Download Chapter 6 - Z scores/normal curve presentations

Suppose you take the SAT test and the
ACT test. Not using the chart they
provide, can you directly compare your
SAT Math score to your ACT math score?
Why or why not?
We need to standardized these scores so
that we can compare them.
z score
• Standardized score
• Has m = 0 & s = 1
z 
x  mx
sx
Let’s explore . . .So what does the z-score
tell you?
Suppose the mean and standard deviation of a
distribution are m = 50 & s = 5.
If the x-value is 55, what is the z-score?
1
If the x-value is 45, what is the z-score?
-1
If the x-value is 60, what is the z-score?
2
What do these z scores mean?
-2.3
1.8
6.1
-4.3
2.3 s below the mean
1.8 s above the mean
6.1 s above the mean
4.3 s below the mean
Jonathan wants to work at Utopia
Landfill. He must take a test to see if
he is qualified for the job. The test
has a normal distribution with m = 45
and s = 3.6. In order to qualify for the
job, a person can not score lower than
2.5 standard deviations below the
mean. Jonathan scores 35 on this test.
Does he get the job?
No, he scored 2.78 SD below the mean
Sally is taking two different math
achievement tests with different means
and standard deviations. The mean score
on test A was 56 with a standard deviation
of 3.5, while the mean score on test B was
65 with a standard deviation of 2.8. Sally
scored a 62 on test A and a 69 on test B.
On which test did Sally score the best?
She did better on test A.
Density Curves
•
•
•
•
Can be created by smoothing histograms
ALWAYS on or above the horizontal axis
Has an area of exactly one underneath it
Uses m & s to represent the mean & standard
deviation
• Describes the proportion of observations that fall
within a range of values
• Is often a description of the overall distribution
Normal Curve
• Bell-shaped, symmetrical curve
• Transition points between cupping
upward & downward occur at m + s
and m – s
• As the standard deviation increases,
Let’s use our calculator to
the curve flattens & spreads
graph some normal curves
• As the standard deviation decreases,
the curve gets taller & thinner
Empirical Rule
• Approximately 68% of
the
Can ONLY be used
observations are withinwith
1s normal
of m curves!
• Approximately 95% of the
observations are within 2s of m
• Approximately 99.7% of the
observations are within 3s of m
• See p. 181
The height of male students at
PWSH is approximately normally
distributed with a mean of 71 inches
and standard deviation of 2.5 inches.
a) What percent of the male students
are shorter than 66 inches? About 2.5%
b) Taller than 73.5 inches? About 16%
c) Between 66 & 73.5 inches?
About 81.5%
Remember the bicycle problem? Assume that the
phases are independent and are normal
distributions. What percent of the total setup
times will be more than 44.96 minutes?
First, find the mean
& standard
deviation for the
total setup time.
Phase
Mean
SD
Unpacking
Assembly
Tuning
3.5
21.8
12.3
0.7
2.4
2.7
2.5%