Category: Mathematics

Discuss your rationale for your choice of the real-world optimization problem. How were you able to identify that it was an example of the CPP? Why is this example relevant?

Chinese Postman Problem

Select a real-world optimization problem that is an example of the Chinese Postman Problem (CPP).

Discuss your rationale for your choice of the real-world optimization problem. How were you able to identify that it was an example of the CPP? Why is this example relevant?

Describe how you created the graph modeling the real-world scenario corresponding to the problem.

Identify if the graph has a Euler path and a Euler circuit. Justify your answer.

Explain in detail how you solved the problem.

Describe an alternative optimal solution to the problem. If no other solution exists, explain why not.

Consider another variable that could be optimized in the problem. How would your answers to Part I be affected by this change?

Discuss the advantages of using graph theory to solve this problem.
Requirements:

Compare the calculated value from step VI to this actual value. Explain why (or why not) you would have expected your prediction (calculated value) to be close to the actual value.

Data Analysis

Draw a trend line for the scatter plot. Use the following scatter plot of the ordered pairs (fat grams, total calories) as an example.

Calculate the slope of the trend line. (Choose two points on the line and find vertical change over horizontal change.)

Graphing calculators and spread sheets have features with which to draw trend lines and determine the equation. You may choose to use one of these options. If you use technology, indicate what steps were taken to arrive at your equation.

Using the slope and y-intercept, write the equation of the trend line ( y = mx + b).

Choose a “calories from fat” value that is not in your collected data set and that is at least 10 fat calories away from any collected value. Use the equation calculated in step V to predict the number of fat grams in an item having that number of fat calories. Be sure to show your work.

Search for an item in a fast food menu having the same number of fat calories as the one you chose above. (If you cannot find the exact value, get as close as you can.) Compare the calculated value from step VI to this actual value. Explain why (or why not) you would have expected your prediction (calculated value) to be close to the actual value.

Explain why the creator of the misleading graph would want to create the graph in the first place.Are numbers on the axis spaced out proportionally or have they been varied to create a dramatic impression?

Misleading Graphs

Is there a title that explains what the graph is displaying?

Are numbers on the axis spaced out proportionally or have they been varied to create a dramatic impression?

Is the graph too loud? Does it have too many components that it distracts from content?

Are there sources cited to know where the data came from?

Use the internet to find a misleading graph. Key Terms to Search: Misleading Graphs

Provide a screenshot of the graph
Cite the Source
Explain why the graph is misleading
Analysis

Explain how you would fix the graph so it is not misleading.

Explain why the creator of the misleading graph would want to create the graph in the first place.

How large of a sample is required in order to be 90% sure that the sample proportion is off by no more than 2%? Does a parent’s education level have anything to do with their involvement at their child’s school?

How large of a sample is required in order to be 90% sure that the sample proportion is off by no more than 2%?

Does a parent’s education level have anything to do with their involvement at their child’s school?

At the 0.01 level of significance, test the claim that educational attainment and volunteering at school are independent.

The president of a college wants to determine the mean number of units that college students take per semester. How large of a sample is required in order to be 99% sure that a sample mean will be off by no more than 1.25 units? An initial study suggested that the standard deviation is approximately 2.1 units.

Test the claim that the mean time required for high school students to run 1 mile is greater than 7 minutes at the 0.05 level of significance. Here are the results of a random sample of
25 students.
7.3 7.7 9.2 8.8 7.6 7.2 6.6 6.4 8.0
7.5 7.5 7.7 8.1 8.6 6.8 6.9 7.5
7.3 7.8 8.2 10.4 11.6 7.2 7.7 7.0

A sample of 200 smokers were asked at what age they started smoking. The meanage was 18.3 years. The population standard deviation is 4.35 years. Construct a 90% confidence interval for the mean age that all smokers begin smoking.

A random sample of 400 Spanish adults revealed that 143 were smokers. A random sampleof 300 American adults revealed that 70 were smokers. At the 0.05 level of significance, test the claim that the proportion of Spanish adults that smoke is greater than the proportion of American adults that smoke.

