Download Honors Advanced Algebra

Honors Advanced Algebra
Statistics
Target Goals
By the end of this chapter, you should be able to…
•
Find the mean, median, and mode for a set of data. (Day 1)
_____ got it
_____needs work
_____ no clue
•
Use the mean, median, and mode to interpret data. (Day 1)
_____ got it
_____needs work
_____ no clue
•
Find the range and interquartile range for a set of data. (Day 2)
_____ got it
_____needs work
_____ no clue
•
Determine if any values in a set of data are outliers. (Day 2)
_____ got it
_____needs work
_____ no clue
•
Represent data using box-and-whisker plots. (Day 2)
_____ got it
_____needs work
_____ no clue
•
Calculate the standard deviation for a set of data. (Day 3)
_____ got it
_____needs work
_____ no clue
•
Read and interpret data from stem-and-leaf plots. (Day 3)
_____ got it
_____needs work
_____ no clue
•
Solve problems involving normally distributed data. (Day 4)
_____ got it
_____needs work
_____ no clue
Honors Advanced Algebra Assignment Guide
Statistics
Central Tendancy: Median, Mode, and Mean
Target Goals: Find the mean, median, and mode for a set of data.
Use the mean, median, and mode to interpret data.
HW #1
Worksheet
Variation: Range, Interquartile Range, and Outliers & Box-and-Whisker Plots
Target Goals: Find the range and interquartile range for a set of data.
Determine if any values in a set of data are outliers.
Represent data using box-and-whisker plots.
HW #2
Worksheet
Variation: Standard Deviation & Stem-and-Leaf Plots
Target Goals: Calculate the standard deviation for a set of data.
Read and interpret data from stem-and-leaf plot.
HW #3
Worksheet
The Normal Distribution
Target Goals: Solve problems involving normally distributed data.
HW #4
Worksheet
Statistics Review
HW #5
TBA
Tentative Test Date: _______________________
Honors Advanced Algebra with Trigonometry
Central Tendency: Median, Mode, & Mean
Notes (Day 1)
Name: _____________________
Date: ______________________
Period: ____________________
Target Goals:
• Find the mean, median, and mode for a set of data.
• Use the mean, median, and mode to interpret data.
The most commonly used averages are the mean, median, and mode.
 The mean of a set of data is the sum of all the values divided by the number of values.
mean ( X ) = average
 The median of a set of data is the middle value; if there are two middle values, it is the average of those
values.
median (M) = think “middle”
 The mode of a set of data is the most frequent value in that set; some sets of data have multiple modes
and other have no mode.
mode = most common
Extreme values (called outliers) are those data values that are vastly different from the central group of data
values. Consider the following list of student test scores (out of 100):
26, 86, 94, 97, 97
Here, the mean = 80
the median = 94
and the mode = 97
Now consider the list without 26%.
Here, the mean = 93.5
the median = 95.5
and the mode = 97
As you can see, every data value affects the value of the mean, so when extreme values are included in a set of
data, the mean may become less representative of the set; however, the values of the median and mode are not
affected by extreme values in the set.
Examples
1. Two partners in a company have salaries of $50,000 each. Of their eight employees, two earn $12,700 each,
two earn $9,600 each, and four earn $8,700 each. Find the mean, median, and mode for the ten wages.
2. Susan took 6 minutes on a quiz, Tom took 8 minutes on the same quiz, Mary took 9 minutes, Al took 12
minutes, and Eric did not finish the quiz. Find the mean, median, and mode.
(Answer to Ex 1: mean = $17,940 median = $9600 mode = $8700)
(Answer to Ex. 2: mean = cannot calculate due to Eric, median = 9, mode = no mode)
Assignment #1: Worksheet
Honors Advanced Algebra with Trigonometry
Central Tendency: Median, Mode, & Mean
Assignment #1
Name: _____________________
Date: ______________________
Period: ____________________
Find the mean, median, and mode for each set of data.
1. {4.8, 5.7, 2.1, 2.1, 4.8, 2.1}
2. {11, 10, 13, 12, 12, 13, 15}
3. {100, 45, 105, 98, 97, 101}
4. A die was tossed 25 times with the results shown below. Find the mean, median, and mode for the tosses.
5
6
1
3
5
6
1
6
6
6
3
4
5
6
1
2
4
2
1
4
2
5
4
4
1
5. Find the mean, median, and mode of the hourly wages of 200 employees. One hundred earn $5.00 per hour,
ten earn $6.25 per hour, ten earn $7.75, twenty earn $4.50 per hour, and sixty earn $5.90 per hour.
