Suppose that diastolic blood pressure readings of adult males have a bell-shaped distribution with a mean of 84 mmHg and a standard deviation of 9 mmHg. Using the empirical rule, what percentage of adult males have diastolic blood pressure readings that are greater than 102 mmHg? Please do not round your answer.

The empirical rule, also known as three-sigma rule or 68-95-99.7 rule, "is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)".

Let X the random variable who represent the diastolic blood pressure readings of adult males

From the problem we have the mean and the standard deviation for the random variable X. [tex]E(X)=84, Sd(X)=9[/tex]

So we can assume [tex]\mu=84 , \sigma=9[/tex]

On this case in order to check if the random variable X follows a normal distribution we can use the empirical rule that states the following:

• The probability of obtain values within one deviation from the mean is 0.68

• The probability of obtain values within two deviation's from the mean is 0.95

• The probability of obtain values within three deviation's from the mean is 0.997

So we need values such that

[tex]P(X<\mu -\sigma)=P(X <75)=0.16[/tex]

[tex]P(X>\mu +\sigma)=P(X >93)=0.16[/tex]

[tex]P(X<\mu -2*\sigma)P(X<66)=0.025[/tex]

[tex]P(X>\mu +2*\sigma)P(X>102)=0.025[/tex]

[tex]P(X<\mu -3*\sigma)=P(X<57)=0.0015[/tex]

[tex]P(X>\mu +3*\sigma)=P(X>211)=0.0015[/tex]

So for this case the answer would be:

[tex]P(X>\mu +2*\sigma)P(X>102)=0.025[/tex]