Question

Heights of men on a baseball team have a​ bell-shaped distribution with a mean of 181 cmand a standard deviation of 6 cm.Using the empirical​ rule, what is the approximate percentage of the men between the following​ values?

a. 169cm and 193cm
b. 175cm and 187cm
a. nothing​%of the men are between 169cm and 193cm.​(Round to one decimal place as​ needed.)
b. nothing​%of the men are between 175cm and 187cm.​(Round to one decimal place as​ needed.)

Answer 1

a) 95.4​% of the men are between 169cm and 193cm

b) 68.2​% of the men are between 175cm and 187cm

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.
`E(X)=181, Sd(X)=6`

So we can assume
`\mu=181 , \sigma=6`

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:

`Z=(X-\mu)/(\sigma)`

IF we find the z score for 169 and 193 we got:

`Z=(169-181)/(6)=-2`

`Z=(193-181)/(6)=2`

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:

`Z=(X-\mu)/(\sigma)`

IF we find the z score for 169 and 193 we got:

`Z=(175-181)/(6)=-1`

`Z=(187-181)/(6)=1`

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