6. The union and the company executives are currently negotiating a raise in salaries for all of the MicroTech
employees. Three of the employees have salaries of $300,000 each. However, a majority of the employees
have salaries of about $30,000 per year.
a. You are a vice-president and would like to show that the current salaries are reasonable. Would you
quote the mean, median, or mode as the “average” salary to justify your claim? Why?
b. You are the union representative for your department and maintain that a pay raise is in order.
Which of the mean, median, and mode would you quote to justify your claim? Why?
1. mean = 3.6
median = 3.45
mode = 2.1
6a. mean, 6b. mode
2. mean = 12.28
median = 12
mode = 12 & 13
3. mean = 91
median = 99
mode = no mode
4. mean = 3.72
median = 4
mode = 6
5. mean = $5.42
median = $5.00
mode = $5.00
Honors Advanced Algebra with Trigonometry
Variation: Range, IQR, Outliers & Box-and-Whisker Plots
Notes (Day 2)
Name: _____________________
Date: ______________________
Period: ____________________
Target Goals:
• Find the range and interquartile range for a set of data.
• Determine if any values in a set of data are outliers.
• Represent data using box-and-whisker plots.
Dispersion is the variation within a set of data. Consider the set {18, 20, 25, 30, 34, 35, 90}
The range of a set of data is the difference between the greatest and least values in the set.



The range in the given set above = ______________.
Is the range affected by extreme values?
Is the range a good measure of variation?
Quartiles are the values in a set of data that separate the data into four equal parts.
•
median/ M – one of the quartiles
- separates the data into two equal parts
- 50% of the data is below M, 50% of the data is above M
- The median in the given set above is __________.
•
lower quartile/Q1 – quartile that is less than the median
- separates the data below the median into two equal parts
- 25% of the data is below Q1, 75% of the data is above Q1
- Q1 in the given set above is __________.
•
upper quartile/Q3 – quartile that is greater than the median
- Separates the data greater than the median into two equal parts
- 75% of the data is below Q3, 25% of the data is above Q3
- Q3 in the given set above is __________.
To find the quartiles:
1. arrange the data set in ascending order
2. find the median – the data value that separates the data into two equal parts
3. find Q1 – the data value that separates the data set below the median into two equal parts
4. find Q3 – the data value that separates the data set above the median into two equal parts
Analyzing Data with the TI-Nspire
Enter Data:
Go to the House and open a New Document [1]
Add a Lists & Spreadsheet [4] page
Arrow up to the top cell in column A and short name the list (such as “ht”)
Enter the first value from your data set in the cell at column A and row 1
Continue entering data in column A
Calculate statistics and/or graph the box and whiskers plot (see below)
Calculate Statistics:
Select a cell in column A (if not already selected)
Go to Menu: Statistics [4]: Stat Calculations [1]: One-Variable Statistics [1]
Click OK in both dialogue boxes
The interquartile range (IQR) is the difference between the upper and lower quartiles
An outlier is any value in a set of data that is at least 1.5 interquartile ranges beyond the upper or lower quartile.
An outlier can be thought of as a very extreme value.
Ex 1: The mean daily temperatures in San Francisco for each month of the year are:
49°, 52°, 53°, 55°, 58°, 61°, 62°, 63°, 64°, 61°, 55°, 49°
Find the range, quartiles, interquartile range, and any outliers for the temperatures by hand and by graphing
calculator.
A box-and-whisker plot displays the quartiles and the extreme values of a set of data using a number line. It is
a way to represent numerical data and can be drawn horizontally or vertically.
To make a box-and-whisker plot:
1. find the min, Q1, M, Q3, and max
2. determine if any outliers exist
3. draw a number line and plot the quartiles as the box, the median splits the box, the min and max as
the whiskers, and outliers as dots (if they exist)
Graphing a Box and Whiskers Plot with the TI-Nspire:
Go to Ctrl + Page
Add a Data & Statistics [5] page
Move cursor to bottom center and Click to add variable
Select name of the data set and Enter
Move cursor to center of page and go to Ctrl Menu: Box Plot [1]
Move the cursor and hover over the box and whiskers to see values of the minimum, Q1, median, Q3 and
maximum. Outliers will be displayed by a point.
