Transcript Chapter 1 - Pendleton County Schools

Assignments 2.1-2.4
Due Wednesday 12, 2014
Assignments 2.5-2.6
Due Friday 14, 2014
Sta220 - Statistics
Mr. Smith
Room 310
Class #5
Section 2.6-2.7
Revisit 2.6
Sanitation Inspection of Cruise
Ships
a. Find the mean and standard deviation of the
sanitation scores.
b. Calculate the intervals 𝑥 ± 𝑠, 𝑥 ± 2𝑠,
𝑎𝑛𝑑 𝑥 ± 3𝑠
c. Find the percentage of measurement in the
data set that fall within each of the intervals
in part b. Do these percentages agree with
either Chevyshev’s rule or the empirical rule?
Sanitation Inspection of Cruise
Ships
Sanitation Inspection of Cruise
Ships
a. Find the mean and standard deviation of the
sanitation scores.
Sanitation Inspection of Cruise
Ships
b. Calculate the intervals 𝑥 ± 𝑠, 𝑥 ± 2𝑠,
𝑎𝑛𝑑 𝑥 ± 3𝑠
x ± s = (90.736, 100.662)
x ± 2s = (85.773, 105.625)
x ± 3s = (80.81,110.588)
x = 95.699
s = 4.963
Sanitation Inspection of Cruise
Ships
c. Find the percentage of measurement in the
data set that fall within each of the intervals in
part b. Do these percentages agree with either
Chevyshev’s rule or the empirical rule?
x ± s = (90.736, 100.662)
169
186
≈ .907 = 90.7%
x ± 2s = (85.773, 105.625) )
179
186
≈ .962 = 96.2%
x ± 3s = (80.81,110.588) )
182
186
≈ 978. = 97.8%
Land Purchase Decision
On the basis of this information, what is the
buyer’s decision?
Lesson Objectives
You will be able to:
1. Interpret percentiles(2.7)
2. Determine and interpret quartiles (2.7)
3. Determine and interpret z-scores (2.7)
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Lesson Objective #1-Interpret percentiles(2.7)
Central tendency and variability describe the
general nature of quantitative data.
Descriptive measures of the relationship of
measurement to the rest of the data are called
measures of relative standing.
One measure of the relative standing of
measurement is its percentile ranking.
For any set of n measurements (arranged in
ascending or descending order), the pth
percentile is a number such that p% of the
measurements fall below that number and
(100 − 𝑝%) fall above it.
The pth percentile is a value such
that p percent of the observations
fall below or at that value.
Location of 90th percentile for test grades
Copyright © 2013 Pearson
Education, Inc.. All rights
reserved.
EXAMPLE
Interpret a Percentile
The Graduate Record Examination (GRE) is a
test required for admission to many U.S.
graduate schools. The University of Pittsburgh
Graduate School of Public Health requires a
GRE score no less than the 70th percentile for
admission into their Human Genetics MPH or
MS program.
(Source: http://www.publichealth.pitt.edu/interior.php?pageID=101.)
Interpret this admissions requirement.
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EXAMPLE
Interpret a Percentile
In general, the 70th percentile is the score such
that 70% of the individuals who took the exam
scored worse, and 30% of the individuals scores
better. In order to be admitted to this program,
an applicant must score as high or higher than
70% of the people who take the GRE. Put
another way, the individual’s score must be in
the top 30%.
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Lesson Objective #2 Determine and interpret
quartiles (2.7)
Quartiles divide data sets into fourths, or four equal parts.
• The 1st quartile, denoted Q1, divides the bottom 25% the
data from the top 75%. Therefore, the 1st quartile is
equivalent to the 25th percentile.
• The 2nd quartile divides the bottom 50% of the data from
the top 50% of the data, so that the 2nd quartile is
equivalent to the 50th percentile, which is equivalent to the
median.
• The 3rd quartile, denoted Q3, divides the bottom 75% of
the data from the top 25% of the data, so that the 3rd
quartile is equivalent to the 75th percentile.
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EXAMPLE
Finding and Interpreting Quartiles
A group of Brigham Young University—Idaho students
(Matthew Herring, Nathan Spencer, Mark Walker, and
Mark Steiner) collected data on the speed of vehicles
traveling through a construction zone on a state
highway, where the posted speed was 25 mph. The
recorded speed of 14 randomly selected vehicles is
given below:
20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40
Find and interpret the quartiles for speed in the
construction zone.
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EXAMPLE
Finding and Interpreting Quartiles
Step 1: The data is already in ascending order.
Step 2: There are n = 14 observations, so the median,
or second quartile, Q2, is the mean of the 7th and 8th
observations. Therefore, M = 32.5.
Step 3: The median of the bottom half of the data is the
first quartile, Q1.
20, 24, 27, 28, 29, 30, 32
The median of these seven observations is 28.
Therefore, Q1 = 28. The median of the top half of the
data is the third quartile, Q3. Therefore, Q3 = 38.
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Interpretation:
• 25% of the speeds are less than or equal to the first
quartile, 28 miles per hour, and 75% of the speeds are
greater than 28 miles per hour.
• 50% of the speeds are less than or equal to the second
quartile, 32.5 miles per hour, and 50% of the speeds are
greater than 32.5 miles per hour.
• 75% of the speeds are less than or equal to the third
quartile, 38 miles per hour, and 25% of the speeds are
greater than 38 miles per hour.
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Sodium in Cereal
Sodium in Cereal
0
50 70 100 130 140 140 150 160 180
180 180 190 200 200 210 210 220 290 340
Lesson Objective #3 Determine and interpret zscores (2.7)
Z-Scores
Interpretation of z-Scores from Mound-Shaped
Distributions of Data
• Approximately 68% of the measurements will
have a z-score between -1 and 1
• Approximately 95% of the measurements will
have a z-score between -2 and 2.
• Approximately 99.7% (almost all) of the
measurements will have a z-score between -3
and 3.
Figure 2.30 Population z-scores for a mound–
shaped distribution
Copyright © 2013 Pearson
Education, Inc.. All rights
reserved.
Example: z-Score
Suppose a sample of 2,00 high school seniors’
verbal SAT scores is selected. The mean and
standard deviation are
𝑥 = 550
𝑠 = 75
Suppose Joe Smith’s Score is 475. What is his
sample z-score?
Figure 2.29 Verbal SAT scores of high school
seniors
𝑥 − 𝑥 475 − 550
𝑧=
=
= −1. 0
𝑠
75
Copyright © 2013 Pearson
Education, Inc.. All rights
reserved.
This tells us that Joe Smith’s score is 1.0 standard
deviations below the sample mean; in short, his
sample z-score is -1.0
EXAMPLE Using Z-Scores
The mean height of males 20 years or older is
69.1 inches with a standard deviation of 2.8
inches. The mean height of females 20 years or
older is 63.7 inches with a standard deviation of
2.7 inches. Data based on information obtained from
National Health and Examination Survey.
N
N
(69.1, 2.8)
( , )
N
(63.7, 2.7)
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EXAMPLE Using Z-Scores
Shaquille O'Neal’s height is 7’1’’ .
Candace Parker’s height is 6’4’’.
Who is relatively taller?
http://en.wikipedia.org/wiki/Shaquille_O'Neal
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http://www.wnba.com/news/rookieofmonth_080603.html
EXAMPLE Using Z-Scores
Shaquille O'Neal’s height is 7’1’’ thus his
z-score is 85  69.1 15.9
2.8

2.8
 5.68
Candace Parker’s height is 6’4’’ which produces
a z-score of 76  63.7  12.3  4.56
2.7
2.7
O’Neal is relatively taller as he has a larger
z-score.
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