Download Normal Distributions

Warm Up

May 20th
Please pick up the 11-2 Enrichment sheet from
the cart and get started (both sides)
Tougher Probability

Carson is not having much luck lately. His car will
only start 80% of the time and his moped will only
start 60% of the time.
 Draw a tree diagram to illustrate the situation.
 Use the diagram to determine the chance that
 Both will start
 He has to take his car.
 He has to take the bus.
Tougher Probability



A box contains 3 red, 2 blue and 1 yellow marble.
Draw a tree diagram to represent drawing 2 marbles.
With replacement
Without replacement
Find the probability of getting two different
colors:
 If replacement occurs
 If replacement does not occur
Tougher Probability

A bag contains 5 red and 3 blue marbles. Two marbles
are drawn simultaneously from the bad. Determine
the probability that at least one is red.
Probability Check
1.
2.
13/24
47/70
1/5
6. 2/5
7. 3/5
8. 4/5
9. 7/18
10. 11/18
11. 5/6
12. 5/9
5.
144π/5000 = .0905
15. 3/625 = .0048
16. (144π + 24)/5000 = .0953
17. 1-(144π/5000) = .9095
14.
22.
a) 1,1 2,1
1,2 2,2
1,3 2,3
1,4 2,4
b) 16
c) 3/16
3,1
3,2
3,3
3,4
4,1
4,2
4,3
4,4
Probability & Trials

Ch. 19 Dice Simulation
Normal Distribution

Bell-shaped curve defined by the mean and
standard deviation of a data set.
Characteristics of a Normal
Distribution

What do the 3 curves have in common?
Characteristics of a Normal
Distribution

The curves may have different mean and/or
standard deviations but they all have the
same characteristics
 Bell-shaped continuous curve
 Symmetrical about the mean
 Mean, median and mode are the same and
located at the center
 It approaches, but never touches the x axis
 Area under the curve is always 1 (100%)
Is it Normal?
Empirical Rule

If data follows a normal distribution…
68% of it will be within 1 standard
deviation
 95% of it will be within 2 standard
deviations
 99% of it will be within 3 standard
deviations

Empirical Rule
Empirical Rule
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?
Examples (cont.)
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?
You Try!
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?

Back to Heights Examples
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?
Standard Deviations
How to be more specific…

A standard normal distribution is the set of
all z-scores (or z-values).
 It represents how many standard
deviations a certain data point is away from
the mean.
 The z-score is positive if the data value lies
above the mean and negative if it’s below
the mean.
How to find Z-Scores
Examples

Find z if X = 24, µ = 29 and σ = 4.2

You Try! Find Z if X = 19, μ = 22, and σ = 2.6
Back to Heights Examples
More Specific
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?
You Try!
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.
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?
Back to Heights 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
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?
Area & Probability
2nd
 DISTR (Vars button)
 normalcdf(minimum z value, maximum z value)
Back to h) If you pick a student at random,
what is the probability they will be between
65 and 69 inches tall? (remember mean = 67
and SD = 2.5)

More Examples


The temperatures for one month for a city in
California are normally distributed with mean
= 81 degrees and sd = 6 degrees. Find each
probability and use a graphing calculator to
sketch the corresponding area under the
curve.
a. P(70 < x < 90)
More Examples


The scores on a standardized test are
normally distributed with mean = 72 and sd =
11. Find each probability and use a graphing
calculator to sketch the corresponding area
under the curve.
Find: P(65 < x < 85)
You Try Again!
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.
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 about these?

The temperatures for one month for a city in
California are normally distributed with m =
81 degrees and s = 6 degrees. Find P (x > 95)
What about this one?

The scores on a standardized test are
normally distributed with mean = 72 and sd =
11. Find P (x < 89)
You Try!
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
i) What is the probability they will be more than
70 inches tall?

j) What is the probability they will be less than
61 inches tall?
You Try Again!
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.
i) What is the probability it will have more than
1.13 liters?

j) What is the probability it will have less than
1.04 liters?