Download Bell Ringer - Lake County Schools

Bell Ringer
1.
Mrs. Hall’s class had a test score average of 81.2%. The standard
deviation was 5.8%. Assuming the scores had a normal
distribution, what range did approximately 68% of the students
score?
2.
Which of the following show approximately normal
distribution? Choose all that apply.
A.
B.
C.
Z Scores
April 7, 2016
 Also known as a standard score
 Indicates how many standard deviations an element is
from the mean
 Can be placed on normal distribution curves (AKA bell
curves)
Z Scores
 Range from -3 to +3
 Used to compare results from a test to a normal
population
 A z-score can tell you where a person’s weight is compared
to the average population’s mean weight
 You need to know…
 Mean, µ
 Standard Deviation, ơ
𝑥−𝜇
𝑧=
𝜎
 Example: A test score is 190. The mean score was 150. The
standard deviation was 25. How many standard deviations away
from the mean is the test score?
Z Score
Formula
 Determine each variable.
 X=190
 µ=150
 ơ=25
𝑧=
𝑧=
𝑥−𝜇
𝜎
190−150
25
 𝑧 = 1.6
 The test score 190 is 1.6 standard deviations above the mean.
 You take the SAT and score 1100. The mean score for
the SAT is 1026 and the standard deviation is 209. How
well did you score on the SAT compared to the average
test taker?
𝑧=
SAT Scores
𝑧=
𝑥−𝜇
𝜎
1100−1026
209
 𝑧 = 0.354
 You scored 0.354 standard deviations above the
average
 In order to know what percentage of test takers scored
below you, use a z-table
0.6368
How to Read a
Z-Table
1.
Find the first digit/decimal place of your z-score on the left column.
2.
Find the hundredths place for your answer for the z-score in the top row
3.
Where they intersect gives you the decimal representation for what
percent scored below you
Therefore, on the SAT you scored above 63.68% of other test takers.
 The price of printers in a store have a mean of $240 and
a standard deviation of $50. The printer that you
eventually chose costs $340.
Printer Prices
 What is the z-score for the price of your printer?
𝑧=
340 − 240
=2
50
 How many standard deviations above the mean was your
printer?
The price of my printer was 2 standard deviations
above the mean price
Height
 Joey is 63 inches tall. The mean height for
boys at his school is 68.1 inches, and the
standard deviation of the boys’ heights is
2.8 inches. What is the z-score for Joey’s
height? Round to the nearest hundredth.
What does the value mean? Using the ztable, what percent of boys at Joey’s
school are shorter than Joey?
 Z-Score: -1.82
 The z-score means that Joey is 1.82
standard deviations below the average
height for boys at his school.
 Only 3.44% of the boys at Joey’s school
are shorter than him.
 You can use the Z-table to determine the probability of
an event happening.
 Be careful with the question! It is different depending
on if they are asking for less than or greater than!
Z-Scores and
Probability
 Z-table is for values less than. If they ask for greater
than, then you must subtract from z-table value from 1
Highway
Speeds
 Vehicle speeds at a highway location have a
normal distribution with a mean of 65 mph
and a standard deviation of 5 mph. What is
the probability that a randomly selected car
is going 73 mph or less?
 Draw a sketch
 Determine the z-score for 73 mph.
 Z-score=1.60
 Use a z-table to determine the probability.
 There is a 94.52% chance the car is going less
than 73 mph.
Pulse Rate
 Suppose pulse rates of adult females have a normal curve
distribution with a mean of 75 bpm and a standard deviation of 8.
What is the probability that a randomly selected female has a
pulse rate greater than 85?
 Draw a sketch
 Determine the z-score for 85 bpm.
 Z-score=1.25
 Use a z-table to determine the probability.
 BE CAREFUL! We must subtract from 1!
 The probability that a randomly selected female will have a pulse
rate above 85 bpm is 10.56%.
 A swimmer named Amy specializes in the 50-meter backstroke. In
competition, her mean time for the event is 39.7 seconds, and the
standard deviation of her times is 2.3 seconds. Assume that Amy’s
times are approximately normally distributed. Estimate the
probability that Amy’s time is between 37 and 44 seconds. Then
estimate the probability that Amy’s time is more than 45 seconds.
Challenge:
Swimming
Probability that her time is
between 37 and 44 seconds:
Approximately 84.8%
Probability that her time is
more than 45 seconds:
Approximately 3%