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Quantitative Reasoning is the method of using mathematics to
solve real world problems. This can be anything from trying to figure out how
long it will take to fill a swimming pool with water based on the size of the
pool to figuring out consensus problems of how many people participated doing
what. The focus of the lesson is translating real world problems into solvable
mathematics problems that can be answered. Click here for Chapter 2. Let's look at an example question:
A political scientist surveys 24 of the current 84 representatives in a state's legislature.
What is the size of the sample:
What is the size of the population:
When conducting surveys it's important to be able to distinguish between the sample and the population.
The sample is the group being studied in this case the sample is 24.
The population is the entire group that you are drawing conclusions about whether they are surveyed or not which is 84 making the proper answer to this question:
What is the size of the sample:24
What is the size of the population:84
Similar question:
Next Question:
The city of Raleigh has 11100 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 350 randomly selected registered voters was conducted. 208 said they'd vote for Brown, 110 said they'd vote for Feliz, and 32 were undecided.
Describe the population the surveyors are really interested in.
The reason the
answer is All registered voters in Raleigh is because the surveyors conducted a
telephone poll of 350 randomly selected registered voters in Raleigh and it
wasn’t talking about unregistered votes or anything else but the registered
voters.
Next Question:
The city of Raleigh has 6200 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 500 randomly selected registered voters was conducted. 219 said they'd vote for Brown, 274 said they'd vote for Feliz, and 7 were undecided.
Describe the population actually represented by this survey.
The reason the answer is All registered voters with telephones in Raleigh is because the pool was a telephone poll of registered voters.
The city of Raleigh has 11200 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 400 randomly selected registered voters was conducted. 138 said they'd vote for Brown, 228 said they'd vote for Feliz, and 34 were undecided.
Describe the population actually represented by this survey.
Describe the population actually represented by this survey.
The key is when trying to interpret data where a lot of unneeded data is given look for keywords like telephone poll of registered voters often they give you so much extra data which makes you slip up but if you break that information down the answer becomes clear.
Next Question:
The city of Raleigh has 11400 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 300 randomly selected registered voters was conducted. 170 said they'd vote for Brown, 101 said they'd vote for Feliz, and 29 were undecided.
Describe the sample for this survey.
Describe the sample for this survey.
The reason the answer is The 300 voters surveyed is because the telephone poll of 300 randomly selected registered voters was conducted. That is the subset of the registered voters that was used for this question.
Next Question:
The city of Raleigh has 11100 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 200 randomly selected registered voters was conducted. 84 said they'd vote for Brown, 110 said they'd vote for Feliz, and 6 were undecided.
Give the sample statistic for the proportion of voters surveyed who said they'd vote for Brown. Note: The proportion should be a fraction or decimal, not a percent.
This sample statistic suggests that we might expect of the 11100 registered voters to vote for Brown.
Give the sample statistic for the proportion of voters surveyed who said they'd vote for Brown. Note: The proportion should be a fraction or decimal, not a percent.
This sample statistic suggests that we might expect of the 11100 registered voters to vote for Brown.
To come to the conclusion of these answers you would take the number of the people who said they would vote for brown and the total number of voters selected which in this case would be:
To get the sample statistic you would divide 84/200 which gives you 0.42 then:
0.42×11,100=4,662
That gives you the sample statistic.
Next Question:
The city of Raleigh has 12000 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 250 randomly selected registered voters was conducted. 122 said they'd vote for Brown, 115 said they'd vote for Feliz, and 13 were undecided.
Describe the target population for this survey.
Describe the target population for this survey.
Again another one in which the poll was taken randomly from selected registered voters.
Next Question:
The city of Raleigh has 6400 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 550 randomly selected registered voters was conducted. 321 said they'd vote for Brown, 182 said they'd vote for Feliz, and 47 were undecided.
Describe the sample for this survey.
Describe the sample for this survey.