To test the claim that the mean blood glucose level of women is 100 mg/dL, a researcher takes a random sample of 135 women. They had a mean blood glucose level of 108.3 mg/dL.

(The population standard deviation is 3.4 mg/dL). Test the claim at the 0.05 level of significance.

A random sample of 50 bags of blue corn tortilla chips had a mean weight of 9.06 ounces, with a standard deviation of 0.05 ounces. A random sample of 50 jalapeno tortilla chips had a mean weight of 9.01 ounces, with a standard deviation of 0.03 ounces. At the 0.05 level of significance, test the claim that the mean fill of blue corn tortilla chip bags is the same as the mean fill of jalapeno tortilla chips.

A group of 9 concertgoers was selected at random. Before the concert they were given a hearing test, and then given another one after the concert. (The volume varied during the test, and the subject also had to state which ear the sound was in.) Here are the number of correctly identified sounds out of 10, both before and after the concert.
Person A B C D E F G H I
Before 9 10 9 8 8 9 9 9 8
After 8 8 9 6 6 7 10 8 5

At the 0.05 level of significance, test a person’s hearing is worse after being affected by the noise of a concert.

Construct a 90% confidence interval for the mean highway mileage for hybrid cars. A random sample of 8 hybrid cars had the following highway mileages in mpg.
36 41 37 45 50 40 32 39

Hole selections for professional golf courses are changed each day of a PGA tournament. Are any of the days set up to be more difficult or easier? Here are the scores of 7 golfers on the four days of a tournament.
Round 1 Round 2 Round 3 Round 4
63 65 68 65
66 67 68 67
66 67 70 67
70 65 66 69
67 68 69 67
68 67 70 67
68 67 67 70
At the 0.01 level of significance, test the claim that the mean scores produced by the four different rounds are equal.

In a survey of 500 women, 120 said that Valentine’s Day was their favorite occasion to receive flowers. Construct an 88% confidence interval for the proportion of all women who feel that Valentine’s Day is their favorite occasion to receive flowers

What is the rate per gallon for an apartment that uses under 100 gallons per day?Which equation calculates total water cost for water use between 100 and 150 gal?

Consider the example above.
Looking at the graph above, how much does a person earn for a day if the person worked 9 hours?
$80
$95
$100
No one line represents the function. To graph, the first piece’s line is:
0-8 hours: y = 10x
The second piece’s line starts at the first piece’s end, meaning x is 8 and y is 80. What is that line’s slope?
10
15
3)
Drawing a line with slope 15, starting at x of 8 and y of 80, can be done by drawing a second point, obtained by going 1 unit to the right and up by how many units?

1
15
95
4)
No one equation can represent the piecewise linear function. Instead, the function is described like this:
0-8 hours: y = 10x
8+ hours: y = 80 + 15(x – ?)
(The 8+ means “8 or more”).
1
8
5)
For more than 8 hours worked, the total earnings are y = 80 + 15(x – 8). A person works a long 20 hour day. How many dollars does the person earn for that long day?
$200
$260
$380
6)
Suppose the company above pays double-time (20 dollars/hour) for above 12 hours. The equations are then:
0-8 hours: y = 10x
8-12 hours: y = 80 + 15(x – 8)
12+ hours: y = ___ + 20(x – 12)
80
140
155
What is the rate per gallon for an apartment that uses under 100 gallons per day?
0.2 cents/gal
0.5 cents/gal
1 cent/gal
2)
What is the rate per gallon for an apartment that uses over 150 gallons per day?
0.2 cents/gal
0.5 cents/gal
1 cent/gal
3)
Knowing that using 150 gallons in a day costs 45 cents, and every gallon above 150 gal costs 1 cent, what would 400 gallons cost? Hint: Add the cost for those additional 400 – 150 = 250 gal to the 45 cents.
295 cents
400 cents
4)
Which equation calculates total water cost for water use between 100 and 150 gal?
y = 0.2x
y = 0.5x
y = 20 + 0.5x
y = 20 + 0.5(x – 100)

What does the equation y = 3 look like when graphed?For y = 3x + 1, plotting a point when x is 1 yields what y?

