Consider a data value X that represents the weights of bodybuilders. Find the z-values corresponding to X=180 and X=190.
B
b
Consider a data value X that represents the weights of bodybuilders. Find the z-values corresponding to X=195.
A
a
≈ 638 bodybuilders
B
b
≈ 22.4 %
Practice makes perfect
a
We have been told that the data is normally distributed with a mean of μ = 190.6 and a standard deviation of σ = 5.8. We want to find what percent of bodybuilders weigh between 180 and 190 pounds.
P( 180 < X < 190)
To do that, we will use the z-values.
Formula for z-values
The z-value for a data value X in a set of normally distributed data is given by z= X-μσ, where μ is the mean and σ is standard deviation.
In our case, X represents the weights of the bodybuilders. We want to find the corresponding z-value for X= 180 and X= 190. Let's start with X= 180.
The z-values corresponding to X= 180 and X= 190 are z=- 1.828 and z=- 0.103. The percentage of bodybuilders who weigh between 180 and 190 pounds is equal to the area between these z-values. To find this area, we can use a graphing calculator. Push 2ND and VARS. Then, scroll down to the second option and push ENTER.
Now, set lower to the z-value z=- 1.828 and upper to the z-value z=- 0.103. Next, press PASTE and calculate the area by pressing ENTER.
The area for - 1.828 < z < - 0.103 is about 0.425, which tells us that about 42.5 % of 1500 bodybuilders weigh between 180 and 190 pounds. Let's calculate the corresponding number of bodybuilders.
Therefore, about 638 bodybuilders weigh between 180 and 190 pounds.
b
This time we want to find the probability that a randomly selected bodybuilder has a weight greater than 195 pounds. We can do this by using the z-values again.
z=X-μ/σ
Once again, let X represent the weights of bodybuilders. Let's find the z-value corresponding to X= 195 pounds. We know that the mean μ is equal to 190.6 and that he standard deviation σ is 5.8. Let's substitute these values.
We found that X=195 corresponds to the z-value z=0.759. Therefore, the probability that a randomly selected bodybuilder weighs more than 195 pounds is the area for z-values greater than 0.759. To find this, we will again use a graphing calculator. Push 2ND and VARS. Then, scroll down to the second option and push ENTER.
We set lower to the z-value z=0.759. Since the majority of the values of standard normal distribution are within ± 4 of the mean, we can set the upper z-value to 4. Next, press PASTE and calculate the area by pressing ENTER once again.
We conclude that the probability that a randomly selected bodybuilder has a weight greater than 195 pounds is about 0.224 or 22.4 %.
