Transcript File

Presenting Statistical Data
The heights (in centimeters) of a class of 10th grade
students is shown to the right.
A frequency distribution is a table that shows how
many times each data value appears in a set.
Presenting Statistical Data
The heights (in centimeters) of a class of 10th grade
students is shown to the right.
Part 1 – Create a frequency
distribution.
Presenting Statistical Data
The heights (in centimeters) of a class of 10th grade
students is shown to the right.
A histogram can be used to display the
frequency of data in certain convenient
intervals. It does not display the individual data.
Presenting Statistical Data
The heights (in centimeters) of a class of 10th grade
students is shown to the right.
Part 2 – Create a histogram.
Presenting Statistical Data
The heights (in centimeters) of a class of 10th grade
students is shown to the right.
Part 3 – Use the histogram to answer the questions:
• What interval contains the most students?
• What interval contains the fewest students?
• How many students are 175 cm or taller?
• How many students are less than 170 cm tall?
Presenting Statistical Data
The heights (in centimeters) of a class of 10th grade
students is shown to the right.
A stem-and-leaf plot displays the actual data as
the first part of the number (stem) and last digit
(leaf).
Presenting Statistical Data
The heights (in centimeters) of a class of 10th grade
students is shown to the right.
Part 4 – Create a stem-andleaf plot.
Presenting Statistical Data
The heights (in centimeters) of a class of 10th grade
students is shown to the right.
Part 5 - Numbers used to describe a set of data
are called statistics. Three different statistics used
to measure the central tendency of a distribution
are mean, median, and mode.
Presenting Statistical Data
The heights (in centimeters) of a class of 10th grade
students is shown to the right.
The mode is the number that occurs most
frequently.
Find the mode of this data.
Presenting Statistical Data
The heights (in centimeters) of a class of 10th grade
students is shown to the right.
The median is the middle number of the ordered
set of data.
Find the median of this data.
Presenting Statistical Data
The heights (in centimeters) of a class of 10th grade
students is shown to the right.
The mean is the arithmetic average of the
numbers.
Find the mean of this data.
Presenting Statistical Data
Part 6 - Use the distribution of
the number of sit-ups performed
by students in a PE class in the
stem-and leaf plot to:
Find the mode:
Find the median:
Find the mean:
Learning Log Summary
LT 1 – I can display data using a frequency
distribution, histogram, and stem-and-leaf plot
and compute the measures of central
tendency.
A histogram / stem-and-leaf plot is…
The measures of central tendency are…
Closure
Homework
pg. 711 ~ 1-13 (All)
Presenting Statistical Data
The accompanying stem-andleaf plot represents Ben’s test
scores for the year.
Find the median of the data:
Find the median of the lower half of the data:
Find the median of the upper half of the data:
Analyzing Statistical Data
The accompanying stem-andleaf plot represents Ben’s test
scores for the year.
Dispersion describes how spread a data set is.
Analyzing Statistical Data
The accompanying stem-andleaf plot represents Ben’s test
scores for the year.
One measure of dispersion is the range. It is the
distance between the largest and smallest
numbers in the data set.
The range of Ben’s test scores is…
Analyzing Statistical Data
The accompanying stem-andleaf plot represents Ben’s test
scores for the year.
A box-and-whisker plot shows the median, first
and third quartiles, and the range of a
distribution.
Analyzing Statistical Data
Ex) The accompanying boxand-whisker plot represents the
cost, in dollars, of twelve CD’s.
Which cost is the upper quartile?
What is the median?
How many CDs cost between
$14.50 and $26?
What is the range of the costs?
Which cost represents the 100th percentile?
How many CDs cost less than $14.50?
Analyzing Statistical Data
The average number of hours of
sleep of several people is shown
in the table.
Measures of dispersion describe how scattered a set of
data is.
Along with a measure of central tendency, you can
provide a good description of a data set.
Analyzing Statistical Data
The average number of hours of
sleep of several people is shown
in the table.
If x1, x2, x3, ... , xn are n numbers and M is their mean,
then the variance of the distribution is…
x1  M   x2  M   ... x n  M 
2
2
n
2
Analyzing Statistical Data
The average number of hours of
sleep of several people is shown
in the table.
Ex) Find the variance of the distribution of the number of
hours slept by participants in the table.
Analyzing Statistical Data
The average number of hours of
sleep of several people is shown
in the table.
The standard deviation is the square root of the
variance.
Analyzing Statistical Data
Ex) The stem-and-leaf plot to
the right shows the number of
yards gained by several
students on the football team.
a) Find the mean of the distribution and explain its
meaning.
