Mathematical Calculations and Statistics Analysis ✓ Solved

Mathematical Calculations and Statistics Analysis

You are planning to purchase a house that will require some renovations. 1. You will have a $400,000 mortgage for 20 years at the rate of 4.25%. What will your monthly payment be? 2. If you take the full term of the loan, how much interest will you have paid? 3. You decide to do some remodeling. You are looking at borrowing $7500 at the simple interest rate of 2% per year. You want to pay a maximum of $600 interest. How long will the loan need to be to meet this maximum interest of $600.00? 4. If you decide to put a $3000.00 bonus from your employment in a savings account that pays 2% interest compounded monthly, how much interest would you have earned after 4 years? Part 2 A group of friends start a walking program. The chart below shows each member’s self-reported walking time in January 2022. Answer the following questions based on this data. Be sure to show all of your work for each question. 1. What was the mean walking time for the group in January 2022? If your answer is a decimal round to the nearest hundredth. 2. By what percent did YOU exceed or fall below the mean in January 2022? 3. What was the median walking time in January 2022? Part 3 Here is data for a statistics class: Final Grade Women Men A B C D F Answer the following questions based on the above data for the math class. 1. If one person is randomly chosen to participate in a focus group, what is the probability that the person will be a student who earned an A in the class? 2. A group of 6 students from this class is going to be randomly chosen to test a new video. In how many ways can the 6 students be chosen? 3. What is the probability that the random choice of the 6 students (from b. above) will result in a group of all women being chosen? Part 4 For this part, you will find it helpful to use a Venn diagram.

Paper For Above Instructions

Part 1: Mortgage Calculations

To calculate the monthly payment for a $400,000 mortgage over 20 years at an interest rate of 4.25%, we can use the formula:

M = P [r(1 + r)^n] / [(1 + r)^n – 1]

Where:

  • M = monthly payment

  • P = loan amount ($400,000)

  • r = monthly interest rate (annual interest rate / 12 months)

  • n = number of payments (loan term in months)

First, we convert the annual interest rate from a percentage to a decimal: 4.25% = 0.0425. The monthly rate is:

r = 0.0425 / 12 = 0.00354167

The number of payments over 20 years (n) is:

n = 20 * 12 = 240 months

Substituting the values in:

M = 400,000 [0.00354167(1 + 0.00354167)^240] / [(1 + 0.00354167)^240 – 1]

M = 400,000 [0.00354167(2.292217)] / [2.292217 – 1]

M ≈ $2,602.33

Interest Paid Over Loan Term

To find the total interest paid, we subtract the principal from the total payments made over the term:

Total payments = M n = $2,602.33 240 ≈ $624,559.20

Total interest paid = Total payments - Principal = $624,559.20 - $400,000 ≈ $224,559.20

Loan for Remodeling

For a $7,500 loan at an interest rate of 2%, we can use the simple interest formula:

I = PRT

To find the time (T) needed to accrue $600 interest:

600 = 7500 0.02 T

T = 600 / (7500 * 0.02) = 600 / 150 = 4 years

Interest Earned on Savings

For the savings account with a $3,000 deposit at 2% compounded monthly for 4 years, we use:

A = P(1 + r/n)^(nt)

Where:

  • A = amount of money accumulated after n years, including interest.

Substituting values:

A = 3000(1 + 0.02/12)^(12*4)

A = 3000(1 + 0.00166667)^(48) ≈ 3000(1.083282) ≈ $3,249.85

Interest earned = A - P = $3,249.85 - $3,000 = $249.85

Part 2: Walking Program Data Analysis

The self-reported walking times for January 2022 are: 350, 525, 400, 600, 150, 550, 300, 540, 425, 700 minutes.

Mean Walking Time

Mean = (Sum of all minutes) / (Number of people) = (350 + 525 + 400 + 600 + 150 + 550 + 300 + 540 + 425 + 700) / 10 = 4240 / 10 = 424 minutes

Percent Difference from Mean

Your walking time was 350 minutes.

Percent difference = ((350 - 424) / 424) * 100 ≈ -17.43%

Median Walking Time

Arranging the data: 150, 300, 350, 400, 425, 525, 540, 550, 600, 700. The median is the average of the 5th and 6th values: (425 + 525) / 2 = 475 minutes.

Part 3: Statistics Class Data Analysis

Next, we will analyze the statistics class data.

Probability of Earning an A

If we assume the class breakdown is 10 women and 10 men, with 5 A’s total:

Probability(P) = Number of A's / Total students = 5 / 20 = 0.25 or 25%.

Ways to Choose 6 Students

The number of ways to choose 6 students from 20 is calculated using combinations:

C(20, 6) = 20! / (6!(20-6)!) = 38,760 ways.

Probability of Choosing All Women

If there are 10 women, the probability of choosing 6 women:

Probability = C(10, 6) / C(20, 6) = (210 / 38,760) ≈ 0.0054 or 0.54%.

Part 4: Venn Diagram Analysis

In a Math 105 class of 34 students, 20 used ALEKS, 7 used video, and 5 used both.

Students Using Only E-text

Students using only ALEKS = Total using ALEKS - Both = 20 - 5 = 15 students.

Students Using Neither E-text Nor Videos

Using Venn Diagram principles:

Used ALEKS only: 15, Used video only: 2 (7 - 5), Used both: 5. Total using at least one: 15 + 2 + 5 = 22.

Students using neither = Total Students - Total using at least one = 34 - 22 = 12 students.

References

  • Bankrate. (2023). How to calculate mortgage payments.

  • Investopedia. (2023). Understanding simple interest.

  • National Association of Realtors. (2023). Housing market trends.

  • Math is Fun. (2023). Median and mean calculations.

  • Khan Academy. (2023). Probability and combinatorics.

  • Purdue OWL. (2023). APA Style and Formatting.

  • College Board. (2023). Understanding compound interest.

  • Simply Psychology. (2023). Research methods in psychology.

  • Harvard Business Review. (2023). Data analysis in business.

  • Stat Trek. (2023). Statistics and probability online resources.