Math Problem Statement

(THIS IS A SIMPLE LINEAR REGRESSION PROBLEM. YOU WILL NEED TO USE DIFFERENT PARTS OF THIS SCENARIO FOR DIFFERENT PROBLEMS. BE SURE TO KEEP ALL WORK HANDY SO THAT YOU WILL BE READY FOR ANOTHER QUESTION RELATED TO IT.)

In a study on marriage satisfaction, researchers sought to explore whether a woman should spend time "shopping around" before settling down with the right partner. They investigated length of time in years spend prior to dating and level of marriage satisfaction ten years into the marriage.

x y

(Length of time in (Level of marriage years spent dating satisfaction ten years prior to marriage) into marriage: 0 to 100%) 10 47 4 43 6 58 1 72 5 87 Use the regression equation to find the level of marriage satisfaction if the length of dating time is 6 years. Question 17 options:

65.3%

51.9%

44.9%

59.5%

38.7%

Solution

To solve this problem using simple linear regression, we'll follow these steps:

1. Identify the general form of the linear regression equation:

y=mx+by = mx + b Where:

  • yy is the predicted value (level of marriage satisfaction).
  • xx is the independent variable (length of time spent dating prior to marriage).
  • mm is the slope of the line.
  • bb is the y-intercept.

2. Gather the data points:

xyx \quad \quad y (10,47),(4,43),(6,58),(1,72),(5,87)(10, 47), (4, 43), (6, 58), (1, 72), (5, 87)

3. Calculate the slope mm and y-intercept bb.

Step 1: Compute the necessary sums:

x=10+4+6+1+5=26\sum x = 10 + 4 + 6 + 1 + 5 = 26 y=47+43+58+72+87=307\sum y = 47 + 43 + 58 + 72 + 87 = 307 x2=102+42+62+12+52=100+16+36+1+25=178\sum x^2 = 10^2 + 4^2 + 6^2 + 1^2 + 5^2 = 100 + 16 + 36 + 1 + 25 = 178 xy=(10×47)+(4×43)+(6×58)+(1×72)+(5×87)=470+172+348+72+435=1497\sum xy = (10 \times 47) + (4 \times 43) + (6 \times 58) + (1 \times 72) + (5 \times 87) = 470 + 172 + 348 + 72 + 435 = 1497

Step 2: Compute the slope mm using the formula:

m=n(xy)(x)(y)n(x2)(x)2m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} Where nn is the number of data points (5 in this case).

Substitute the values: m=5(1497)(26)(307)5(178)(26)2m = \frac{5(1497) - (26)(307)}{5(178) - (26)^2} m=74857982890676m = \frac{7485 - 7982}{890 - 676} m=4972142.32m = \frac{-497}{214} \approx -2.32

Step 3: Compute the y-intercept bb using the formula:

b=ym(x)nb = \frac{\sum y - m(\sum x)}{n} Substitute the values: b=307(2.32×26)5b = \frac{307 - (-2.32 \times 26)}{5} b=307+60.325b = \frac{307 + 60.32}{5} b367.32573.46b \approx \frac{367.32}{5} \approx 73.46

4. Use the regression equation to predict yy when x=6x = 6:

y=2.32(6)+73.46y = -2.32(6) + 73.46 y=13.92+73.46y = -13.92 + 73.46 y59.54%y \approx 59.54\%

So, the predicted level of marriage satisfaction after 6 years of dating is approximately 59.5%.

Final Answer:

59.5%\boxed{59.5\%}

Would you like to go into more details or have any other questions?

Here are five related questions:

  1. How does the slope influence the relationship between years of dating and satisfaction?
  2. What would be the marriage satisfaction prediction for someone dating for 3 years?
  3. Can you interpret what the y-intercept represents in this context?
  4. How would the prediction change if the slope were positive?
  5. How does simple linear regression differ from multiple regression?

Tip: Always check if the relationship between variables appears linear before applying linear regression.

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Math Problem Analysis

Mathematical Concepts

Statistics
Linear Regression
Algebra

Formulas

y = mx + b
m = (n(Σxy) - (Σx)(Σy)) / (n(Σx^2) - (Σx)^2)
b = (Σy - m(Σx)) / n

Theorems

Linear regression theorem

Suitable Grade Level

Grades 10-12