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/mul.) 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?

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:

[tex]\approx68\%[/tex] of the data lies within 1 standard deviation of the mean

.

[tex]\approx95\%[/tex] of the data lies within 2 standard deviations of the mean

.

[tex]\approx99.7\%[/tex] 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.

[tex]z=\frac{x-\mu}{\sigma}[/tex]

[tex]z=\frac{126.3-248.5}{61.1}[/tex]

[tex]z=\frac{-122.2}{61.1}[/tex]

[tex]z=-2[/tex]

[tex]z=\frac{370.7-248.5}{61.1}[/tex]

[tex]z=\frac{122.2}{61.1}[/tex]

[tex]z=2[/tex]

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.

[tex]z=\frac{x-\mu}{\sigma}[/tex]

[tex]z=\frac{65.2-248.5}{61.1}[/tex]

[tex]z=\frac{-183.3}{61.1}[/tex]

[tex]z=-3[/tex]

[tex]z=\frac{431.8-248.5}{61.1}[/tex]

[tex]z=\frac{183.3}{61.1}[/tex]

[tex]z=3[/tex]

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.

MrRoyal MrRoyal · Answer 2 · 2021-10-28T13:53:21+02:00

The blood platelet count is an illustration of normal distribution

Approximately 95% of the data lies within 2 standard deviations of the mean.
There are approximately 99.7% of women with platelet count between 65.2 and 431.8

The given parameters are:

[tex]\mathbf{\mu = 248.5}[/tex]

[tex]\mathbf{\sigma = 61.1}[/tex]

(a) The percentage within 2 standard deviation of mean or between 126.3 and 370.7

Start by calculating the z-score, when x = 126.3 and x = 370.7

[tex]\mathbf{z = \frac{x - \mu}{\sigma}}[/tex]

So, we have:

[tex]\mathbf{z = \frac{126.3 -248.5}{61.1}}[/tex]

[tex]\mathbf{z = \frac{-122.2}{61.1}}[/tex]

[tex]\mathbf{z = -2}[/tex]

Also:

[tex]\mathbf{z = \frac{370.7 -248.5}{61.1}}[/tex]

[tex]\mathbf{z = \frac{122.2}{61.1}}[/tex]

[tex]\mathbf{z = 2}[/tex]

The empirical rule states that:

Approximately 95% of the data lies within 2 standard deviations of the mean.

Hence, there are approximately 95% of women with platelet count within 2 standard deviations of the mean.

(b) The percentage with platelet count between 65.2 and 431.8

Start by calculating the z-score, when x = 65.2 and x = 431.8

[tex]\mathbf{z = \frac{x - \mu}{\sigma}}[/tex]

So, we have:

[tex]\mathbf{z = \frac{65.2 -248.5}{61.1}}[/tex]

[tex]\mathbf{z = \frac{-183.3}{61.1}}[/tex]

[tex]\mathbf{z = -3}[/tex]

Also:

[tex]\mathbf{z = \frac{431.8 -248.5}{61.1}}[/tex]

[tex]\mathbf{z = \frac{183.3}{61.1}}[/tex]

[tex]\mathbf{z = 3}[/tex]

The empirical rule states that:

Approximately 99.7% of the data lies within 3 standard deviations of the mean.

Hence, there are approximately 99.7% of women with platelet count between 65.2 and 431.8

Read more about normal distributions at:

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