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Lesson: Least Squares Regression Line

In this lesson, we will learn how to find and use the least squares regression line equation.

Video

17:15

Sample Question Videos

03:14

Worksheet • 25 Questions • 1 Video

Q1:

The scatterplot shows a set of data for which a linear regression model appears appropriate.

The data used to produce this scatterplot is given in the table shown.

π‘₯ 0.5 1 1.5 2 2.5 3 3.5 4
𝑦 9.25 7.6 8.25 6.5 5.45 4.5 1.75 1.8

Calculate the equation of the least squares regression line of 𝑦 on π‘₯ , rounding the regression coefficients to the nearest thousandth.

  • A 𝑦 = 1 0 . 6 5 7 βˆ’ 2 . 2 3 1 π‘₯
  • B 𝑦 = 1 0 . 2 3 5 βˆ’ 1 . 0 7 8 π‘₯
  • C 𝑦 = 6 . 8 1 9 βˆ’ 0 . 5 2 5 π‘₯
  • D 𝑦 = 9 . 9 7 3 βˆ’ 2 . 1 5 0 π‘₯
  • E 𝑦 = 4 . 0 9 4 + 0 . 6 8 6 π‘₯

Q2:

The table shows the price of a barrel of oil and the economic growth. Using the information in the table, estimate the economic growth if the price of a barrel of oil is 35.40 dollars.

Price of a Barrel of Oil in Dollars 26 13.30 22.90 12.40 26.70 17.90 23.60 37.40
Economic Growth Rate 1.8 0.4 3.7 2.3 3.2 2.7 0.5 0.3
  • A1.5
  • B2.4
  • C0.2
  • D2.5

Q3:

Given that points ( 3 , βˆ’ 9 ) and ( 2 , βˆ’ 4 ) lie on a regression line 𝑦 on π‘₯ , which of the following points does not lie on the same line?

  • A ( 1 6 , βˆ’ 6 9 )
  • B ( βˆ’ 1 0 , 5 6 )
  • C ( 2 0 , βˆ’ 9 4 )
  • D ( 1 2 , βˆ’ 5 4 )

Q4:

Two variables 𝑋 and π‘Œ have a correlation coefficient of π‘Ÿ and their mean and standard deviations are denoted by 𝑋 , π‘Œ , 𝑠 𝑋 , and 𝑠 π‘Œ , respectively. Which of the following is the formula for calculating the slope, 𝑏 , of the least squares regression line π‘Œ = π‘Ž + 𝑏 𝑋 ?

  • A π‘Ÿ 𝑠 𝑠 π‘Œ 𝑋
  • B 𝑠 π‘Ÿ β‹… 𝑠 π‘Œ 𝑋
  • C π‘Ÿ 𝑠 𝑠 𝑋 π‘Œ
  • D 𝑠 𝑠 𝑋 π‘Œ
  • E 𝑠 𝑠 π‘Œ 𝑋

Q5:

Using the information in the table, find the regression line Μ‚ 𝑦 = π‘Ž + 𝑏 π‘₯ . Round π‘Ž and 𝑏 to 3 decimal places.

Cultivated Land in Feddan 126 13 104 180 38 161 14 99 55 177
Production of a Summer Crop in Kilograms 160 40 80 340 260 200 280 280 140 100
  • A Μ‚ 𝑦 = 0 . 2 0 1 π‘₯ + 1 6 8 . 5 6 3
  • B Μ‚ 𝑦 = 1 6 8 . 5 6 3 π‘₯ + 0 . 2 0 1
  • C Μ‚ 𝑦 = 0 . 2 0 1 π‘₯ + 2 0 7 . 4 3 7
  • D Μ‚ 𝑦 = 0 . 0 3 4 π‘₯ + 1 6 8 . 5 6 3

Q6:

Using the information in the table, find the regression line Μ‚ 𝑦 = π‘Ž + 𝑏 π‘₯ . Round π‘Ž and 𝑏 to 3 decimal places.

