Download 2.2 Normal Distributions

2.2 Normal Distributions
Normal Distributions:
-All have the same overall shape: Symmetric, Unimodal, and Bell-Shaped
-Exact density curve is given by its 𝜇 and 𝜎
- Mu and median is the same
-Notation: 𝑁(𝜇, 𝜎)
68-95-99.7 Rule
68% of observations fall
within 𝜎 of the 𝜇
95% of observations fall
within 2 𝜎 of the 𝜇
99.7% of observations
fall within 3 𝜎 of the 𝜇
Example 1
The length of wear on Spinning Tires is normally distributed with a mean of 60,000 miles and a standard
deviation of 5,000 miles. Draw and label the Normal Distribution. Shade the region under the curve that
represents the fraction of tires that last between 50,000 miles and 70,000 miles. What fraction of tires
does that represent?
Standard Normal Distribution: standardizing a variable that has any normal distribution
-Normal distribution with 𝑁(0,1)
𝑧=
𝑥−𝜇
𝜎
-Table A is a table of areas under the standard normal curve
-Table entry for each values z, is the area under the curve to the left of z.
Example 2
a. Find the proportion of observations from the standard Normal distribution that are less than 2.22.
b. Find the proportion of observations from the standard Normal distribution that are greater than
-2.15.
**Always sketch a curve, mark the z-value, and shade the area of interest. Don’t turn off your Brain!!!
Then: Standardize to put the problem in terms of z.
Use the table: Subtract when necessary
Conclusion: write in context of the problem.
Example 3
The level of cholesterol in the blood is important because high cholesterol levels may increase risk of
heart disease. The distribution of blood cholesterol levels in a large population of people of the same age
and sex is roughly Normal.
For 14-year-old boys, the mean is 𝜇 = 170 𝑚𝑖𝑙𝑙𝑖𝑔𝑟𝑎𝑚𝑠 𝑜𝑓 𝑐ℎ𝑜𝑙𝑒𝑠𝑡𝑒𝑟𝑜𝑙 𝑝𝑒𝑟 𝑑𝑒𝑐𝑖𝑙𝑖𝑡𝑒𝑟 𝑜𝑓 𝑏𝑙𝑜𝑜𝑑 (𝑚𝑔/𝑑𝑙)
and the standard deviation is 𝜎 = 30 𝑚𝑔/𝑑𝑙 . Levels above 240 mg/dl may require medical attention.
a.
What percent of 14-year-old boys have more than 240 mg/dl of cholesterol?
b. What percent of 14-year-old boys have blood cholesterol between 170 and 240 mg/dl?
z-scores outside the range covered by Table A: below -3.40 then use the area of 0
above 3.49 then use the area of 1
We can use the table backwards as well:
Example 4
Scores on the SAT Verbal test in recent years follow approximately the 𝑁(505, 110) distribution. How
high must a student score in order to place in the top 10% of all students taking the SAT?