Question

Suppose that diastolic blood pressure readings of adult males have a bell-shaped distribution with a mean of 8080 mmHg and a standard deviation of 1111 mmHg. Using the empirical rule, what percentage of adult males have diastolic blood pressure readings that are at least 6969 mmHg? Please do not round your answer.

Answer 1

For this case we want this probability:

`P(X >69)`

And we can use the z score formula given by:

`z = (x -\mu)/(\sigma)= (69-80)/(11)= -1`

We know that within one deviation from the mean we have 68% of the values so then on the tails we need to have (1-0.68)/2 = 0.16 or 16% so then we can conclude that P(Z<-1) =0.16 and by the complement rule:

`P(z>1) = 1-0.16 = 0.84`

So we expect about 84% of readings of at least 69 mm Hg

Explanation:

Previous concepts

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 variable of interest.

From the problem we have the mean and the standard deviation for the random variable X.
`E(X)=80, Sd(X)=11`

So we can assume
`\mu=80 , \sigma=11`

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

Solution to the problem

For this case we want this probability:

`P(X >69)`

And we can use the z score formula given by:

`z = (x -\mu)/(\sigma)= (69-80)/(11)= -1`

We know that within one deviation from the mean we have 68% of the values so then on the tails we need to have (1-0.68)/2 = 0.16 or 16% so then we can conclude that P(Z<-1) =0.16 and by the complement rule:

`P(z>1) = 1-0.16 = 0.84`

So we expect about 84% of readings of at least 69 mm Hg

Answer 2

84% of adult males have diastolic blood pressure readings that are at least 69 mmHg

Explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 80

Standard deviation = 11

Using the empirical rule, what percentage of adult males have diastolic blood pressure readings that are at least 69 mmHg?

The normal probability distribution is symmetric, which means that 50% of the measures are above the mean and 50% are below.

69 = 80 - 11

So 69 is one standard deviation below the mean.

By the Empirical Rule, 68% of the measures below the mean, are within 1 standard of the mean. So

50% + 0.68*50% = 84%

84% of adult males have diastolic blood pressure readings that are at least 69 mmHg