Download Normal Distribution Guided Notes

NORMAL DISTRIBUTION
EXPERIMENT!
Person 1: Looks at clock and says “START” and “STOP” for
5 seconds
Person 2: Hits start and stop on stopwatch when person 1
tells them to
Person 3: Record time, rounded to the nearest 0.1 second,
in the table to the right
Switch after 10 trials.
+Enter all data values in the calculator at the front.
+Record all data values in the frequency graph on the
board.
Characteristics of a normal curve:
1.
2.
3.
4.
5.
Examples and Non-Examples:
EMPIRICAL RULE:
Says that if your data follows a normal distribution…
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What it looks like:
Times
Examples:
The heights of the 880 students at East Meck High
School are normally distributed with a mean of 67
inches and a standard deviation of 2.5 inches
a. Draw and label the normal curve.
b. 68% of the students fall between what two
heights?
c. What percent of the students are between 59.5
and 69.5 inches tall?
d. Approximately how many students are more
than 72 inches tall?
e. If a student is 62 inches tall, how many standard
deviations from the mean are they?
f.
If a student is 71 inches tall, how many standard
deviations from the mean are they?
g. If you pick a student at random, what is the
probability that they will be between 62 and 72
inches tall?
h. If you pick a student at random, what is the
probability they will be between 65 and 69
inches tall?
i.
j.
A machine used to fill water bottle dispenses slightly
different amounts into each bottle. Suppose the volume
of water in 120 bottles is normally distributed with a
mean of 1.1 liters and a standard deviation of 0.02 liter.
a. Draw and label the normal curve.
b. 95% of the water bottles fall between what two
volumes?
c. What percent of the bottles have between 1.08
and 1.12 liters?
d. Approximately how many bottles of water are
filled with less than 1.06 liters?
e. If a water bottle has 1.16 liters, how many
standard deviations from the mean is it?
f.
If a water bottle has 1.07 liters, how many
standard deviations away from the mean is it?
g. If you pick a water bottle at random, what is the
probability that it will be between 1.06 and 1.12
liters?
h. If you pick a water bottle at random, what is the
probability it will be between 1.05 and 1.11
liters?
What is the probability they will be more than
70 inches tall?
i.
What is the probability it will have more than
1.13 liters?
What is the probability they will be less than 61
inches tall?
j.
What is the probability it will have less than
1.04 liters?
Z-Score/Standard Score:
What it is:
How to find it:
Example 1: Find z if X = 24,  = 29 and  = 4.2
You Try: Find Z if X = 19, μ = 22, and σ = 2.6
To find area under the curve/probability in your calculator:
2nd  Distr  normalcdf( minimum z, maximum z)
The temperatures for one month for a city in
The scores on a standardized test are normally
California are normally distributed with mean = 81
distributed with mean = 72 and sd = 11. Find each
degrees and sd = 6 degrees. Find each probability
probability and use a graphing calculator to sketch the
and use a graphing calculator to sketch the
corresponding area under the curve.
a. Find: P(65 < x < 85)
corresponding area under the curve.
a. P(70 < x < 90)
b. P (x > 95)
b. P (x < 89)
HOMEWORK (Complete on a separate sheet of paper and draw a normal curve for each situation FIRST)
Find the missing values:
As part of a noise pollution study, researchers measured the sound
a. Z if X = 52, μ = 43, and σ = 3.7
level in decibels of a busy city street for 3 days. According to the
b. Z if X = 32, μ = 38, and σ = 2.8
study, the average noise was 82 decibels with a standard deviation
of 6 decibels. Assume that the data are normally distributed.
a. If a normal conversation is held at about 64 decibels,
determine the number of hours during the study that the
noise level was this low.
b. Determine the percent of the study during which the noise
was between 76 decibels and 88 decibels.
The life of a certain brand of AA battery is
normally distributed with mean = 8 hours
and sd = 1.5 hours. Find each probability.
a. The battery will last less than 6 hours
b. The battery will last more than 12
hours
c. The battery will last between 8 and 9
hours
The average blood cholesterol level in adult Americans is 203
mg/dL (milligrams per deciliter) with a standard deviation of 38.8
mg/dL. Find each probability. Assume that the data are normally
distributed.
a. A blood cholesterol level below 160 mg/dL, which is
considered low and can lead to a higher risk of stroke
b. A blood cholesterol level above 240 mg/dL, which is
considered high and can lead to a higher risk of heart
disease
c. A blood cholesterol level between 180 and 200 mg/dL,
which is considered normal
Dion commutes 290 miles each week for work. His car averages 29.6 miles per gallon with a standard deviation of
5.4 miles per gallon. Assume that the data are normally distributed.
a. Approximate the number of miles that Dion’s car gets a gas mileage of 35 miles per gallon or better.
b. For what percentage of Dion’s commute does his car have a gas mileage between 24.2 miles per gallon and
40.4 miles per gallon?