MATH 144 Grand Canyon University Unit Conversions Algebra Discussion

According to the National Center for Health Statistics, the mean height of an American male is 69.3 inches and the mean height of an American female is 63.8 inches. The standard deviation for both genders is 2.7 inches.

• According to Chebyshev’s Theorem 75% of the data for your gender lies between what two heights?
• If height is assumed to be normal, what percentage of the data lies between those same two heights?

Look in the Guided Worksheets on page 49 for more information.

A Microsoft Excel spreadsheet is required for this DQ. Chapter from book attached

Note from Professor:

Dear class,

1. Read example 13 on page 03-25 to review z score and SD.

2. Read example 14 on page 03-30 to under the SD of a normal distribution.

3. Find page 49 in the guided worksheets under the Course Materials to understand “Chebyschev’s Theorem.” : For any data set at least 75% of the data values are within 2 standard deviations of the mean and at least 89% are within 3 SD’s. Use this information to answer the first part of this DQ topic. (attached)

4.For the second part of DQ, use the graph on the top of the page 49 in guided worksheets to find the percentages.

Al materials attached

Guided Worksheets
Eric Gaze
Thinking Quantitatively
Communicating with Numbers
Eric Gaze
Table of Contents
Quantitative Literacy
Q-1
Chapter 1
Quantitative Reasoning/Introduction to Excel
1
Function and TV Loan
5
Car Loan
9
Descriptive Statistics
13
Chapter 2
Ratios!
17
Weighted Averages
21
Proportionality
23
CPI
25
PE and Money Ratios
27
Z-Scores
31
Histogram and z-scores
33
Chapter 3
Units, Conversions, Scales, and Rates
37
Rates, Canceling Units, and a Clever Equation
41
Z-Scores, Standard Error, and the Normal Curve
43
Normal Distributions
49
Chapter 4
Percentages
51
Chapter 5
Proportionality and Linear Functions
55
Linear Functions
61
Divorce Rate
65
Chapter 6
Exponential Growth
69
Exponential Puzzlers
73
Chapter 7
Logarithms
77
AP…R AP…Y Oh My!
79
Log Scales and Cumulative Frequency Distributions
83
Chapter 8
Correlation
85
Line of Best Fit
89
Chapter 9
RNPPF
93
Investing
97
Chapter 10
Logic
103
IPO Investing
107
Quantitative Reasoning
A piece presented on the Bloomberg View explores the data regarding “How Americans Die.”
Let’s take a closer look.
1,150
1118.5
1,100
1,050
1,000
967.3
950
900
850
823.7
800
750
Males
Everyone
2010
2008
2006
2004
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
1972
1970
700
1968
Mortality Rate per 100,000 (Males/Females/All)
1. On the first slide use the statistics for 1968: 823.7, 967.3, and 1,118.5, in sentences.
Female
Data from: http://www.bloomberg.com/dataview/2014-04-17/how-americans-die.html
The mortality rate for the entire U.S. population in 1968 was 967.3 deaths per 100,000
people.
The mortality rate for U.S. women in 1968 was 823.7 deaths per 100,000 women.
The mortality rate for U.S. men in 1968 was 1,118.5 deaths per 100,000 men.
Note the second quantity of this ratio is not always people! If it was you could the male
and female rates to get the total. Also note the 967.3 is not the average of the male and
female rates, there are more women in the population so the overall rate skews towards
the female rate.
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2. The presentation tells us the overall rate “fell by about 17%” from 1968 to 2010, from
967.3 to 799.5.
1,150
1,100
1,050
1,000
950
900
799.5
812
850
800
750
784.4
Males
Everyone
2010
2008
2006
2004
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
1972
1968
700
1970
Mortality Rate per 100,000 (Males/Females/All)
a. Verify this and quantify the change for the rates for men and women in a similar
fashion.
Female
Data from: http://www.bloomberg.com/dataview/2014-04-17/how-americans-die.html
The total change is, divide this by the original 967.3 to get -17.3%.
The total change is, divide this by the original 823.7 to get -4.4% for women.
The total change is
, divide this by the original 1,118.5 to get -27.4% for men.
b. Why can we compare the 1970 and 2010 statistics, even though the population
has increased over this period?
Because we are using rates per 100,000 which take into the population sizes.
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3. Slide 1 says the decline in mortality rates stops in the mid 1990’s and slide 2 attributes
this to the aging of the population.
a. What is the logic behind this argument?
30
11.82%
Share of Population
25
20
7.03%
15
10
6.01%
Over 75
65-74
2010
2008
2006
2004
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
1972
1970
1968
5
55-64
Data from: http://www.bloomberg.com/dataview/2014-04-17/how-americans-die.html