Analyzing Statistical Data
Ex) The stem-and-leaf plot to
the right shows the number of
yards gained by several
students on the football team.
b) Find the standard deviation of the distribution.
Learning Log Summary
LT 2 – I can compute measures of dispersion
and use them describe a distribution.
A measure of dispersion tells you…
To find the variance and standard deviation…
Closure
Homework
pg. 717 ~ 1-13 (All)
The Normal Distribution
Data can be distributed in different ways:
The Normal Distribution
When it is distributed around a central value:
The Normal Distribution
The Normal Distribution
The data is distributed in the following way:
Likely to be within 1
standard deviation (68
out of 100 should be)
Very likely to be within
2 standard deviations
(95 out of 100 should
be)
Almost certainly within
3 standard deviations
(997 out of 1000 should
be)
The Normal Distribution
Ex) A set of student test scores has a mean of 75
points and a standard deviation of 8 points.
 What percentage of students scored between 67
and 83?
 19 out of 20 students has a score in between what
two numbers?
 What percentage of students scored below a 59?
The Normal Distribution
Ex) Out of 300 high school students with normally
distributed heights, 95% of students are between 61 and
73 inches tall.
 How many students are between 64 and 70 in tall?
 What percentage of students are taller than 70 in?
 How many students are less than 61 in tall?
Ex. 1: Understanding Mean &
Standard Deviation
1.
Which normal curve has a greater mean?
2.
Which normal curve has a greater standard deviation?
The Normal Distribution
 Consider the normal curves shown below. Which
normal curve has the greatest mean? Which normal
curve has the greatest standard deviation? Justify
your answers.
Learning Log Summary
LT 3 – I can recognize a normal distribution
and use it to determine the percentage of the
data that falls within a given range.
A normal distribution is…
To find the percentage of data within a certain
number of standard deviations of the mean…
Closure
Homework
Normal Distribution Worksheet
Correlation
 A survey of several professionals in a neighborhood
asked the number of years they have been working
and what their current salary is.
a)How do you expect years of experience to influence
salary?
Correlation
b) Draw a scatter plot of the data found in the table to
the right.
Experience
in Years
Salary
(in thousands)
0
5
20
25
10
35
5
15
10
30
20
40
60
50
50
55
30
50
30
70
Correlation
A statistic called the correlation coefficient is used to
characterize how closely the points in a scatter plot
cluster about a line.
Given a set of ordered pairs, the correlation coefficient,
denoted by ‘r’, is:
r
M xy  M x  M y
 x  y
where M x and  x are the mean and S.D. of the x-values,
M y and  y are the mean and S.D. of the y-values,
And
M xy is the mean of the products of the ordered pairs.
Correlation
A statistic called the correlation coefficient is used to
characterize how closely the points in a scatter plot
cluster about a line.
Correlation
c) Using the scatter plot created for experience and
salary, describe the nature of the correlation.
Correlation
c) Using the scatter plot created for experience and
salary, describe the nature of the correlation.
Correlation
d) Suppose x represents experience and y represents
salary. Determine the correlation coefficient of the
ordered pairs from the table given that M x  15.5 ,
M y  45.5,  x  11.06 ,  y  14.57 , and M xy  837.5 .
Correlation
 When correlation between two variables is high, you
can draw a regression line, which is a line that best
fits the known values of the variables.
An example is shown below:
Correlation
 The regression line contains the point M x , M y  and
has a slope of
 y
r 
x

 .

Correlation
e) Using part (d), determine an equation for the
regression line of the salary data.
Correlation
f) Find the expected salary of someone who works for
45 years.
Learning Log Summary
LT 4 – I can draw
a scatterplot of data and
use it to determine the correlation coefficient
and regression line that models the data.
A correlation coefficient is…
To find the correlation coefficient and regression
equation…
Closure
Homework
Pg. 727 ~ 1-8 (all)
Performance Task due Monday!
Fundamental Counting Principle
Intro: You are at an awards dinner and are about to
place your food order. The card placed at your seat
describes the menu options as follows:
How many possible dinner orders are there? Explain or
show how you know.
Fundamental Counting Principle
If there are “m” ways to make a first choice and “n”
ways to make a second choice, etc. then there are
m· n ways to make both choices.
Fundamental Counting Principle
Ex 1) You go to Best Buy to purchase a new television. You have
the following choices: LCD or plasma; screen size 27”, 32”, 36”, 41”,
51”, or 63” and manufacturer Sony, Vizio or Phillips. How many
different televisions does the store have to offer?
