Solving Problems Using the Normal Distribution
Completion requirements
D. Solving Problems Using the Normal Distribution
You now have the tools to solve a variety of problems related to normally distributed data. It's time to put the pieces together.
Example 1
In the early 19 th century, a study measured the chest sizes of Scottish militiamen. The data obtained is listed in the table below.
1. Draw a histogram to determine whether this data is reasonably normal.
The data is symmetrical and bell-shaped. It appears to be fairly normal.
2. Determine the mean and standard deviation of the data.
Use technology to determine the mean and standard deviation.
µ = 39.83
s = 2.05
3. Suppose one of the men is selected at random. Determine the probability that his chest is
a. larger than 42 inches.
a. larger than 42 inches.
Use the mean and standard deviation to sketch a normal curve that represents the problem.
The shaded region represents the area corresponding to the probability
that a randomly selected man will have a chest size larger than 42.
Converting the measure 42 to a
z
-score will allow you to determine this area.
Now determine the unknown area using technology or a z-score table.
Using a z -score table:
The area to the left of a z -score of 1.06 is 0.8554, so the area to the right is
1 - 0.8554 = 0.1446.
There is approximately a 14.5% chance that the man will have a chest larger than 42 inches.
Using a z -score table:
The area to the left of a z -score of 1.06 is 0.8554, so the area to the right is
1 - 0.8554 = 0.1446.
There is approximately a 14.5% chance that the man will have a chest larger than 42 inches.
3. Suppose one of the men is selected at random. Determine the probability that his chest is
b. between 35 and 39 inches.
b. between 35 and 39 inches.
Use the mean and standard deviation to sketch a normal curve that represents the problem.
Determine the
z
-scores of each measurement.
Now determine the unknown area using a
z
-score table or technology.
Using technology:
The area between the two values is 0.3335. This means there is approximately a 33.4% chance the randomly selected man will have a chest between 35 and 39 inches.
The area between the two values is 0.3335. This means there is approximately a 33.4% chance the randomly selected man will have a chest between 35 and 39 inches.
4. Suppose the measurements of Scottish militiamen were taken again 10
years later and that the findings remained very similar. If 5000 men
were measured, 1000 of the men would have a chest size measuring over
what value?
The conditions are similar, so you can assume µ = 39.83 and s = 2.05.
The conditions are similar, so you can assume µ = 39.83 and s = 2.05.
Sketch a normal curve to represent the problem.
The unknown
z
-score can be found using an inverse function with technology or by using a
z
-score table.
Using the z -score table:
If 20% of the data is above the unknown z -score, then 1 - 0.2 = 0.8 or 80% of the data is below it. Find the value closest to 0.8 on the inside of one of the z -score tables to determine the z -score.
Using the z -score table:
If 20% of the data is above the unknown z -score, then 1 - 0.2 = 0.8 or 80% of the data is below it. Find the value closest to 0.8 on the inside of one of the z -score tables to determine the z -score.
The value 0.7995 is the closest one to 0.8 in the table, so the unknown
z
-score is approximately 0.84.
Now use the
z
-score formula to determine the unknown
x
-value.
1000 of the 5000 militiamen will have a chest size over approximately 41.55 inches.
A video demonstration of the solution for Example 1 is provided.