Respuesta :
Answer:
84% of adult males have diastolic blood pressure readings that are at least 69 mmHg
Step-by-step 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
Answer:
For this case we want this probability:
[tex] P(X >69)[/tex]
And we can use the z score formula given by:
[tex] z = \frac{x -\mu}{\sigma}= \frac{69-80}{11}= -1[/tex]
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:
[tex] P(z>1) = 1-0.16 = 0.84[/tex]
So we expect about 84% of readings of at least 69 mm Hg
Step-by-step 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. [tex]E(X)=80, Sd(X)=11[/tex]
So we can assume [tex]\mu=80 , \sigma=11[/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
Solution to the problem
For this case we want this probability:
[tex] P(X >69)[/tex]
And we can use the z score formula given by:
[tex] z = \frac{x -\mu}{\sigma}= \frac{69-80}{11}= -1[/tex]
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:
[tex] P(z>1) = 1-0.16 = 0.84[/tex]
So we expect about 84% of readings of at least 69 mm Hg