Fundamental Counting Principle
Ex 2) In how many ways can a 5-question multiple choice quiz
(with options A-D for each choice) be answered if it is ok to leave
questions blank?
Fundamental Counting Principle
Ex 3) How many odd 2-digit whole numbers are less than 70?
Fundamental Counting Principle
Intro (Pt 2):
To get to school, Rita can either ride her bike or take the bus.
There are three possible bus routes and two routs she knows on her
bike. How many ways can she get to school? Explain or show how
you know.
Fundamental Counting Principle
Mutually exclusive events are events that do not influence or
cause one another. They can not happen simultaneously.
For example, flipping a coin and rolling a die are mutually
exclusive.
Picking a card from a deck and picking another card without
replacing the first are not mutually exclusive events.
Fundamental Counting Principle
If the possibilities being counted can be grouped into
mutually exclusive cases, then the total number of
possibilities is the sum of the number of possibilities in
each case.
Fundamental Counting Principle
Ex 4) How many positive integers less than 100 can be
written using the digits 6, 7, 8, and 9?
Fundamental Counting Principle
Ex 5) How many license plates of 3 symbols (letters and
digits) can be made using at least one letter?
Fundamental Counting Principle
Ex 6) How many seven-digit phone numbers can be
created if the first digit must be 8, the second digit must
be 5, and the third digit must be 2 or 3?
Learning Log Summary
LT 5 – I can use the Fundamental Counting
Principle to determine the number of ways
various choices can be made.
The Fundamental Counting Principle says…
If there are m ways to make a first selection…
Closure
Homework
Pg. 732 ~ 1-10 (all)
Q3 Benchmark Warm-Up
1) Which of the following represents log 3 20 ?
A) log10 3  log10 20
B)
log10 20
log10 3
C) log10 3  log10 20
D)
log10 3log10 20
Q3 Benchmark Warm-Up
2) What is the standard form of the conic below?
x 2  4 y 2  2 x  16 y  11  0
A)
x  12   y  22
4
C)
1
 y  22  x  12
1
4
1
1
B)
x  12   y  22
1
 y  22  x  12
1
4
D)
1
1
4
Q3 Benchmark Warm-Up
3) Find the first 3 terms in the expansion of:
1  2 x 5
A)
1  10 x  40 x 2
B)
1  8 x  24 x 2
C)
1  10 x  40 x 2
D)
1  8 x  24 x 2
Q3 Benchmark Warm-Up
4) Find the sixth term of the sequence:
2 , 6 ,18 , ...
A)
486
B)
38
C)
144
D)
27
Q3 Benchmark Warm-Up
1
5) Solve for x: log x 
2
A)
5
B)
5
C)
10
D)
20
Q3 Benchmark Warm-Up
6) Solve for x:
18
A) log
4
4
C) log
18
4 x  18
B)
log 18
log 4
D)
log 4
log 18
Q3 Benchmark Warm-Up
7) Find an explicit formula for the sequence:
7,13 ,19 , 25,...
A) an  6n
B) an  6n  7
C) an  6n  1
D) an  6n  1
Fundamental Counting Principle Review
 For many computer tablets, the owner can set a 𝟒digit pass code to lock the device.
a) How many digits could you choose from for the first number of
the pass code?
b) How many digits could you choose from for the second number
of the pass code? Assume that the numbers can be repeated.
c) How many different 𝟒-digit pass codes are possible?
Explain/show how you got your answer.
Fundamental Counting Principle Review
The store at your school wants to stock sweatshirts that
come in four sizes (small, medium, large, xlarge) and in
two colors (red and white). How many different types
of sweatshirts will the store have to stock?
Fundamental Counting Principle Review
The call letters for all radio stations in the United States
start with either a 𝑊 (east of the Mississippi river) or a 𝐾
(west of the Mississippi River) followed by three other
letters that can be repeated. How many different call
letters are possible?
Learning Log Summary
LT 5 – I can use the Fundamental Counting
Principle to determine the number of ways
various choices can be made.
The Fundamental Counting Principle says…
If there are m ways to make a first selection…
Permutations
 Suppose that the 𝟒-digit pass code a computer
tablet owner uses to lock the device cannot have
any digits that repeat. For example, 𝟏𝟐𝟑𝟒 is a valid
pass code. However, 𝟏𝟏𝟐𝟑 is not a valid pass code
since the digit “𝟏” is repeated.
 An arrangement of four digits with no repeats is an
example of a permutation. A permutation is an
arrangement in a certain order (a sequence).