Cultivated Land in Feddan 146 14 113 23 112 55 36 73 92 6
Production of a Summer Crop in Kilograms 260 300 280 200 280 380 300 20 240 80
  • A Μ‚ 𝑦 = 0 . 4 2 2 π‘₯ + 2 0 5 . 7 2 6
  • B Μ‚ 𝑦 = 2 0 5 . 7 2 6 π‘₯ + 0 . 4 2 2
  • C Μ‚ 𝑦 = 0 . 4 2 2 π‘₯ + 2 6 2 . 2 7 4
  • D Μ‚ 𝑦 = 0 . 0 7 9 π‘₯ + 2 0 5 . 7 2 6

Q7:

Given the quadratic regression model 𝑦 = 3 π‘₯ βˆ’ 5 π‘₯ + 2 2 , calculate the value of 𝑦 when π‘₯ is equal to 2.7.

Q8:

The following table shows the relation between the lifespan of cars in years and their selling price in thousands of pounds. Find the equation of the line of regression in the form Μ‚ 𝑦 = π‘Ž + 𝑏 π‘₯ , writing π‘Ž and 𝑏 to 3 decimal places.

Car’s Lifespan ( π‘₯ ) 5 2 2 3 5 5 1 2
Selling Price ( 𝑦 ) 71 83 60 90 93 70 41 45
  • A Μ‚ 𝑦 = 6 . 8 2 8 π‘₯ + 4 7 . 7 8 8
  • B Μ‚ 𝑦 = 4 7 . 7 8 8 π‘₯ + 6 . 8 2 8
  • C Μ‚ 𝑦 = 6 . 8 2 8 π‘₯ + 9 0 . 4 6 3
  • D Μ‚ 𝑦 = 0 . 7 3 6 π‘₯ + 4 7 . 7 8 8

Q9:

For a given data set, ο„š π‘₯ = 4 7 , ο„š 𝑦 = 4 5 . 7 5 , ο„š π‘₯ = 3 2 9  , ο„š 𝑦 = 3 8 9 . 3 1 2 5  , ο„š π‘₯ 𝑦 = 3 1 0 . 2 5 , and 𝑛 = 8 . Calculate the value of the regression coefficient 𝑏 in the least squares regression model 𝑦 = π‘Ž + 𝑏 π‘₯ . Give your answer correct to three decimal places.

  • A 𝑏 = 0 . 7 8 4
  • B 𝑏 = 0 . 9 8 9
  • C 𝑏 = 0 . 6 1 6
  • D 𝑏 = βˆ’ 0 . 1 7 6
  • E 𝑏 = βˆ’ 0 . 1 8 8

Q10:

The latitude ( π‘₯ ) and the average temperatures in February ( 𝑦 , measured in ∘ C ) of 10 world cities were measured. The calculated least squares linear regression model for this data was 𝑦 = 3 5 . 7 βˆ’ 0 . 7 1 3 π‘₯ .

What is the interpretation of the value of βˆ’ 0 . 7 1 3 in the model?

  • AFor every additional degree of latitude, the average temperature decreased by 0 . 7 1 3 ∘ C .
  • BIt is the average temperature in February for a city of latitude 0 (on the equator).
  • CFor every additional degree of latitude, the average temperature increased by 0 . 7 1 3 ∘ C .
  • DIt is the 𝑦 -intercept of the regression line.
  • EFor every additional 0.713 degrees of latitude, the average temperature decreased by 1 ∘ C .

What is the interpretation of the value of 35.7 in the model?

  • AIt is the average temperature in February for a city of latitude 0 (on the equator).
  • BFor every additional 0.713 degrees of latitude, the average temperature decreased by 1 ∘ C .
  • CFor every additional degree of latitude, the average temperature decreased by 0 . 7 1 3 ∘ C .
  • DIt is the gradient of the regression line.
  • EFor every additional degree of latitude, the average temperature increased by 0 . 7 1 3 ∘ C .

