Question

Assume the upper arm length of males over 20 years old in the United States is approximately Normal with mean 39.3 centimeters (cm) and standard deviation 2.4 cm. Use the 68 – 95 – 99.7 rule to answer the given questions.a. What range of lengths covers almost all. 99.7% of this distribution?b. What percent C of men over 20 have upper arm lengths less than 36.9 cm?

Answer 1

Answer:

a) For this case we can use this:

$P(X$

$P(X>\mu +3*\sigma)=P(X>46.5)=0.0015$

So the range of lengths would be 32.1 and 46.5 since on the middle of these two values we have 1-0.0015-0.0015=0.997 or 99.7 % of the values.

b) $P(X>36.9)$

And using the complement rule we have:

$P(X>36.9)= 1-P(X$

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. $E(X)=39.3, Sd(X)=2.4$

So we can assume $\mu=39.3 , \sigma=2.4$

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 have the following probabilities using this rule:

$P(X$    

$P(X>\mu +\sigma)=P(X >41.7)=0.16$  

$P(X$    

$P(X>\mu +2*\sigma)P(X>44.1)=0.025$

$P(X$

$P(X>\mu +3*\sigma)=P(X>46.5)=0.0015$

Part a

For this case we can use this:

$P(X$

$P(X>\mu +3*\sigma)=P(X>46.5)=0.0015$

So the range of lengths would be 32.1 and 46.5 since on the middle of these two values we have 1-0.0015-0.0015=0.997 or 99.7 % of the values.

Part b

For this casee we want this probability:

$P(X>36.9)$

And using the complement rule we have:

$P(X>36.9)= 1-P(X$