 How many different 𝟒-digit pass codes are possible if
digits cannot be repeated?
Permutations
 How many ways can you arrange “n” items?
n ! n  n  1  n  2   ...  3  2 1
“n factorial”
 How many ways can you arrange 5 books on a shelf?
Permutations
 Ex 1) Suppose a password requires three distinct
letters. Find the number of permutations for the three
letters in the code, if the letters may not be repeated.
Permutations
 Ex 2) The high school track has 𝟖 lanes. In the 𝟏𝟎𝟎 meter
dash, there is a runner in each lane. Find the number of
ways that 𝟑 out of the 𝟖 runners can finish first, second,
and third.
Permutations
 Ex 3) A home security system has a pad with 𝟗 digits (𝟏 to 𝟗). Find the
number of possible 𝟓-digit pass codes:
 if digits can be repeated.
 if digits cannot be repeated.
Permutations
 Now imagine you are making a passcode out of different characters.
There are “n” options, and you need to create a password that is “r”
items long. Write an expression for the number of passcodes you could
create:
Permutations
 To find the number of permutations of n objects,
taken r at a time:
n!
n Pr 
n  r !
Permutations

Ex 4) How many different ways can 5 of the 𝟏𝟓 numbered pool balls be placed in a line on
the pool table?

Ex 5) Ms. Smith keeps eight different cookbooks on a shelf in one of her kitchen cabinets.
How many ways can the eight cookbooks be arranged on the shelf?

Ex 6) How many distinct 𝟒-letter groupings can be made with the letters from the word
champion if letters may not be repeated?

Ex 7) There are 𝟏𝟐 different rides at an amusement park. You buy five tickets that allow you
to ride on five different rides. In how many different orders can you ride the five rides? How
would your answer change if you could repeat a ride?
Learning Log Summary
LT 6 – I can find the number of permutations of
the elements of a set and apply it to real-world
problems.
A permutation is…
To find the number of permutations of n objects,
taken r at a time…
Closure
Homework
Pg. 732 ~ 1-10 (all)
Combinations
Intro: Seven speed skaters are competing in an Olympic race. The firstplace skater earns the gold medal, the second-place skater earns the
silver medal, and the third-place skater earns the bronze medal. In how
many different ways could the gold, silver, and bronze medals be
awarded? The letters A, B, C, D, E, F, and G will be used to represent these
seven skaters.
a) How can we determine the number of different possible outcomes?
How many are there?
Combinations
Intro: Seven speed skaters are competing in an Olympic race. The firstplace skater earns the gold medal, the second-place skater earns the
silver medal, and the third-place skater earns the bronze medal. In how
many different ways could the gold, silver, and bronze medals be
awarded? The letters A, B, C, D, E, F, and G will be used to represent these
seven skaters.
b) Now consider a slightly different situation. Seven speed skaters are
competing in an Olympic race. The top three skaters move on to the next
round of races. How many different "top three" groups can be selected?
How is this situation different from the first situation? Would you expect
more or fewer possibilities in this situation? Why?
Combinations
A permutation is…
A combination is…
Combinations
 To find the number of combinations of n objects, taken r at a
time:
n!
n Cr 
r !n  r !
 Consider the figure skating problem, and use the formula to find
the number of ways the “top three” can be formed.
Combinations
 Ex 1) How many ways can a committee of 𝟓 students
be chosen from a student council of 𝟑𝟎 students? Is
the order in which the members of the committee
are chosen important?
Combinations
 Ex 2) A football team of 𝟓𝟎 players will choose two
co-captains. How many different ways are there to
choose the two co-captains?
Combinations
 Ex 3) There are seven people who meet for the first
time at a meeting. They shake hands with each
other and introduce themselves. How many
handshakes have been exchanged?
Combinations
 Ex 4) At a particular restaurant, you must choose two
different side dishes to accompany your meal. If
there are eight side dishes to choose from, how many
different possibilities are there?
Combinations
 Ex 5) Brett has ten distinct t-shirts. He is planning on
going on a short weekend trip to visit his brother in
college. He has enough room in his bag to pack four
t-shirts. How many different ways can he choose four
t-shirts for his trip?
Combinations
 Ex 6) How many three-topping pizzas can be
ordered from the list of toppings below? Did you
calculate the number of permutations or the number
of combinations to get your answer? Why did you
make this choice?
sausage
pepperoni
meatball
onions
olives
pineapple
ham
green
peppers
mushrooms
bacon
spinach
hot peppers
Learning Log Summary
LT 7 – I can find the number of combinations
of the elements of a set and apply it to realworld problems.
