Finding Specified Data Values In Exercises 31–38, answer the q...

Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Red Blood Cell Count The red blood cell counts (in millions of cells per microliter) for a population of adult males can be approximated by a normal distribution, with a mean of 5.4 million cells per microliter and a standard deviation of 0.4 million cells per microliter.

a. What is the minimum red blood cell count that can be in the top 25% of counts?

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says, the daily sugar intake in grams for a group of adults is normally distributed with a mean meu equal to 50 g and a standard deviation sigma equal to 6 g. What is the minimum daily sugar intake that falls in the top 25% of all daily sugar intake amounts? And we're given 4 possible choices as our answers. For choice A, we have 54 g. For choice B, we have 50 g. Choice C, we have 51 g, and for choice D, we have 58 g. So, in this problem, we're asked to find the minimum daily sugar intake that falls in the top 25% of all daily sugar intake amounts. So the top 25% means that we're looking for the 75th percentile. And the reason why we're looking for the 75th percentile is because if 25% of the daily sugar intake amounts fall above the value that we're looking for, that means that 75% fall below it. So, in order to find the sugar intake value, we first need to find the Z score for the 75th percentile. So, for the seventy-fifth percentile, our accumulative probability P of capital Z is less than lowercase z. It's going to be equal to, and here we just need to convert 75% into a decimal by divided by 100, so we'll have 0.75. And when we use our standard normal distribution tables to look up the Z value, we see that we get a Z value that is going to be approximately equal to 0.67. So now we need to convert the Z score into a daily sugar intake amount. And for this, we call your formula for calculating a Z score, and that is Z is equal to the quantity of X minus U in quantity, divided by sigma. Where Z here is going to be your Z score, X is going to be our daily sugar intake amount, U here is our mean, and sigma is our standard deviation. So we'll substitute in the values that we have. For Z, we have 0.67, and that's going to be equal to the quantity of X, which is the value that we're looking for. minus from mu, we're told the problem that that turns out to be 50 g, so we'll have X minus 50, and that quantity is going to be divided by our standard deviation sigma, which we're told, and the problem is 6. We'll divide this by 6. So now we just need to solve this equation for X. So first thing we need to do is multiply both sides of the equation by 6 in order to get rid of our fraction. Doing so we'll get 4.02 is equal to x minus 50. And to solve for X, we'll just add 50 to both sides, and then we'll get X is equal to 54. So here we just round it to the nearest whole number. And the units here for this is going to be grams, since both are mean and standard deviation were given in grams. And so this here is the answer that we're actually looking for for this problem. And if we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Normal Distribution

Mean and Standard Deviation

Percentiles

Watch next