Ex 2: Make a box-and-whisker plot for the temperature example above.
Assignment #2: Worksheet
Honors Advanced Algebra with Trigonometry
Variation: Range, IQR, Outliers & Box-and-Whisker Plots
Assignment #2
Name: _____________________
Date: ______________________
Period: ____________________
Find the range, quartiles, and interquartile range for each set of data.
1. {4, 1, 3, 7, 7, 5, 4, 1, 8, 20, 2, 11, 7, 7, 1}
2. {1055, 1075, 1095, 1125, 1005, 975, 1123, 1100, 1145, 1025, 1075}
3. Find any outliers for the set of data in #1.
4. Find any outliers for the set of data in #2.
5. The sales of the 15 largest American businesses are given below. Find the range, quartiles, and interquartile
range for the sales figures. Then determine if there are any outliers.
Company
Amoco
Chevron
Chrysler
Du Pont
Exxon
Ford Motor
General Electric
General Motors
Sales (in billions)
21
25
35
33
80
92
49
121
Company
IBM
Mobil
Occidental Petroleum
Phillip Morris
Proctor and Gamble
Shell Oil
Texaco
Sales (in billions)
60
48
19
26
19
21
34
6. The number of calories in a regular serving of French fries at different restaurants are listed below. Make a
box-and-whisker plot of the data.
Restaurant
Burger Chef
Burger King
Carl’s Jr.
Dairy Queen
Friendly’s
Calories
250
240
220
200
125
Restaurant
Hardee’s
McDonald’s
Roy Rogers
Wendy’s
Calories
239
211
240
327
7. The table below shows the median ages of men and women at the time of their first marriage for the decades
of 1890 through 1990. Find the range, quartiles, interquartile range, and determine if there are any outliers
for both the men and the women. Then make a box-and-whisker plot for each.
Year
Men
Women
Year
Men
Women
26.1
22.0
1950
22.8
20.3
1890
25.9
21.9
1960
22.8
20.3
1900
25.1
21.6
1970
23.2
20.8
1910
24.6
21.2
1980
24.7
22.0
1920
24.3
21.3
1990
26.2
25.1
1930
24.3
21.5
1940
Honors Advanced Algebra with Trigonometry
Variation: Standard Deviation
Notes (Day 3)
Name: _____________________
Date: ______________________
Period: ____________________
Target Goals:
• Calculate the standard deviation for a set of data.
• Read and interpret data from stem and leaf plots.
Standard deviation is the average measure of how much each value in a set of data differs from the mean.
- symbolized by σ (Greek lower-case sigma) or SD
- the most commonly used measure of variation
- describes the “spread” of the data set
Definition of standard deviation: From a set of data with n values, if xi represents a value such that i and n are
positive integers and 1 ≤ i ≤ n , and x represents the mean, then the standard deviation can be found as follows:
n
σ=
(x i − x)2
∑
i =1
n
Do not be intimidated by this formula!! Let’s look at how standard deviation looks written out for the set
{4, 5, 6, 6, 9} with mean 6:
To find the standard deviation of a set of data:
1. find the mean
2. find the difference between each value in the set of data and the mean
3. square each difference
4. find the mean of the squares
5. take the principle square root of this mean
Evaluate the standard deviation for the set above: ________________ (Round to 3 decimal places)
Finding the Standard Deviation with the TI-Nspire:
Enter data on a Lists & Spreadsheet [4] page
Calculate statistics using Menu: Statistics [4]: Stat Calculations [1]: One-Variable Statistics [1]
Find the value of the standard deviation σ x in the spreadsheet
Check the standard deviation for the set above: _________________ (Round to 3 decimal places)
Note: To earn credit on any homework, quizzes, or tests, you will ALWAYS have to set up the standard
deviation formula for at least 3 terms in each set (use “…” to signify more than 3 terms)! Once you have set up
each problem, then you may let your calculator do the work.
Examples:
**Round each standard deviation to 3 decimal places.
1.
Find the standard deviation for:
Stem I Leaf 2 I 9 represents 29
2I9
3I6789
4I0578
5I1
2.
Find the standard deviation for: {200, 476, 721, 579, 152, 158}
3.
The winning average speeds, in miles per hour, of the Daytona 500 from 1975 to 1988 are
154, 152, 153, 160, 144, 178, 170, 154, 156, 151, 172, 148, 177, and 138.