Q11:

Using the information in the table, estimate the value of 𝑦 when π‘₯ = 1 3 . Give your answer to the nearest integer.

π‘₯ 23 9 24 15 7 12
𝑦 22 24 25 13 21 9

Q12:

Using the information in the table, find the error in 𝑦 if π‘₯ = 2 2 . Give your answer to the nearest integer.

π‘₯ 26 22 28 15 30 10 25 29
𝑦 5 4 12 7 14 10 13 15

Q13:

An ice cream salesman records data on the number of ice creams sold each day and the temperature at midday during the April-November period. He fits a linear regression model of the form 𝑦 = π‘Ž + 𝑏 π‘₯ to the data. Would you expect the regression coefficient 𝑏 to be positive or negative in this context?

  • Anegative
  • Bpositive

Q14:

The table shows the price of a barrel of oil and the economic growth. Using the information in the table, find the regression line Μ‚ 𝑦 = π‘Ž + 𝑏 π‘₯ . Round π‘Ž and 𝑏 to 3 decimal places.

Price of One Barrel of Oil in Dollars 50.40 55.30 63 70.70 83.60 94.10 102.50 118
Economic Growth Rate βˆ’ 1 0.5 0.5 1 2.8 3.9 4.9 5
  • A Μ‚ 𝑦 = 0 . 0 9 2 π‘₯ βˆ’ 5 . 1 3 2
  • B Μ‚ 𝑦 = βˆ’ 5 . 1 3 2 π‘₯ + 0 . 0 9 2
  • C Μ‚ 𝑦 = 0 . 0 9 2 π‘₯ + 9 . 5 3 2
  • D Μ‚ 𝑦 = 0 . 0 0 4 π‘₯ βˆ’ 5 . 1 3 2

Q15:

The table shows the price of a barrel of oil and the economic growth. Using the information in the table, find the regression line Μ‚ 𝑦 = π‘Ž + 𝑏 π‘₯ . Round π‘Ž and 𝑏 to 3 decimal places.

Price of One Barrel of Oil in Dollars 54.20 58.90 69.20 78.80 82.90 96.80 109.90 111.90
Economic Growth Rate βˆ’ 0 . 7 0.1 0.8 1.3 2.2 3.8 4.1 5.2
  • A Μ‚ 𝑦 = 0 . 0 9 4 π‘₯ βˆ’ 5 . 6 8 6
  • B Μ‚ 𝑦 = βˆ’ 5 . 6 8 6 π‘₯ + 0 . 0 9 4
  • C Μ‚ 𝑦 = 0 . 0 9 4 π‘₯ + 9 . 8 8 6
  • D Μ‚ 𝑦 = 0 . 0 0 3 π‘₯ βˆ’ 5 . 6 8 6

Q16:

The table shows the price of a barrel of oil and the economic growth. Using the information in the table, find the regression line Μ‚ 𝑦 = π‘Ž + 𝑏 π‘₯ . Round π‘Ž and 𝑏 to 3 decimal places.

Price of One Barrel of Oil in Dollars 51 56.10 60.90 71.70 83.70 94.30 107.20 119.30
Economic Growth Rate βˆ’ 0 . 6 0.4 0.6 1.3 2.3 3.7 4.4 5.3
  • A Μ‚ 𝑦 = 0 . 0 8 4 π‘₯ βˆ’ 4 . 5 8 9
  • B Μ‚ 𝑦 = βˆ’ 4 . 5 8 9 π‘₯ + 0 . 0 8 4
  • C Μ‚ 𝑦 = 0 . 0 8 4 π‘₯ + 8 . 9 3 9
  • D Μ‚ 𝑦 = 0 . 0 0 3 π‘₯ βˆ’ 4 . 5 8 9

Q17:

The table shows the price of a barrel of oil and the economic growth. Using the information in the table, find the regression line Μ‚ 𝑦 = π‘Ž + 𝑏 π‘₯ . Round π‘Ž and 𝑏 to 3 decimal places.