The aging population refers to a growing proportion of seniors in the population
from 18.76% 55 and older in 1968 to 24.86% in 2010. So people living longer
would first drive mortality rates down, but as we get a higher proportion of older
people in the population, the mortality rates would stop dropping and level out as
older people die sooner than younger people.
b. Interpret the 11.82% for 2010 in slide 2 and compare to the 25% on the vertical
axis.
In 2010 11.82% of the population was in the 55 – 64 age bracket, and 25% of the
population was 55 and older.
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4. Looking at slide 4 which line stands out from the rest? What do you think accounts for
this difference?
Mortality Rate per 100,000 (1968 = 100)
110
100
80.16
90
75.93
77.15
76.87
80
70
69.12
60
67.27
64.03
50
40
55-64
25-44
2010
45-54
2005
1990
1985
65-74
2000
75-84
1995
Over 85
1980
1975
1970
30
Below 25
Data from: http://www.bloomberg.com/dataview/2014-04-17/how-americans-die.html
The 25-44 mortality rate line shoots up in the mid-1990’s. This was due to AIDS:
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5. Interpret the statistics 80.16 and 64.03 for 1985 on slide 4. Hint: Compare to slide 3
statistics shown here.
20,000
Mortality Rate per 100,000
15,710.8
Over 85
15,000
75-84
65-74
10,000
55-64
6,398.6
45-54
25-44
2,862.9
5,000
1294.2
Below 25
2010
2005
2000
1995
1990
1985
1980
1975
1970
519.3
160.3
102.9
Data from: http://www.bloomberg.com/dataview/2014-04-17/how-americans-die.html
Looking at the slide 3 1985 statistics we see a mortality rate of 15,710.8 deaths per
100,000 for over 85, and 102.9 deaths per 100,000 for under 25. Slide 4 has set the rates
for 1968 (19,598.5 and 160.7 respectively) equal to 100. The 80.16 in slide 4 comes from
the proportion:
Telling us that the rate for the over 85 group in 1985 is 80.16% of the 1968 rate.
Similarly the rate for the under 25 group in 1985 is 64.03% of the 1968 rate.
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1,200
1,046.01
1,000
800
600
400
236.66
200
Deaths from Drugs
Deaths from Suicide
2010
2008
2006
2004
2002
2000
1998
1996
1994
1992
179.62
1990
Deaths from Drugs and Suicide 45-54 (1990 = 100)
6. Deaths from drugs in the 45 to 54 year old populations have increased by what factor
from 1990 to 2010 as shown in slide 11?
Population
Data from: http://www.bloomberg.com/dataview/2014-04-17/how-americans-die.html
so a factor of 10.
7. Does slide 17 indicate Medicare spending has been increasing or decreasing since 2010?
Increase in Medicare Spending
(billions of dollars)
30
25
20
15
$5B
10
$6B
5
October 2011
October 2012
October 2013
Medicare Spending on Alzheimer’s and Other Dementias
Other Medicare Spending
Data from: http://www.bloomberg.com/dataview/2014-04-17/how-americans-die.htmlare spending has
been increasing, the increase has been decreasing.
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Quantitative Reasoning / Introduction to Excel
Why go to college? What is the PURPOSE of a college education? List 3 specific purposes:
1. Career (#1 for students)
2. Life Skills
3. Critical Thinking (#1 for faculty)
What is critical thinking? List 3 characteristics of critical thinking:
1. Asking informed questions!
2. Weighing both sides of an argument.
3. Recognize and define problems.
Introduction to Excel
Screenshots from Microsoft® Excel®. Used by permission of Microsoft Corporation.
1. What formula is entered in cell E3?
=C3*D3
2. How do you fill this formula down?
Highlight the cell, click on fill handle and drag down.
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3. What happens to the cell references in the formula when you fill down?
The numbers increase, C3 to C4 etc.
4. What built-in function can be used in cell E9? What formula using this function is in cell
E9?
The SUM function. =SUM(E3:E7)
5. If you format the Tax in cell E10 to show zero decimal places what happens to the output
in cell E11?
Nothing it stays the same
Screenshots from Microsoft® Excel®. Used by permission of Microsoft Corporation.
6. How do you change the name of the sheet tabs?
Double-click on the sheet tab (CTRL-Click on a Mac), the name will be highlighted,
change the name.
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7. What button do you hit in the menu to format the numbers as currency? What button
makes borders?
The $ icon. The windowpane looking icon below the Bold B in the screen shot above.
8. To create the chart what cell range was highlighted in the worksheet?
B2:E7
9. What type of chart is this?
Concert Revenue
Hot Dog
Cookie
Popcorn
Water
Soda
$-
$50.00
$100.00
Concert 1
$150.00
Concert 2