This one we can see the 550 randomly selected registered voters was conducted meaning the ones chosen was the 550 voters surveyed.
Next Question:
The city of Raleigh has 11200 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 200 randomly selected registered voters was conducted. 117 said they'd vote for Brown, 71 said they'd vote for Feliz, and 12 were undecided.
Give the sample statistic for the proportion of voters surveyed who said they'd vote for Brown. Note: The proportion should be a fraction or decimal, not a percent.
This sample statistic suggests that we might expect
Give the sample statistic for the proportion of voters surveyed who said they'd vote for Brown. Note: The proportion should be a fraction or decimal, not a percent.
This sample statistic suggests that we might expect
Similar to the last one we take 117 who voted for brown over the 200 randomly selected and divide 117/200= 0.585 then we take 0.585 × 11200 = 6,552
Next Question:
A researcher interested in Springfield citizens' shopping habits surveys a randomly selected group of 200 Walmart shoppers. 76% of those surveyed indicated that price was more important to them than where an item was produced. The researcher concluded that "about three quarters of the people in Springfield are more concerned with cost than where an item is made."
This conclusion might be invalid because:
This conclusion might be invalid because:
The conclusion is invalid because of the amount surveyed. In order to make valid conclusion about population, it’s
very important to use proper sampling technique so that the sample represents
the larger group that is being studied. Otherwise the findings might not be
applicable to the complete population of the sample.
Next Question:
You want to estimate how many fish there are in a pond. Suppose you capture 270 fish, tag them, and throw them back into the pond. After a couple of days, you go back to the pond and capture 540 fish, of which 27 are tagged.
An estimate for the number of fish in the pond is fish
An estimate for the number of fish in the pond is fish
N= Total number of fish in the pond.
M= 270 (the number of fish tagged and released)
n=540 (the number of fish in the second sample)
m=27 (the number of fish tagged in the second sample)
(M/N)=(m/n)
(270/N) =(27/540)
Cross-Muliply:
270*540 = 27 * N
145,800 = 27 * N
Now, divide both sides by 27 to isolate N:
N = 145,800 / 27
N= 5,400
Similar question:
You want to estimate how many fish there are in a pond. Suppose you capture 270 fish, tag them, and throw them back into the pond. After a couple of days, you go back to the pond and capture 405 fish, of which 27 are tagged.
An estimate for the number of fish in the pond is fish
Next Question:
The city of Raleigh has 10900 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 400 randomly selected registered voters was conducted. 152 said they'd vote for Brown, 219 said they'd vote for Feliz, and 29 were undecided.
Give the sample statistic for the proportion of voters surveyed who said they'd vote for Brown. Note: The proportion should be a decimal rounded to 3 decimal places.
This sample statistic suggests that we might expect of the 10900 registered voters to vote for Brown.
We do the same only round the proportion 3 decimal places 152/400= 0.380 x 10900 = 4142
Similar questions:
The city of Raleigh has 6000 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 500 randomly selected registered voters was conducted. 219 said they'd vote for Brown, 268 said they'd vote for Feliz, and 13 were undecided.
Give the sample statistic for the proportion of voters surveyed who said they'd vote for Brown. Note: The proportion should be a fraction or decimal, not a percent.
$0.438$
This sample statistic suggests that we might expect $2,628$ of the 6000 registered voters to vote for Brown.
The city of Raleigh has 6900 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 400 randomly selected registered voters was conducted. 202 said they'd vote for Brown, 190 said they'd vote for Feliz, and 8 were undecided.
Give the sample statistic for the proportion of voters surveyed who said they'd vote for Brown. Note: The proportion should be a fraction or decimal, not a percent.
$0.505$
This sample statistic suggests that we might expect $3484.5$ of the 6900 registered voters to vote for Brown.
The city of Raleigh has 11000 registered voters. There are two candidates for city council in an upcoming election: Brown and Feliz. The day before the election, a telephone poll of 300 randomly selected registered voters was conducted. 111 said they'd vote for Brown, 164 said they'd vote for Feliz, and 25 were undecided.
