Respuesta :
Answer:
a) 95.4% of the men are between 169cm and 193cm
b) 68.2% of the men are between 175cm and 187cm
Step-by-step explanation:
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 Heights of men on a baseball team.
From the problem we have the mean and the standard deviation for the random variable X. [tex]E(X)=181, Sd(X)=6[/tex]
So we can assume [tex]\mu=181 , \sigma=6[/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 2
• The probability of obtain values within two deviation's from the mean is 0.95 4
• The probability of obtain values within three deviation's from the mean is 0.997
a. 169cm and 193cm
We can find the deviations from the mean using the z score formula given by:
[tex]Z=\frac{X-\mu}{\sigma}[/tex]
IF we find the z score for 169 and 193 we got:
[tex]Z=\frac{169-181}{6}=-2[/tex]
[tex]Z=\frac{193-181}{6}=2[/tex]
Since we are within 2 deviations from the mean and using the empirical rule we have 95.4% of the data
95.4% of the men are between 169cm and 193cm
b. 175cm and 187cm
We can find the deviations from the mean using the z score formula given by:
[tex]Z=\frac{X-\mu}{\sigma}[/tex]
IF we find the z score for 169 and 193 we got:
[tex]Z=\frac{175-181}{6}=-1[/tex]
[tex]Z=\frac{187-181}{6}=1[/tex]
Since we are within 1 deviations from the mean and using the empirical rule we have 68.2% of the data
68.2% of the men are between 175cm and 187cm
