Math Problem Statement

What is the​ least-squares regression line with the point left parenthesis 9 comma 13 right parenthesis included in the data​ set? Data Set
x y 3 6 4 5 5 7 6 4 7 6 8 9 8 8 10 8 11 9 11 7 12 10 13 12 13 10 14 11 ​(This is a reading assessment question. Be certain of your answer because you only get one attempt on this​ question.) . . . Question content area right Part 1 ModifyingAbove y with caretequals    enter your response herexplus    enter your response here ​(Type integers or decimals rounded to four decimal places as​ needed.)

Solution

To find the least-squares regression line, we first need to calculate the slope mm and the y-intercept bb of the line y^=mx+b\hat{y} = mx + b. The formulas for the slope and intercept are:

m=n(xy)(x)(y)n(x2)(x)2m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}

b=ym(x)nb = \frac{\sum y - m(\sum x)}{n}

where:

  • nn is the number of data points
  • x\sum x is the sum of all xx-values
  • y\sum y is the sum of all yy-values
  • xy\sum xy is the sum of the product of each xx and yy
  • x2\sum x^2 is the sum of the squares of each xx

Step 1: Calculate the necessary sums.

Given the data set (including the point (9,13)(9, 13)): (3,6),(4,5),(5,7),(6,4),(7,6),(8,9),(8,8),(9,13),(10,8),(11,9),(11,7),(12,10),(13,12),(13,10),(14,11)(3, 6), (4, 5), (5, 7), (6, 4), (7, 6), (8, 9), (8, 8), (9, 13), (10, 8), (11, 9), (11, 7), (12, 10), (13, 12), (13, 10), (14, 11)

  1. Calculate nn, x\sum x, y\sum y, xy\sum xy, and x2\sum x^2:
    • n=15n = 15
    • x=3+4+5+6+7+8+8+9+10+11+11+12+13+13+14=134\sum x = 3 + 4 + 5 + 6 + 7 + 8 + 8 + 9 + 10 + 11 + 11 + 12 + 13 + 13 + 14 = 134
    • y=6+5+7+4+6+9+8+13+8+9+7+10+12+10+11=135\sum y = 6 + 5 + 7 + 4 + 6 + 9 + 8 + 13 + 8 + 9 + 7 + 10 + 12 + 10 + 11 = 135
    • xy=(3)(6)+(4)(5)+(5)(7)+(6)(4)+(7)(6)+(8)(9)+(8)(8)+(9)(13)+(10)(8)+(11)(9)+(11)(7)+(12)(10)+(13)(12)+(13)(10)+(14)(11)=1304\sum xy = (3)(6) + (4)(5) + (5)(7) + (6)(4) + (7)(6) + (8)(9) + (8)(8) + (9)(13) + (10)(8) + (11)(9) + (11)(7) + (12)(10) + (13)(12) + (13)(10) + (14)(11) = 1304
    • x2=32+42+52+62+72+82+82+92+102+112+112+122+132+132+142=1270\sum x^2 = 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 8^2 + 9^2 + 10^2 + 11^2 + 11^2 + 12^2 + 13^2 + 13^2 + 14^2 = 1270

Step 2: Calculate the slope mm and intercept bb.

Using the formulas:

m=15(1304)134(135)15(1270)1342=19560180901905017956=147010941.3449m = \frac{15(1304) - 134(135)}{15(1270) - 134^2} = \frac{19560 - 18090}{19050 - 17956} = \frac{1470}{1094} \approx 1.3449

b=1351.3449(134)15=135180.216615=45.2166153.0144b = \frac{135 - 1.3449(134)}{15} = \frac{135 - 180.2166}{15} = \frac{-45.2166}{15} \approx -3.0144

Step 3: Write the equation of the least-squares regression line.

The equation is:

y^=1.3449x3.0144\hat{y} = 1.3449x - 3.0144

Rounding the slope and intercept to four decimal places, the least-squares regression line is approximately:

y^=1.3449x3.0144\hat{y} = 1.3449x - 3.0144

Would you like further details or have any questions about this calculation?

Related Questions:

  1. How does including an outlier affect the least-squares regression line?
  2. What is the meaning of the slope in a least-squares regression line?
  3. How can you determine if a least-squares regression line is a good fit for the data?
  4. What is the interpretation of the y-intercept in a regression line?
  5. How do you calculate the coefficient of determination, R2R^2, for this data set?

Tip:

When rounding intermediate calculations, keep more decimal places than the final answer to avoid rounding errors in the final result.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Linear Regression
Algebra

Formulas

Slope formula: m = [n(∑xy) - (∑x)(∑y)] / [n(∑x^2) - (∑x)^2]
Y-intercept formula: b = [∑y - m(∑x)] / n
Least-squares regression line formula: ŷ = mx + b

Theorems

Least-Squares Regression Theorem

Suitable Grade Level

Grades 10-12