Give the sample statistic for the proportion of voters surveyed who said they'd vote for Brown. Note: The proportion should be a decimal rounded to 3 decimal places.
$0.37$
This sample statistic suggests that we might expect $4,070$ of the 11000 registered voters to vote for Brown.
Tailgate Project:
Tailgate
Mission
You and your closest 9
friends decide to tailgate before attending the Baseball game. You each give $5.00 for food and
beverages. You are in charge of shopping
at your local supermarket. Based on your
budget, decide what are your best options are for purchasing hamburgers and/or
hot dogs, beverages, condiments, and snacks.
You need to justify (unit analysis) why you made the decisions you
did. Space is given below each item to
show the unit analysis.
You must complete
the unit price for the entire worksheet regardless of whether or not you use that
item in your tailgate. Must
show calculations/formula for credit. Any answer given with no calculations
shown will result in no credit for that answer.
This
is an exercise that tests your skills in conversions, planning, and general
math skills.
You MUST spend no less
than $45 and no more than $50 with a mixed variety of items in
different categories. You can look up conversions on the internet they are
readily available
Assumptions:
·
All paper products,
utensils, and ice are provided by someone else in the group.
·
You must stay
within budget and cannot get money from any other source.
·
All hamburgers
must weigh approximately 1/4 of a pound precooked.
·
HINT: Use unit conversions to compare what item is
a “better deal”.
Beverages
Coke Products
(Coke, diet Coke, Bar’s, Sprite, Fanta, Nestea, Minute Maid)
Ex. I will do the first
one for you
1 liter = 33.814 oz
So, the 2-liter bottle
contains 67.628 oz
Since you get 2 for $3.00
each bottle is $1.50
If you take $1.5 / 67.628
oz that equals $.0222 per oz. always round to the nearest cent (2 decimal
places) you get $.02 per oz or 2 cents per oz
BEVERAGES
Coke Products:
| Description | Quantity | Volume/container | Cost | Unit Price ($per oz) |
| 2 liter bottles | 1 | 2L | 2/$3.00 | $.02/oz |
| 12 pack of cans | 12 | 12 oz | 3/$9.00 | $.02/oz |
| 6 pack of bottles | 6 | 24 oz | 3/$8.00 | $.02/oz |
- 3/$9.00 gets broken down as a cost for 1 by dividing $9/3=3 Multiple 12x12oz=144 $3.00/144=$0.02 rounded stays the same.
- 3/$8.00 gets broken down as a cost for 1 by dividing $8/3=$2.66 rounded $2.67 Multiple 6x24=144 $2.67/144=$0.018 rounded to $0.02.
Pepsi Products:
| Description | Quantity | Volume/container | Cost | Unit Price ($per oz) |
| 2 liter bottles | 1 | 2L | 2/$3.00 | $.02/oz |
| 12 pack of cans | 12 | 12 oz | 3/$12.00 | $.03/oz |
- 1 liter=33.814 so 2x33.814=67.628 1x67.628=67.628 Now divide 3/2=$1.50 Divide $1.50/67.628=$0.02
- 12/3=$4. Multiply 12x12=144 Divide 4/144=0.027 rounded is $0.03
Bottle Water:
| Description | Quantity | Volume/bottle | Cost/case | Unit Price ($per oz) |
| 8 pack Dansani | 8 | 16.9 oz | $3.69 | $0.03/oz |