For y = 3x + 1, where does the line cross the y axis?
0
1
3
2)
For y = 3x + 1, plotting a point when x is 1 yields what y?
3
4
3)
For y = 3x + 1, the y-intercept is 1, so a point can be drawn at x = 0 and y = 1, namely up 1 on the y axis. A second point can be drawn by going to the right 1 and going up ___.
1
3
4)
For y = -2x + 3, what is the slope?
-2
2
3
5)
What does the equation y = 3 look like when graphed?
A vertical line
A horizontal line
Cannot be graphed
Feedback?

A common use of linear equations is to graph various options to help choose one. An example is gym membership options. A graph helps a person visualize the options and choose the best given the expected membership duration.

Month
Total cost ($)
50
100
150
200
250
1
2
3
4
5
6
7
8
9
10
11
12
25
1
y = 25x
1
15
y = 15x + 50
1
10
y = 10x + 100

Consider the example above.

Which line has the most steep slope?
Monthly
Standard
Platinum

Which option is cheapest for 4 months?
Monthly
Standard
Platinum

Which option is cheapest for 6 months?
Monthly
Standard
Platinum

Which option is cheapest for 12 months?
Monthly
Standard
Platinum

What was the equation for the total cost of the with-phone contract?At 6 months, the two options cost the same (the breakeven point). What is that cost?

What was the equation for the total cost of the with-phone contract?

CheckShow answer
2)
Shay chose to plot points for x = 10 and x = 20 because those values were round numbers and so a bit easier to calculate and draw. For the no-phone contract, what did Shay determine as the y value when x was 10?
$
CheckShow answer
3)
The graph of both contracts shows the with-phone contract to be how much cheaper at the start, when months is 0?
$
CheckShow answer
4)
The breakeven point, where the no-phone contract’s total cost becomes cheaper, is at how many months?

5)
The graphs not only show the breakeven point (15 months), but give a person a quick way to see the cost difference for any number of months. At 24 months, a person might visually see the blue line at about 1200 and the purple line at about 1000. What is the overall savings for the no-phone contract after 24 months?
A home loan monthly payment is often a person’s largest expense, perhaps consuming 30% or more of monthly earnings. A home loan has an interest rate, like 6%, that lasts for the loan’s duration, which might be 30 years. If rates go down, a newer loan might have a lower interest rate, like 4.5%. Replacing a home loan with a new lower-interest loan is called refinancing. Refinancing may cost thousands of dollars, so a task is determining whether refinancing is worth that cost.Initially (months = 0), what is the cost difference between the current loan versus refinancing? Answer by looking where each line touches the y axis.
0
3000
2)
At 2 months, what is the difference in cost of the current loan versus refinancing? Answer by looking at each line’s height when x is 2.
2000
4000
3)
At 6 months, the two options cost the same (the breakeven point). What is that cost?
0
12000
4)
Which option will be cheaper after 1 year?
Current loan
Refinanced loan
5)
The graphs below show the difference between the current loan and the refinanced loan over 10 years (120 months). What is the savings from the refinance after 10 years?

What does your result in cell A23 mean? Describe it in cell A24. Is the correlation evaluated in cell A15 positive or negative. Put your answer in cell A23.

Exercises

Is the correlation evaluated in cell A15 weak, moderate or strong. Put your answer in cell A22.

Is the correlation evaluated in cell A15 positive or negative. Put your answer in cell A23.

What does your result in cell A23 mean? Describe it in cell A24.

The p value in cell A16 tells you something about a hypothesis. State the null hypothesis in cell A25.

Look at the p value in cell A16. Does it mean that we reject the null hypothesis or that we do not reject it, and why. Put your answer in cell A26.

Compare the t value in cell A17 to the table t value in cell A18.

How does this comparison tell you if you should reject the null hypothesis or not. Put your explanation in cell A27.