Download Normal Distribution Notes

Math 3
11-9: The Normal Distribution
Unit 4 Day 2
normal distribution – has data that vary randomly about the mean; has a “normal curve” – also known as the
bell
curve
Exampe 1) The following normal distribution describes the repair times for a local auto shop. Answer the
questions below:
48
66
84
156
102
120
138
156
a) What is the mean repair time for the auto shop?
b) What percentage of repairs take between 94 and 120 minutes?
c) What percentage of the repairs take less than 2 hours?
d) What percentage of repairs are between 94 minutes and 138 minutes?
e) If the shop did 200 repairs this week, approximately how many of them took more than 2 hours?
Example 2) Sketch a normal curve for the following distribution: mean = 45, standard deviation = 5
You try! Sketch a normal curve for the following distribution: mean = 60, standard deviation = 15
z-score – the number of standard deviations away from the mean a specific observation is
Examples – ALWAYS SKETCH A NORMAL CURVE AND LABEL THE X-AXIS.
1. A forest products company claims that the amount of usable lumber in its harvested trees averages 172
cubic feet and has a standard deviation of 12.4 cubic feet. Assume that these amounts have
approximately a normal distribution.
a. What proportion of trees contain more than 150 cubic feet?
Step 1: Draw the normal curve and label the x-axis. Shade appropriately for more than
150 cubic ft3.
Step 2: Find the z-score corresponding to 150 ft3. Interpret this in terms of standard deviations
away from the mean.
Step 3: Hit 2nd-VARS-2:normalcdf(…then enter (lower bound, upper bound—use an incredibly large number,
such as 9^99 for upper bound). The answer is the proportion corresponding to your z-score. Write as a
percentage.
b. What proportion of trees contain between 175 and 190 cubic feet?
Repeat steps 1 through 3.
2. Healthy 10-week-old domesticated kittens have average weight 24.5 oz. with a standard deviation of
5.25 oz. The distribution is approximately normal.
a. A kitten is designated as dangerously underweight when, at 10 weeks, it weighs less than 10.0 oz.
What proportion of healthy kittens will designated as dangerously underweight?
b. What is the median weight of the kittens?
Step 1: Find the percentile that corresponds with the median of a data set.
Step 2: Hit 2nd-VARS-3:InvNorm( and type in the percentile.
Step 3: This is a z-score. Plug the mean, standard deviation, and z-score into the z-score formula and solve for
the missing x-value.
Extra Practice with z-scores
1. Suppose that there are 100 franchises of Betty's Boutique in similar shopping malls across
America. The gross Saturday sales of these boutiques are approximately normally distributed
with a mean of $4610 and a standard deviation of $370. Draw a normal curve below and
label the horizontal axis.
(a) Find the z-scores of each of the following gross Saturday sales amounts: $3870, $4425, and
$5535.
(b) What percentage of Betty's Boutique franchises had gross Saturday sales between $4425 and
$5535? Use the z-scores you found in part (a) and the normalcdf function in the calculator.
(c) What percentage had gross Saturday sales between $3870 and $5535? Use the z-scores you
found in part (a) and normalcdf function in the calculator.
(d) What percentage of stores had gross Saturday sales less than $5535?
2. Suppose that a certain insect has a mean lifespan of 5.6 days with a standard deviation of 1.2
days. Assume that the lifespan of this insect is approximately normally distributed. Draw a
normal curve below and label the horizontal axis.
(a) Calculate the percentage of insects with a life-span between 3.2 and 8 days.
(b) Calculate the percentage of insects with a life-span between 2.6 and 8.6 days.
(c) What percentage of insects will live longer than 3.32 days?
(d) What percentage of insects will live less than 6.56 days?