$200.00
Concert 3
Stacked bar chart
Play with Excel and create some more charts. Have Fun!
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$250.00
Function and TV Loan
Definition: A function is a relationship between quantities referred to as inputs and
outputs, in which every collection of inputs is paired up with one and only one output.
1. Determine which of the following are functions:
INPUTS
OUTPUT
FUNCTION?
States
Senators
Senators
States
States
Number of senators
People
People
Anyone they been married
to…
Number of spouses
US Citizens
Social Security Numbers
No, 2 senators (outputs) per
state (input).
Yes, each senator (input) has 1
and only 1 state (output).
Yes, constant function each
state (input) has same output
(2).
No, the output will be multiple
people for some inputs.
Yes, the output is a single
number for each person.
Yes, one-to-one function.
SS#’s
US Citizens
Yes, one-to-one function.
People
Birthdays
Yes
Birthdays
People born on that day
No
Students in this class
Shoe size
Yes
Shoe Size
Students in this class with that
shoe size
No
2. Come up with a function between two quantities which remains a function when you
switch the inputs and outputs…
Input: Number of gallons of water
Output: Weight of the water
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Next we will explore some functions associated with taking out a loan.
Financial Literacy Vocabulary for Loans
•
Principal: The amount of money borrowed from the lender.
•
Interest: The money or fee the lender charges you for borrowing money.
•
Period: The length of time before your next payment is due and interest is charged;
typically 1 month for most loans.
•
Balance: What you owe at the end of each period factoring in any interest and payments
made.
•
Interest Rate: The ratio of interest charged to amount owed, typically represented as a
percentage which is a rate per 100. An interest rate of 6% means you will be charged $6
for every $100 you owe. Ratios will be covered in Chapter 2, rates in Chapter 3 and
percentages in Chapter 4.
•
Annual Percentage Rate (APR): The interest rate for a period of 1 year.
•
Periodic Rate: The interest rate for a period other than 1 year, it is the APR divided by
the number of periods in a year: APR/n. A 6% APR computed monthly will give a
6%/12 = 0.5% periodic rate.
•
Annual Percentage Yield (APY): Given a periodic rate, your interest will compound.
The APY is the ratio of the total interest charged for the year to the original principal.
Credit Card Loan
Let us assume you buy a Sony flat screen TV for your dorm room that costs $1,000. You make
the purchase with a store credit card that has a 12% Annual Percentage Rate (APR). You do not
have to make any monthly payments for the first year (sounds good!), but they will charge interest
at the end of each month. How much do you owe at the end of the first year?
3. What is the Principal?
$1,000
4. What is the monthly interest rate?
1%
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5. What is the interest charged for the first month?
$10
Double-click on the fill handle of
the highlighted cells to quickly fill
down to month 12 (you must have
months 1-12 already filled in).
Screenshots from Microsoft® Excel®. Used by permission of Microsoft Corporation.
Relative
Mixed
Mixed
Absolute
Cell Reference Handbook
Changes the row number when you fill up/down and changes the
column letter when you fill left/right.
$A4 Changes the row number when you fill up/down and fixes the
column letter when you fill left/right.
A$4 Fixes the row number when you fill up/down and changes the
column letter when you fill left/right.
$A$4 Fixes the row number when you fill up/down and fixes the column
letter when you fill left/right.
A4
6. What formula is in cell C4? D4?
=B4*$G$6 and =B4+C4
7. Which formula involves an absolute cell reference?
=B4*$G$6
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8. Why do we need to fill in the second row before we fill the formulas down?
To link the beginning of month 2 to the end of Month 1, which are the same value.
9. What do you owe at the end of the first year?
$1,126.83
10. What is the APY?
12.683%
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Car Loan
You have just bought a new VW Jetta for $17,254.38. You put down $2,254.38 of your savings
so you only have to borrow $15,000. The auto dealership gets you a loan from a bank for 5 years
at 6%, which you agree to and sign. What will be your fixed monthly payment?
The periodic payment is a function of 4 inputs:
INPUTS
Principal (P)
APR
Number of periods in 1 year (n)
Number of years of the loan (t)
OUTPUT
Periodic payment (PMT)
APR
n
PMT =
  APR − nt 
1 − 1 +
 
