Question

he blood platelet counts of a group of women have a​ bell-shaped distribution with a mean of 247.3 and a standard deviation of 60.7. ​(All units are 1000 ​cells/μ​L.) Using the empirical​ rule, find each approximate percentage below. a. What is the approximate percentage of women with platelet counts within 3 standard deviations of the​ mean, or between 65.2 and 429.4

Answer 1

The approximate percentage of women with platelet counts within 3 standard deviations of the​ mean is 99.7%.

Explanation:

We are given that the blood platelet counts of a group of women have a​ bell-shaped distribution with a mean of 247.3 and a standard deviation of 60.7.

Let X = the blood platelet counts of a group of women

The z-score probability distribution for the normal distribution is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = population mean = 247.3

`\sigma` = standard deviation = 60.7

Now, according to the empirical rule;

• 68% of the data values lie within one standard deviation of the mean.

• 95% of the data values lie within two standard deviations of the mean.

• 99.7% of the data values lie within three standard deviations of the mean.

Since it is stated that we have to calculate the approximate percentage of women with platelet counts within 3 standard deviations of the​ mean, or between 65.2 and 429.4, i.e;

z-score for 65.2 =
`(X-\mu)/(\sigma)`

=
`(65.2-247.3)/(60.7)` = -3

z-score for 429.4 =
`(X-\mu)/(\sigma)`

=
`(429.4-247.3)/(60.7)` = 3

So, it means that the approximate percentage of women with platelet counts within 3 standard deviations of the​ mean is 99.7%.