| 12 pack Ice Mountain | 12 | 8 oz | $2.99 | $0.03/oz |
| 24 pack of Nestle | 24 | 8 oz | $2.99 | $0.02/oz |
| 6 pack of Aquafina | 6 | 24 oz | $3.79 | $0.03/oz |
| 24 pack of store brand | 24 | 16.9 oz | $2.49 | $0.01/oz |
- 8x16.9=135.2 3.69/135.2=0.027 rounded to $0.03
- 12x8=96 2.99/96=0.031 is $0.03
- 24x8=192 2.99/192=0.015 rounded is $0.02
- 6x24=144 3.79/144=0.026 rounded is $0.03
- 24x16.9=405.6 2.49/405.6=0.006 rounded is $0.01
Power Drinks:
| Description | Quantity | Volume | Cost | Unit Price ($per oz) |
| Gatorade | 1 | 32 oz | $0.69 | $0.02 |
| Monster | 1 | 16 oz | 4/$5.00 $1.25 | $0.08 |
| Powerade | 1 | 32 oz | $0.79 | $0.02 |
- 1x32=32 0.69/32=0.02
- First solve cost of one by dividing 5/4=1.25 1x16=16 1.25/16=0.078 rounded is $0.08
- 1x32=32 0.79/32=0.024 is $0.02
ENTREE
Hot Dogs:
| Description | Quantity/pack | Weight/pack | Cost/pack | Unit Price ($per oz) |
| Oscar Meyer | 10 | 1 lb | $1.99 | $0.01 |
| Kahn’s Bun Size | 8 | 16 oz | $2.49 | $0.02 |
| Hebrew National | 7 | 12 oz | $3.99 | $0.05 |
| Oscar Meyer | 8 | 1 lb | $3.39 | $0.03 |
| John Morrell | 10 | 16 oz | $1.69 | $0.01 |
1 lb=16 oz
- 10x16=160 1.99/160=$0.01
- 8x16=128 2.49/128=$0.02
- 7x12=84 3.99/84=$0.05
- 8x16=128 3.39/128=$0.03
- 10x16=160 1.69/160=$0.01
Hamburgers:
| Description | Weight/pack | Cost/pack | Notes | Unit Price ($per lb) |
| Ground Sirloin | 1 lb | $5.69 | $5.69 | |
| 4 premade ground angus patties | 1.5 lbs | $5.49 | $3.66 | |
| Tube of ground chuck | 3 lbs | $10.47 | 80%lean/20%fat | $3.49 |
| Tube of ground beef | 5 lb | $13.95 | 73%lean/27%fat | $2.79 |
| Ground turkey | 1 lb | $4.19 | $4.19 |
- $5.69/1=$5.69
- $5.49/1.5=$3.66
- $10.47/3=$3.49
- $13.95/5=$2.79
- $4.19/1=$4.19
Hot Dog Buns:
| Description | Quantity/package | Cost/package | Unit Price ($per bun) |
| Pack of 8 | 8 | $1.00 | $0.13 |
| Pack of 12 | 12 | $2.69 | $0.22 |
- 1.00/8=0.13
- 2.69/12=0.22
Hamburger Buns:
| Description | Quantity/package | Cost/package | Unit Price ($per bun) |
| Pack of 8 | 8 | $1.00 | $0.13 |
| Pack of 8 Sesame Seed | 8 | $1.50 | $0.19 |
| Pack of 12 | 12 | $1.99 | $0.17 |
- 1.00/8=0.13
- 1.50/8=0.19
- 1.99/12=0.17
CONDIMENTS
Mustard:
| Description | Weight/container | Cost/container | Unit Price ($per oz) |
| French’s | 8 oz | $1.47 | $0.18 |
| Hellmann’s | 9.5 oz | $2.59 | $0.27 |
| Store brand | 8 oz | $0.79 | $0.10 |
- 1.47/8=0.18
- 2.59/9.5=0.27
- 0.79/8=0.10
Ketchup:
| Description | Weight/container | Cost/container | Unit Price ($per oz) |
| Hunt’s | 24 oz | $1.00 | $0.04 |
| Heinz | 20 oz | $2.19 | $0.11 |
| Store brand | 24 oz | $1.00 | $0.04 |
- 1.00/24=0.04
- 2.19/20=0.11
- 1.00/24=0.04
Mayonnaise:
| Description | Weight/container | Cost/container | Unit Price ($per oz) |
| Hellmann’s | 16.5 oz | $2.99 | $0.18 |
| Miracle Whip | 12 oz | $3.59 | $0.30 |
| Store brand | 12 oz | $1.99 | $0.17 |
- 2.99/16.5=0.18
- 3.59/12=0.30
- 1.99/12=0.17
American Cheese:
| Description | Quantity/pack | Weight/pack | Cost/pack | Unit Price ($per oz) |
