Red Blood Cell Count Use the normal distribution in Exercise 1...

Question

Red Blood Cell Count Use the normal distribution in Exercise 16.

a. What percent of the adult males have a red blood cell count less than 6 million cells per microliter?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says in a study, hemoglobin levels in adult males are normally distributed with a mean of 15.5 g per deciliter, and a standard deviation of 1.1 g per deciliter. What percentage of adult males have a hemoglobin level less than 17 g per deciliter? And we get four possible choices as our answers. For choice A, we have 91.14%. For choice B, we have 19.14%. For choice C, we have 18.31%. And for choice D, we have 91.37%. So we're asked to find the percentage of adult males that have a hemoglobin level, less than 17 g per deciliter. So the first thing we want to do is convert that 17 g per deciliter into a Z2. And so recall your formula for a Z score, that is going to be Z, it's going to be equal to the quantity of X minus mu in quantity, divided by sigma. Where Z here is our Z score, X is going to be our hemoglobin level, mu is going to be our mean, and sigma is going to be our standard deviation. So, in our case here, our Z score, Z is going to be equal to 4 X we're gonna have 17. Since we have 17 g per desk liter, then we'll have minus for I mean we're told in the problem that we have 15.5 g per desk liter. So we'll have 15.5 in from you, and we're going to take that quantity and divide it by our standard deviation of 1.1 g per desk liter, so we'll divide this by 1.1. And so when we evaluate this expression, We see that it is approximately equal to. 1.3636. So now we need to do is to find our cumulative probability that corresponds to Z being equal to 1.3636. And so we can use this using our standard normal distribution table. However, our Z score here actually falls between two values in our table. So it falls between Z equal to 1.36, which has a corresponding cumulative probability of 0.9131. And the other Z value, it's going to be Z is equal to 1.37. That corresponds to a cumulative probability of. 0.9147. So we're actually going to have to use interpolation in order to find our cumulative probability for our particular Z score. So for our cumulative probability or our particular Z score, we'll have the cumulative probability P of Z less than 1.3636. And that is going to be equal to 2, and here we call, when you are doing interpolation, we're going to first have a fraction here, where the numerator is going to be the difference of our cumulative probabilities that we have in our table, and it's going to be divided by in the denominator, the difference between our Z scores from the table. So we're going to have the quantity in the numerator of 0.9147 minus 0.9131 in quantity, and that's going to be divided by the quantity of 1.37 minus 1.36 in quantity. And then that fraction gets multiplied by the quantity of our Z score that we're looking for, which is going to be 1.3636. Minus the lower bound of our Z interval here. So that's gonna be the 1.36. We'll subtract 1.36 from that in quantity. Then we have to add to this. The cumulative probability of our lower bound, which is in this case, it's going to be 0.9131. And so, when we evaluate this expression, this is going to be approximately equal to. 0.9137. However, we need to actually convert this into a percentage, since the question is asking for a percentage. So, we just need to multiply this by 100%. And in doing so, we get that this is going to be equal to. 91.37%. So that is the answer that we're looking for for this problem. And when we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

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Red Blood Cell Count Use the normal distribution in Exercise 16.

a. What percent of the adult males have a red blood cell count less than 6 million cells per microliter?

Key Concepts

Normal Distribution

Z-Score

Percentile

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Red Blood Cell Count Use the normal distribution in Exercise 16.a. What percent of the adult males have a red blood cell count less than 6 million cells per microliter?