Price of One Barrel of Oil in Dollars 50.60 59.60 68.80 70.40 85.10 92.80 103.70 112.80
Economic Growth Rate 0 0.2 1 2 3 3.3 4.8 5.1
  • A Μ‚ 𝑦 = 0 . 0 8 9 π‘₯ βˆ’ 4 . 7 3 7
  • B Μ‚ 𝑦 = βˆ’ 4 . 7 3 7 π‘₯ + 0 . 0 8 9
  • C Μ‚ 𝑦 = 0 . 0 8 9 π‘₯ + 9 . 5 8 7
  • D Μ‚ 𝑦 = 0 . 0 0 3 π‘₯ βˆ’ 4 . 7 3 7

Q18:

Given the quadratic regression model 𝑦 = βˆ’ π‘₯ + 5 . 2 π‘₯ βˆ’ 2 . 1 2 , calculate the value of 𝑦 when π‘₯ is equal to 3.

Q19:

A city council is investing in improving their bus services. Over a five-year period, they collect data on the amount of money invested in each bus route ( π‘₯ , measured in 100s of dollars) and the percent of bus services that run on time ( 𝑦 , measured in %). They find that the data can be described by the linear regression model 𝑦 = 5 2 . 3 + 2 . 7 π‘₯ .

What is the interpretation of the value of 2.7 in the regression model?

  • AFor every additional $100 of investment, an additional 2.7% of bus services run on time.
  • BFor every additional $52.3 of investment, an additional 2.7% of bus services run on time.
  • CIt represents the percent of bus services that would run on time with no investment.
  • DIt is the 𝑦 -intercept of the regression line.

What is the interpretation of the value of 52.3 in the regression model?

  • AIt represents the percent of bus services that would run on time with no investment.
  • BIt is the gradient of the regression line.
  • CFor every additional $100 of investment, an additional 2.7% of bus services run on time.
  • DIt represents the percent of bus services that would run on time with $100 of investment.

Q20:

The relationship between the distances jumped by competitors in the long jump and high jump during the women’s heptathlon at the 2016 Rio Olympics can be modeled by the regression line .

What is the interpretation of the value 0.218 in the regression model?

  • AFor every extra meter jumped in the long jump, the competitors jumped, on average, an extra 0.218 metres in the high jump.
  • BThis is the predicted high jump result for a competitor who jumped 0 metres in the long jump competition.
  • CFor every extra meter jumped in the high jump, the competitors jumped on average an extra 0.218 metres in the long jump.
  • DIt is the -intercept of the regression line.

What is the interpretation of the value 0.483 in the regression model?

  • AThis is the predicted high jump result, in meters, for a competitor who jumped 0 metres in the long jump competition.
  • BThis is the predicted long jump result, in meters, for a competitor who jumped 0 metres in the high jump competition.
  • CFor every extra meter jumped in the long jump, the competitors jumped, on average, an extra 0.483 metres in the high jump.
  • DIt is the slope of the regression line.
  • EIt is the -intercept of the regression line.

Does the interpretation of the value 0.483 seem reasonable in the context of the data?

  • Ayes
  • BNo, the model has been extrapolated a long way and is therefore unreliable.

Estimate, to the nearest hundredth of a meter, the expected high jump result for a competitor who jumped 6.03 m in the long jump competition.

Q21:

The scatterplot shows the high jump and long jump results achieved by 15 competitors in the women’s heptathlon competition in the 2016 Rio Olympics.

Does a linear model appear to be appropriate for modeling this data set?

  • Ano
  • Byes

Would you expect the regression coefficient of this model to be positive or negative?

  • Anegative
  • Bpositive

The data table shows the numerical data used to produce the scatter diagram.