n  
 
P×
We are going to create the following spreadsheet:
Screenshots from Microsoft® Excel®. Used by permission of Microsoft Corporation.
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Caution! When entering formulas into Excel you must use Order of Operations:
1. Parentheses: Everything entered in parentheses will be computed first. When in doubt use
parentheses, especially for the numerator and denominator of fractions. Parentheses are like
Vitamin C, too much won’t hurt you (you just have to use the rest room a lot) but too little and
your formula will get gangrene and rot.
2. Exponents: Exponents are next, use the ^ symbol above the number 6. Complicated exponents
1
3
need parentheses: 2 = 2^(1/3) .
3. Multiplication and Division: These are tied, Excel will compute from left to right.
4. Addition and Subtraction: These are also tied and will be computed from left to right.
5. PEMDAS: Please Excuse My Dear Aunt Sally is the traditional mnemonic device to help
remember the order of operations.
6. Examples:
a. = 3*5 − 2 = 13
b. = 3* ( 5 − 2 ) = 9
c.
= 5 + 2 *3 = 11 (not 21)
1. Evaluate =2-6/4+2
2. What formula is in cell E2?
=(A2*B2/C2)/(1-(1+B2/C2)^(-C2*D2))
3. In the 10×6 table what are the two inputs (variables) for each of the 60 monthly payments
(outputs)? Note: Assume the principal, $15,000, and the number of months in one year,
12, are both fixed constants.
APR and Term (Number of Years)
4. What are the two specific numeric inputs for the monthly payment in cell G12?
6% and 5 years
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5. What are the two specific numeric inputs, and associated cell references, for the formula
in cell C7? Which of these should vary as we fill the formula down?
1% and 1 year (B7 and C6). The 1% should change.
6. What formula is in cell C7?
=($A$2*$B7/12)/(1-(1+$B7/12)^(-12*C$6))
7. How much total interest do you pay with the $289.99 monthly payment?
$2,399.40 (60*289.99-15000)
8. How much would you save if you switched from the 5 year 6% loan to a 5% APR?
$415.29 ($2,399.40 – $1,984.11)
9. How much would you save if you switched from the 5 year 6% loan to a 3 year 6% loan?
$1,215.12 ($2,399.40 – $1,184.28)
10. If you have horrible credit you might be forced to take out a 6 year 10% loan. How much
interest do you pay over the life of this loan?
$5,007.90
11. What is fixed (column or row) by the following cell references?
$F$4
G$2
T1
Both
Row
Neither
$V85
J$21
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$S99
Descriptive Statistics
We will use the CAT scores sheet in Data Sets:
Screenshots from Microsoft® Excel®. Used by permission of Microsoft Corporation.
Descriptive Statistics: Note that the name of each function in Excel is as given in the spreadsheet,
except for the mean which uses the AVERAGE function, and the range which is MAX – MIN.
Enter the appropriate functions into the spreadsheet to compute the following, and write the
formula you would type into Excel after each definition (note there are 98 scores listed in column
B and the cell range B4:B101 has been named scores).
1. Mean: the arithmetic average.
=AVERAGE(scores) or =AVERAGE(B4:B101)
2. Median: the middle of the data set (half above, half below, 50th percentile).
=MEDIAN(scores)
3. Mode: the most frequently occurring value.
=MODE(scores)
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4. Standard Deviation: the “average distance” of the data values from the mean.
=STDEV(scores)
5. Max: the largest value.
=MAX(scores)
6. Min: the smallest value.
=MIN(scores)
7. Range: the difference between largest and smallest values (Max – Min).
=E8 – E9
8. Count: the number of values (usually referred to as N).
=COUNT(scores)
Histogram
Note the formula bar in the spreadsheet giving the function =COUNTIF(scores,”2
1≤ z < 2 z