Question

The blood platelet counts of a group of women have a bell-shaped distribution with a mean of 257.62 and a standard deviation of 62.1 (All units are 1000 cells/μ L.)

Using the empirical rule, find each approximate percentage below.
What is the approximate percentage of women with platelet counts within 1 standard deviation of the mean, or between 195.5 and 319.7 ?
What is the approximate percentage of women with platelet counts between 71.3 and 443.9 ?
a. Approximately 68 % of women in this group have platelet counts within 1 standard deviation of the mean, or between 195.5 and 319.7.
(Type an integer or a decimal. Do not round.)
b. Approximately ____ % of women in this group have platelet counts between 71.3 and 443.9.
(Type an integer or a decimal. Do not round.)

Answer 1

(a) Approximately 68 % of women in this group have platelet counts within 1 standard deviation of the mean, or between 195.5 and 319.7.

(b) Approximately 99.7% of women in this group have platelet counts between 71.3 and 443.9.

Explanation:

We are given that the blood platelet counts of a group of women have a bell-shaped distribution with a mean of 257.62 and a standard deviation of 62.1

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

So, X ~ Normal(
`\mu=257.62, \sigma^(2) =62.1^(2)`)

Now, the empirical rule states that;

• 68% of the data values lie within the 1 standard deviation of the mean.

• 95% of the data values lie within the 2 standard deviations of the mean.

• 99.7% of the data values lie within the 3 standard deviations of the mean.

(a) The approximate percentage of women with platelet counts within 1 standard deviation of the mean, or between 195.5 and 319.7 is 68% according to the empirical rule.

(b) The approximate percentage of women with platelet counts between 71.3 and 443.9 is given by;

z-score of 443.9 =
`(X-\mu)/(\sigma)`

=
`(443.9-257.62)/(62.1)` = 3

z-score of 71.3 =
`(X-\mu)/(\sigma)`

=
`(71.3-257.62)/(62.1)` = -3

So, approximately 99.7% of women in this group have platelet counts between 71.3 and 443.9.