Download 9.2 Normal Curve Notes

Algebra 2
Unit 9
Day 2
Notes
Name:________________________
The Normal Curve
Date:___________
Block:______
Today:
 You will learn about the normal distribution curve that will include the empirical rule and the area under the
curve.
 At the end of class, you will be able to express the area under the curve as a percentage using z-scores and
your calculator or by using the empirical rule. You will also be able to determine a set of values given an area
under the curve.
Warm-Up: SOL Review
Simplify the following completely.
1.
(4𝑧 7 ) (2𝑧 4 )3
2.
24𝑥 9
12𝑐 4 𝑦 6
3.
(3𝑥 6 )3
∙
20𝑐 2 𝑦 8
Write each expression in exponential form.
4.
3
4
5.
4
73
6.
5
a 9b15
Write each expression in radical form.
7. 3 0.5
8. 5

3
5
9. 4m  15
2
Simplify each expression. Assume all variables are positive.
10. 64 27 x  24 3x 3
3
11.
3
625 xy6
5x 7 y 2
Simplify.
12. 33 16  43 54  3 128
13.
 11  3 7 
2
10𝑐𝑦 3
3𝑐 2 𝑦 2
Let’s Review…
The standard deviation measure the dispersion of the data from its ___________. The _____________ the
deviation, the more spread apart the data.
A z-score measures a score’s relationship to the mean in a group of scores. A z-score of _____________ means the
score is the same as the mean. A z-score can also be ____________ or __________________, indicating whether it is
above or below the mean and by how many standard deviations.
The standard normal distribution is the normal distribution with mean zero and standard deviation one.
Review
1) Write 2 notations that represent the mean of a set of data___________________
2) Describe what a standard deviation is.
3) What symbol is used to represent standard deviation?
4) What is the formula for the z-score?
5) What does the z-score tell us?
6) A positive z-score means_____________________________________
7) A negative z-score means_____________________________________
8) What does it mean when the standard deviation is small?
9) What does it mean when the standard deviation is large?
10) What does variance mean and what is the notation for it?
Example:
Mean = 60
SD = 5

What number is 1 standard deviation
above the mean?

What number is 1 standard deviation
below the mean?

What number is 2 standard deviations
above the mean?

What is the z-score of the number 65 in this normal distribution?

What is the z-score of the number 50 in this normal distribution?

What number has a z-score of z = 2.2 in this normal distribution?

What number has a z-score of z = -1.6 in this normal distribution?
The Bell Curve
The Empirical Rule, also known as the 68-95-99.7 Rule states that almost all data will fall within _________
standard deviations of the mean. Broken down, the empirical rule show that _______ percent will fall within the first
standard deviation, ________ percent within the first two standard deviations and ________ percent will fall within
the first three standard deviations of the mean.
But what happens when you are not exactly 1, 2 or 3 standard deviations away – how do we determine
percent%???
On the Calculator….
normalcdf (1st z-score, 2nd z-score) x 100 =
To find the shaded area below –
2nd DISTR
normalcdf( - hit ENTER
enter cut points: normalcdf(-.5, 1.0) – hit ENTER
Note – theoretically the standard Normal model extends forever in both directions, but you can’t tell the calculator
to use infinity as the right cut point… instead let’s just always use 99 (or -99).
Examples: Use your calculator to find the probability of the given x-value in a normal curve with a mean of 150 and
standard deviation 25.
1. x less than 140
2.
x more than 145
3.
From Percentiles to Scores: z in Reverse
Calculator functions:
invNorm (percent of distribution up to a certain point, written as a decimal) =
What z-score represents the first quartile in a Normal model?
2nd DISTR
invNorm( - hit ENTER
enter desired percentile (in decimals): invNorm(.25) – hit ENTER
x between 80 and 130
Example: In a standard Normal model, what value(s) of z cut(s) off the region described?
a. the lowest 12%
Use invNorm(.12) to answer a
Use invNorm(.7) to answer b
b. the highest 30%
c. the highest 7%
d. the middle 50%
Practice: Determine the area under the standard normal curve for the given z-scores. Sketch your diagram.
3. z is less than 1.3
4. z is more than -1.21
5. z is between -0.36 and 1.42
6. A patient recently diagnosed with Alzheimer’s disease takes a cognitive abilities test and scores a 45. The mean on
this test is 52 and the standard deviation is 5. What is the patient’s percentile rank?
You’ve Got Problems!
1. What percent of a standard Normal model is found in each region?
a. z > 1.5
b. z < 2.25
c. -1 < z < 1.15
d. |𝑧| < 1.28
2. In a standard Normal model, what value(s) of z cut(s) off the region described?
a. the lowest 12%
Use invNorm(.12) to answer a
Use invNorm(.7) to answer b
b. the highest 30%
c. the highest 7%
d. the middle 50%
Determine the area under the standard normal curve for the given z-scores. Sketch your diagram.
3. z is less than 1.3
4. z is less than -2.01
5. z is more than 0.32
6. z is more than -1.21
7. z is between -0.36 and 1.42
9. z is between -1.4 and 1.83
10. Your teacher reported that the class average on the last test was an 80% with a standard
deviation of 10%.
Assuming the students’ grades followed a normal distribution curve, answer each of the following.
a) If one student is selected at random, find the probability that he got less than a 60%.
b) An A on the test was anything between a 90% and a 100%. If a student is selected at random,
find the probability that she passed the test.
c) On the following test, the score distribution was still a normal distribution curve, but this time
the mean was a 95% and the standard deviation was 5%. Kelly got a 93% on the first test but a
98% on the second test. On which test did she perform better?