A combination is…
To find the number of combinations of n objects,
taken r at a time…
The difference between a combination and
permutation is…
Combinations and
Permutations on the
Graphing Calculator
Self-Check
1.
2. Pat has 𝟏𝟐 books he plans to read during the school year. He decides to take 𝟒 of these books with him
while on winter break vacation. He decides to take Harry Potter and the Sorcerer’s Stone as one of the books.
In how many ways can he select the remaining 𝟑 books?
Closure
Homework
Pg. 737 ~ 1-14 (all)
Pg. 740 ~ 1-16 (all)
Permutations (Re-Visited)
Review: How many ways can the letters in WORD be rearranged? Does this require a combination or
permutation? Explain.
Permutations (Re-Visited)
In how many ways can the letters in DAD be rearranged?
a) Write out the possibilities.
b) How many of these permutations are unique?
Permutations (Re-Visited)
Permutations of n Elements (with repeating elements)
If a set of n element has n1 elements of one kind alike,
n2 of another kind alike, etc., then the number of
permutations P of the n elements taken n at a time is:
n!
n1!n2 !...
Permutations (Re-Visited)
Ex 1) Find the number of ways the letters in the word
HUBBUB can be arranged.
Permutations (Re-Visited)
Ex 2) There are 6 plain towels on a laundry line. 3 are
blue, 2 are red, and 1 is white. Find the number of ways
these towels can be arranged on the line.
Learning Log Summary
LT 6 – I can find the number of permutations of
the elements of a set and apply it to real-world
problems.
To find the number of permutations of n objects
where there are repeated objects…
Closure
Homework
Pg. 737 ~ 15-23 (all)
Sample Space and Events
A random experiment is an experiment in which you do
not necessarily get the same outcome when it is
repeated with the same conditions.
Sample Space and Events
The sample space of an experiment is the set of all
possible outcomes of a random experiment.
Sample Space and Events
An event is any subset of the possible outcomes for an
experiment.
Sample Space and Events
Ex 1) For the rolling of a die, specify:
a) The sample space for the experiment.
b) The event that you roll a number greater than 2.
c) The event that you roll an odd number.
Sample Space and Events
Ex 2) Suppose you roll two dice, one red and one blue. An
outcome in this experiment can be represented by the ordered
pair (r,b) where r is the number showing on the red die and b is the
number showing on the blue die.
a) How many items are in the sample space? Explain.
Sample Space and Events
Ex 2) Suppose you roll two dice, one red and one blue. An
outcome in this experiment can be represented by the ordered
pair (r,b) where r is the number showing on the red die and b is the
number showing on the blue die.
b) Specify the event that the red die shows 5.
Sample Space and Events
Ex 2) Suppose you roll two dice, one red and one blue. An
outcome in this experiment can be represented by the ordered
pair (r,b) where r is the number showing on the red die and b is the
number showing on the blue die.
c) Specify the event that the sum of the dice is 7.
Learning Log Summary
LT 8 – I can identify the sample spaces and
events for random experiments.
The sample space of an experiment is…
An event is…
Probability
Intro: If you roll a 6-sided die, what are the chances you
will roll a 5? Explain how you know.
Probability
The probability that an event (E) will occur, given a
sample space with n items, is:
# of items in the Event
P E  
# of items in the Sample Space
Probability
Ex 1) A die is rolled. Find the probability of each event.
a) Event A: The number showing is less than 5.
b) Event B: The number showing is between 2 and 6.
c) Event C: The number showing is even.
Probability
Ex 3) There are 12 food items wrapped in identical aluminum foil
packaging. 9 of them are cheeseburgers and 3 of them are
hamburgers. If two items are selected at random, find the
probability of each event.
Brainstorming Questions…
a) Event A: Both items are hamburgers.
b) Event B: One item will be a hamburger and one will be a
cheeseburger.
Probability
Ex 4) There are 3 red, 2 blue, and 3 yellow marbles in a bag. Jeff
randomly selects one, returns it to the bag, then randomly selects
another. Find the probability of each event.
Brainstorming Questions…
a) Event A: The first marble is blue and the second is yellow.
b) Event B: Both marbles selected are red.
Learning Log Summary
LT 9 – I can calculate the probability that an
event will occur based on the size of the
sample space.
Probability describes…
The probability of an event can be calculated by…
Closure
Homework
Pg. 744 ~ 1-6 (all)
Pg. 748 ~ 1-8 (all)