Find the mean and the standard deviation.
Assignment #3: Worksheet
Honors Advanced Algebra with Trigonometry
Variation: Standard Deviation
Assignment #3
Name: _____________________
Date: ______________________
Period: ____________________
Find the standard deviation for each set of data. Round to 3 decimal places.
1. Stem Leaf
3 I 0 represents 3.0
3 001245
66689
4 11344
5567
2. Stem
4
5
6
7
Leaf
139
2369
44578
247
4 I 1 represents 41
3. The weights in pounds of the starting players for three area high schools’ football teams are given below.
West High: 160, 180, 190, 200, 210, 170, 250, 220, 180, 200, 240
Ridgemont: 160, 190, 210, 230, 240, 220, 150, 190, 210, 160, 240
Grandview: 250, 170, 205, 220, 185, 215, 205, 210, 205, 185, 170
a. Find the standard deviation for the weights of the players on the West High team.
b. Find the standard deviation for the weights of the players on the Ridgemont team.
c. Find the standard deviation for the weights of the players on the Grandview team.
d. Which of the teams has the most variation in weights?
4. Under what circumstances would the standard deviation of a set of data equal to zero?
Honors Advanced Algebra with Trigonometry
The Normal Distribution
Notes (Day 4)
Name: _____________________
Date: ______________________
Period: ____________________
Warm-Up:
1. Find the standard deviation for: {1145, 1100, 1125, 1050, 1175, 835, 1075, 1095}.
2. The ages of a group of people are listed below. Make a box-and-whisker plot for the data.
2I9
3I6689
4I05789
5I12
7I25
2 I 9 represents 29
Target Goals:
• Solve problems using normally distributed data.
One way of analyzing data is to consider ______________________________________________________

Frequency distribution – shows how the data are spread out

Histogram – bar graph that shows a frequency distribution

Normal distribution – bell-shaped, symmetric bar graphs

Bell curve – curve of the graph of a normal distribution that is symmetric and shaped like a bell
Normal distributions have these properties:
1.
the graph is maximized at the mean
2.
about 68% of the values are within one standard deviation from the mean
3.
about 95% of the values are within two standard deviations from the mean
4.
about 99.7% of the values are within three standard deviations from the mean
Examples:
1.
A grading scale is set up for 1000 students’ test scores. It is assumed that the scores are normally
distributed with a mean score of 75 and a standard deviation of 15.
a) How many students will have scores between 45 and 75?
b) If 60 is the lowest passing score, how many students are expected to pass the test?
2.
The weights of a large group of people is assumed to be normally distributed. The mean weight is 136
pounds and the standard deviation is 18 pounds.
a) What percentage of people weigh at most 172 pounds?
b) What percentage of people weigh between 82 and 154 pounds?
Assignment #4: Worksheet
Honors Advanced Algebra with Trigonometry
The Normal Distribution
Assignment #4
Name: _____________________
Date: ______________________
Period: ____________________
1. The diameters of metal fittings made by a machine are normally distributed. The mean diameter is 7.5 cm,
and the standard deviation is 0.5 cm.
a. What percentage of the fittings have diameters between 7.0 cm and 8.0 cm?
b. What percentage of the fittings have diameters between 7.5 cm and 8.0 cm?
c. What percentage of the fittings have diameters greater than 6.5 cm?
d. Of 100 fittings, how many will have a diameter between 6.0 cm and 8.5 cm?
2. The lifetimes of 10,000 light bulbs are normally distributed. The mean lifetime is 300 days, and the standard
deviation is 40 days.
a. How many light bulbs will last between 260 and 340 days?
b. How many light bulbs will last between 220 and 380 days?
c. How many light bulbs will last less than 300 days?
d. How many light bulbs will last more than 300 days?
e. How many light bulbs will last more than 380 days?
f. How many light bulbs will last less than 180 days?
3. Jalisa and Wes were conducting an experiment in statistics. They asked each student in their mathematics
classes to toss a fair coin 100 times and record the number of heads they obtained. They found that the
number of heads was almost distributed normally. The mean number of heads was 50 and the standard
deviation was 5.
a. What percentage of the students who tossed the coins obtained less than 50 heads?
b. What percentage of the students who tossed the coins obtained between 35 and 45 heads?
c. What percentage of the students who tossed the coins obtained between 40 and 60 he