mean that P of capital Z is less than 1.28, is equal to 0.8997, and that P of capital Z is less than 1.29, which is going to be equal to 0.9015. Fantastic. So that will mean, as we should recall a note, we could go ahead and write that Z minus 1.28. Divided by. 1.29 minus 1.28. will be equal to when we plug in our known known variables. We'll get 0.90. So when we plug in our known variables or known values, we'll get is this is all equal to 0.90 minus 0.8997 divided by 0.9015 minus 0.8997. Fantastic. So, therefore, when we rearrange this expression to isolate and solve for Z all by itself, we will find that Z is equal to 0.0003. And then we need to take this value and we need to divide it by 0.0018, and then we need to multiply it by 0.01, and then we need to add 1.28, which when we do this, we will find that it's approximately equal to when we round to four decimal places, 1.2817. Fantastic. So now we need to recall and set up the equation to help us solve for the minimum qualifying score, and we can accomplish this goal by recalling and using the Z score formula. And as you should recall, the Z score formula states that Z is equal to X minus mu divided by sigma, where mu in this case, let's make a note that mu, our mean is equal to 500 and sigma, our standard deviation is equal to 100. So, when we plug in our known variables, we will find that 1.2817 is equal to X minus 500 divided by 100. So when we rearrange this expression to isolate in sulfur X, I'm gonna skip a few steps because it's basic algebra, but when you've correctly solve for X, we will find that X is approximately equal to 629 when we round to the nearest whole number, means initially when you plug this into your calculator, you will find that it's equal to 6. 628.17 Fantastic. So, Once again, we need to round up in order to ensure that the students or that particular student or students are in the top 10 percentile, so that's how we get 629, and that is our final answer. Hooray, we did it. So therefore, without a doubt, the minimum score that is required to qualify for the scholarship, without a doubt has to be 629. And when we look at our multiple choice answers, the correct answer has to be multiple choice answer letter D, which once again is 629. 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

Normal Distribution

Percentiles

Z-scores

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