Question

The blood platelet counts of a group of women have a​ bell-shaped distribution with a mean of 248.5 and a standard deviation of 61.1. ​(all units are 1000 ​cells/mu​l.) using the empirical​ rule, find each approximate percentage below.
a. what is the approximate percentage of women with platelet counts within 2 standard deviations of the​ mean, or between 126.3 and 370.7​?
b. what is the approximate percentage of women with platelet counts between 65.2 and 431.8​?

Answer 1

Answer:

a. Approximately 95% women have platelet counts between 126.3 and 370.7.

b. Approximately 99.7% women have platelet counts between 65.2 and 431.8.

Step-by-step explanation:  

We have been given that the  blood platelet counts of a group of women have a​ bell-shaped distribution with a mean of 248.5 and a standard deviation of 61.1.

a. Empirical rule of normal distribution states:

$\approx68\%$ of the data lies within 1 standard deviation of the mean

.

$\approx95\%$ of the data lies within 2 standard deviations of the mean

.

$\approx99.7\%$ of the data lies within 3 standard deviations of the mean.

Now let us find z-score for our given values as z-score represents that a data point is how many standard deviation away from mean.

$z=\frac{x-\mu}{\sigma}$

$z=\frac{126.3-248.5}{61.1}$

$z=\frac{-122.2}{61.1}$

$z=-2$

$z=\frac{370.7-248.5}{61.1}$

$z=\frac{122.2}{61.1}$

$z=2$

Since our given values are -2 and 2 standard deviations of mean, therefore, approximately 95% of women have platelet counts between 126.3 and 370.7​.

b. Let us find z-score for our given values as z-score represents that a data point is how many standard deviation away from mean.

$z=\frac{x-\mu}{\sigma}$

$z=\frac{65.2-248.5}{61.1}$

$z=\frac{-183.3}{61.1}$

$z=-3$

$z=\frac{431.8​-248.5}{61.1}$

$z=\frac{183.3}{61.1}$

$z=3$

Since our given values are -3 and 3 standard deviations of mean, therefore, approximately 99.7% of women have platelet counts between 65.2 and 431.8.

Answer 2

A) 68% will fall within that range.
b) 99.7% will fall within that range.

The empirical rule states that 68% of data falls within one standard deviation of the mean; 95% falls within 2 standard deviations of the mean; and 99.7% fall within 2 standard deviations of the mean.  For each one of these, subtract them from the mean and divide by 61.1 to see how many deviations away they are.