| Kraf | 24 slices | 16 oz | $4.49 | $0.01 |
| Velveeta | 24 slices | 12 oz | $5.69 | $0.02 |
| Store brand | 16 slices | 12 oz | $2.69 | $0.01 |
- 24x16=384 4.49/384=0.01
- 24x12=288 5.69/288=0.02
- 16x12=192 2.69/192=0.01
SNACKS & DESSERTS:
| Description | Weight/pack | Cost/pack | Unit Price ($per oz) |
| Bakery Cookies | 18 oz | $3.99 | $0.22 |
| Oreo Double Stuf | 15 oz | $2.99 | $0.20 |
| Chips Ahoy | 13 oz | $2.49 | $0.19 |
| Lays Chips | 10 oz | $2.99 | $0.30 |
| Grippos | 8 oz | $2.79 | $0.35 |
| Husman’s | 10 oz | $2.29 | $0.43 |
| Ruffles | 8.5 oz | $2.99 | $0.35 |
| Doritos | 11 oz | $3.29 | $0.30 |
| Store brand pretzels | 15 oz | 3/$5.00 $1.67 | $0.11 |
- 3.99/18=0.22
- 2.99/15=0.20
- 2.49/13=0.19
- 2.99/10=0.30
- 2.79/8=0.35
- 4.29/10=0.43
- 2.99/8.5=0.35
- 3.29/11=0.30
- 5/3=1.67 then 1.67/15=0.11
Project Questions ANSWERS:
1. What beverages would you purchase and in what quantities?
1-12 pack of Coke
1-12 pack of Pepsi
1-24 pack of store brand water
4 bottles of Gatorade
2.How did you arrive at that decision?
I chose the sodas in 12 packs because it was more than enough drinks for everyone rather than going the cheaper route with bottles that will be used to quick. The water was cheaper doing it this way and you get 24 bottles of 16.9 oz rather than spending more money for a smaller quantity of bottles and oz per bottle. I did 4 Gatorade because it was cheaper than Powerade and healthier/cheaper than monster.
3.How much would you spend on beverages?
$12.25
4.How many hot dogs and/or hamburgers, including buns, would you purchase? Please list those along with their quantities.
2-Oscar Meyer 8 packs
2-8 pack of hot dog buns
1- Tube of ground beef
2-Packs of hamburger buns
5. How did you arrive at that decision?
The hot dogs were costly but I wanted them to match up with the bun quantity, I could have gone with the off brand but people are sticklers with their name brand hot dogs so I chose oscar meyer. I chose 2 packs of 8 count buns so that the quantities matched. I did the more expensive beef because it came in a 5 lb quantity and chose 2 packs of buns as well.
6.How much did you spend on hot dogs, hamburgers, and buns?
$24.73
7. What condiments would you purchase?
1-Mustard store brand
1- Ketchup Hunts
1- Mayo store brand
1-Store brand cheese
8. How did you arrive at that decision?
I was fine with picking the cheaper store brands because they were cheaper and allowed me to spend more money towards the more important items.
9. How much did you spend on condiments?
$6.47
10. What snacks would you purchase?
1-Bag of Lays
1-Bag of Doritos
11. How did you arrive at that decision?
These are always crowd pleasers when eating tailgate food. They are more costly than other items, but chips go great with burgers and hot dogs. Our saving on other items allowed us to do splurges like this.
12. How much did you spend on snacks?
$6.28
13.How much did you spend total?
$49.73
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