Long Jump (m) 5.51 5.72 5.81 5.88 5.91 6.05 6.08 6.10 6.16 6.19 6.31 6.31 6.34 6.48 6.58
High Jump (m) 1.65 1.77 1.83 1.77 1.77 1.77 1.8 1.77 1.8 1.86 1.86 1.83 1.89 1.86 1.98

Representing long jump by π‘₯ and high jump by 𝑦 , find the values of 𝑆 , 𝑆 π‘₯ π‘₯ 𝑦 𝑦 , and 𝑆 π‘₯ 𝑦 to the nearest thousandth.

  • A 𝑆 = 1 . 1 9 6 , 𝑆 = 0 . 0 7 7 , 𝑆 = 0 . 2 6 1 π‘₯ π‘₯ 𝑦 𝑦 π‘₯ 𝑦
  • B 𝑆 = 1 . 3 9 2 , 𝑆 = 0 . 1 1 , 𝑆 = 0 . 0 5 7 π‘₯ π‘₯ 𝑦 𝑦 π‘₯ 𝑦
  • C 𝑆 = 0 . 4 0 7 , 𝑆 = 1 . 1 5 7 , 𝑆 = 0 . 4 7 1 π‘₯ π‘₯ 𝑦 𝑦 π‘₯ 𝑦
  • D 𝑆 = 1 . 0 0 2 , 𝑆 = 0 . 1 2 1 , 𝑆 = 0 . 6 3 7 π‘₯ π‘₯ 𝑦 𝑦 π‘₯ 𝑦
  • E 𝑆 = 0 . 1 6 4 , 𝑆 = 0 . 5 5 , 𝑆 = 1 . 2 3 7 π‘₯ π‘₯ 𝑦 𝑦 π‘₯ 𝑦

Hence, calculate the equation of the regression line of 𝑦 on π‘₯ .

  • A 𝑦 = 0 . 2 1 8 π‘₯ + 0 . 4 8 3
  • B 𝑦 = 1 . 2 4 5 π‘₯ + 0 . 4 8 3
  • C 𝑦 = 0 . 4 8 3 π‘₯ + 0 . 2 1 8
  • D 𝑦 = 0 . 3 4 9 π‘₯ + 1 . 1 5 7
  • E 𝑦 = 1 . 1 5 7 π‘₯ + 0 . 3 4 9

Q22:

Amir conducted a statistical experiment to measure the number of goals as a function of the number of soccer games. With the number of soccer games as his independent variable and the number of goals as his dependent variable, the line of best fit had a slope of 2.28. What does this mean?

  • AThe unit of the slope is 2.28 games per goal.
  • BThe unit of the slope is 2.28 goals per game.
  • CFor every goal, 2.28 games were played.

Q23:

A linear model was fitted to three data sets. The residual plot for each data set is shown. For which data set is a linear model appropriate?

  • A B
  • B A
  • C C

Q24:

The table shows the relation between the variables π‘₯ and 𝑦 . Find the equation of the regression line in the form Μ‚ 𝑦 = π‘Ž + 𝑏 π‘₯ . Approximate π‘Ž and 𝑏 to 3 decimal places.

π‘₯ 10 22 22 13 16 21
𝑦 25 18 24 25 12 17
  • A Μ‚ 𝑦 = βˆ’ 0 . 3 7 6 π‘₯ + 2 6 . 6 8 4
  • B Μ‚ 𝑦 = 2 6 . 6 8 4 π‘₯ βˆ’ 0 . 3 7 6
  • C Μ‚ 𝑦 = βˆ’ 0 . 3 7 6 π‘₯ + 1 3 . 6 4 9
  • D Μ‚ 𝑦 = βˆ’ 0 . 0 1 3 π‘₯ + 2 6 . 6 8 4

Q25:

Given the regression line Μ‚ 𝑦 = 7 . 3 π‘₯ βˆ’ 5 . 9 , find the expected value of 𝑦 when π‘₯ = 3 0 .
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