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.

Copyright © 2016 Pearson Education, Inc.

QL-1

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.

Copyright © 2016 Pearson Education, Inc.

QL-2

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.

Copyright © 2016 Pearson Education, Inc.

QL-3

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:

Copyright © 2016 Pearson Education, Inc.

QL-4

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.

Copyright © 2016 Pearson Education, Inc.

QL-5

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.

Copyright © 2016 Pearson Education, Inc.

QL-6

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.

Copyright © 2016 Pearson Education, Inc.

1

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.

Copyright © 2016 Pearson Education, Inc.

2

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!

Copyright © 2016 Pearson Education, Inc.

3

$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

Copyright © 2016 Pearson Education, Inc.

5

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%

Copyright © 2016 Pearson Education, Inc.

6

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

Copyright © 2016 Pearson Education, Inc.

7

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%

Copyright © 2016 Pearson Education, Inc.

8

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.

Copyright © 2016 Pearson Education, Inc.

9

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

Copyright © 2016 Pearson Education, Inc.

10

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

Copyright © 2016 Pearson Education, Inc.

11

$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)

Copyright © 2016 Pearson Education, Inc.

13

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