Question

HELP PLEASE

The blood platelet count of a group of women have bell-shaped distribution with a mean of 245.5 and a standard deviation of 68.2 (all units are 1000 cells/ L) Using the empirical rule, fill in the blanks below (Round to the nearest hundredth):
a. Approximately 95% of healthy women in this group
b. Approximately 99.7% of healthy women in this have blood platelet counts between
group have blood platelet counts between
and(1000 cells/ ML). and (1000 cells/ ML).

Answer 1

a) Approximately 95% of healthy women in this group have blood platelet counts between 109.1 and 381.9 (1000 cells/μL).

b) Approximately 99.7% of healthy women in this group have blood platelet counts between 40.9 and 450.1 (1000 cells/μL).

Explanation:

Empirical Rule

The Empirical Rule (also known is the "68-95-99.7" rule) states that nearly all of the data within a normal distribution will fall within three standard deviations of the mean.

• Approximately 68% of the data will fall within one standard deviation of the mean.
• Approximately 95% of the data will fall within two standard deviations of the mean.
• Approximately 99.7% of the data will fall within three standard deviations of the mean.

Given the blood platelet count of a group of women has a bell-shaped distribution with:

• Mean μ = 245.5 (1000 cells/μL)
• Standard deviation σ = 68.2 (1000 cells/μL)

To determine the lower and upper bounds for blood platelet counts for 95% of the women, subtract and add two standard deviations to the mean:

`\begin{aligned}\text{Lower bound}&= \mu - 2\sigma \\&= 245.5 - 2(68.2)\\&=245.5-136.4\\&=109.1\end{aligned} \qquad \begin{aligned}\text{Upper bound}&= \mu + 2\sigma \\&= 245.5 + 2(68.2)\\&=245.5+136.4\\&=381.9\end{aligned}`

Therefore, approximately 95% of healthy women in this group have blood platelet counts between 109.1 and 381.9 (1000 cells/μL).

To determine the lower and upper bounds for blood platelet counts for 99.7% of the women, subtract and add three standard deviations to the mean:

`\begin{aligned}\text{Lower bound}&= \mu - 3\sigma \\&= 245.5 - 3(68.2)\\&=245.5-204.6\\&=40.9\end{aligned} \qquad \begin{aligned}\text{Upper bound}&= \mu + 3\sigma \\&= 245.5 + 3(68.2)\\&=245.5+204.6\\&=450.1\end{aligned}`

Therefore, approximately 99.7% of healthy women in this group have blood platelet counts between 40.9 and 450.1 (1000 cells/μL).