In this worksheet, we will practice finding and using the least squares regression line equation.

**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.

0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | |

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 on , rounding the regression coefficients to the nearest thousandth.

- A
- B
- C
- D
- E

**Q2: **

Using the information in the table, find the regression line . Round and 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
- B
- C
- D

**Q3: **

The table shows the price of a barrel of oil and the economic growth. Using the information in the table, find the regression line . Round and 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 | 0.5 | 0.5 | 1 | 2.8 | 3.9 | 4.9 | 5 |

- A
- B
- C
- D

**Q4: **

The table shows the relation between the variables and . Find the equation of the regression line in the form . Approximate and to 3 decimal places.

10 | 22 | 22 | 13 | 16 | 21 | |

25 | 18 | 24 | 25 | 12 | 17 |

- A
- B
- C
- D

**Q5: **

Using the information in the table, estimate the value of when . Give your answer to the nearest integer.

23 | 9 | 24 | 15 | 7 | 12 | |

22 | 24 | 25 | 13 | 21 | 9 |

**Q6: **

Using the information in the table, find the error in if . Give your answer to the nearest integer.

26 | 22 | 28 | 15 | 30 | 10 | 25 | 29 | |

5 | 4 | 12 | 7 | 14 | 10 | 13 | 15 |

**Q7: **

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 |

- A2.5
- B2.4
- C0.2
- D1.5

**Q9: **

Given the quadratic regression model , calculate the value of when is equal to 2.7.

**Q10: **

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 , writing and to 3 decimal places.

Car’s Lifespan () | 5 | 2 | 2 | 3 | 5 | 5 | 1 | 2 |
---|---|---|---|---|---|---|---|---|

Selling Price () | 71 | 83 | 60 | 90 | 93 | 70 | 41 | 45 |

- A
- B
- C
- D

**Q11: **

For a given data set, , , , , , and . Calculate the value of the regression coefficient in the least squares regression model . Give your answer correct to three decimal places.

- A
- B
- C
- D
- E

**Q12: **

The latitude () and the average temperatures in February (, measured in ) of 10 world cities were measured. The calculated least squares linear regression model for this data was .

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

- AFor every additional 0.713 degrees of latitude, the average temperature decreased by .
- BFor every additional degree of latitude, the average temperature increased by .
- CIt is the -intercept of the regression line.
- DFor every additional degree of latitude, the average temperature decreased by .
- EIt is the average temperature in February for a city of latitude 0 (on the equator).

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).
- BIt is the gradient of the regression line.
- CFor every additional degree of latitude, the average temperature increased by .
- DFor every additional degree of latitude, the average temperature decreased by .
- EFor every additional 0.713 degrees of latitude, the average temperature decreased by .

**Q13: **

Given the quadratic regression model , calculate the value of when is equal to 3.

**Q14: **

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 (, measured in 100s of dollars) and the percent of bus services that run on time (, measured in %). They find that the data can be described by the linear regression model .

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

- AIt is the -intercept of the regression line.
- 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.
- DFor every additional $100 of investment, an additional 2.7% of bus services run on time.

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

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

**Q15: **

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?

- AIt is the -intercept of the regression line.
- 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.
- DFor every extra meter jumped in the long jump, the competitors jumped, on average, an extra 0.218 metres in the high jump.

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.
- BIt is the slope of the regression line.
- CIt is the -intercept of the regression line.
- DFor every extra meter jumped in the long jump, the competitors jumped, on average, an extra 0.483 metres in the high jump.
- EThis is the predicted long jump result, in meters, for a competitor who jumped 0 metres in the high jump competition.

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.

**Q16: **

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
- C

**Q17: **

Given the regression line , find the expected value of when .

**Q18: **

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 to the data. Would you expect the regression coefficient to be positive or negative in this context?

- Apositive
- Bnegative

**Q19: **

Two variables and have a correlation coefficient of and their mean and standard deviations are denoted by , and , respectively. Which of the following is the formula for calculating the slope, , of the least squares regression line ?

- A
- B
- C
- D
- E

**Q20: **

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?

- Ayes
- Bno

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

- Apositive
- Bnegative

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 and high jump by , find the values of , and to the nearest thousandth.

- A
- B
- C
- D
- E

Hence, calculate the equation of the regression line of on .

- A
- B
- C
- D
- E

**Q21: **

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?

- AFor every goal, 2.28 games were played.
- BThe unit of the slope is 2.28 games per goal.
- CThe unit of the slope is 2.28 goals per game.

**Q22: **

A variable has a mean of 67.9 with a standard deviation of 3.1.

A variable has a mean of 29.3 with a standard deviation of 1.2.

Given that the correlation coefficient between and is 0.37, calculate the least squares regression line of on . Round the final values for and to 3 decimal places.

- A
- B
- C
- D
- E

**Q23: **

Use the information in the table to calculate the least squares regression line of on . Write the final values of the correlation coefficient and constant accurate to three decimal places.

1 | 22 | 18 | 396 | 484 | 324 |

2 | 22 | 19 | 418 | 484 | 361 |

3 | 23 | 20 | 460 | 529 | 400 |

4 | 26 | 18 | 468 | 676 | 324 |

5 | 31 | 23 | 713 | 961 | 529 |

6 | 32 | 24 | 768 | 1,024 | 576 |

7 | 34 | 22 | 748 | 1,156 | 484 |

8 | 37 | 25 | 925 | 1,369 | 625 |

9 | 41 | 29 | 1,189 | 1,681 | 841 |

10 | 42 | 27 | 1,134 | 1,764 | 729 |

Sum | 310 | 225 | 7,219 | 10,128 | 5,193 |

- A
- B
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- D
- E

**Q24: **

What happens if some of the points are not on the line of best fit?

- A The line must be altered, moved, or curved to fit through all the points.
- B The experiment or data collection process must be redone because the data is incorrect.
- C These points have been graphed incorrectly and need to be plotted again.
- D The line is not meant to go through all points so long as it finds the pattern within them.

**Q25: **

If and are the standard deviations of and , and is the sample correlation coefficient between and , which of the following is the slope of a simple linear regression ?

- A
- B
- C
- D