Linear Regression Worksheet

**Page 1**

1.

There are two things that should be done before doing regression analysis. They are:

I. Collect the data and then construct a scatter plot to determine the nature of the relationship.

II. Collect the data and construct a histogram.

III. Compute the value of the correlation coefficient to test the significance of the relationship.

IV. Test the significance of the relationship.

I. Collect the data and then construct a scatter plot to determine the nature of the relationship.

II. Collect the data and construct a histogram.

III. Compute the value of the correlation coefficient to test the significance of the relationship.

IV. Test the significance of the relationship.

a. | I and IV only | ||

b. | II and IV only | ||

c. | II and III only | ||

d. | I and III only |

Compute the value of the correlation coefficient to test the significance of the relationship.

Correct answer : (4)

2.

General form of the regression line used in statistics is ____ and the $y$′-intercept and the slope respectively are

a. | $y$′ = $\mathrm{a\; +\; bx}$, $a$, $b$ | ||

b. | $y$′ = $\mathrm{ax}$ ^{2} + $b$, $b$, $a$ | ||

c. | $y$′ = $\mathrm{a\; +\; bx}$ ^{2}, $a$, $b$ | ||

d. | $y$′ = $\mathrm{ax\; +\; b}$, $b$, $a$ |

The

The slope is

Correct answer : (1)

3.

The relation between the sign of the correlation coefficient($r$) and the sign of the slope($b$) of the regression line is

a. | when $r$ is positive, $b$ is also positive and when $r$ is negative, $b$ is also negative | ||

b. | when $r$ is positive, $b$ is negative and when $r$ is negative, $b$ is positive | ||

c. | when $r$ is positive or negative, $b$ is positive | ||

d. | when $r$ is positive or negative, $b$ is negative |

Therefore, if

Correct answer : (1)

4.

Regression should be done only when

a. | $r$ is significant | ||

b. | none of these | ||

c. | regression does not depend on $r$ so can be done disregarding $r$ | ||

d. | $r$ is not significant |

It is meaningless to determine the regression line when

Correct answer : (1)

5.

The formulas for $a$ and $b$ of the regression line $y$′ = $a$ + $\mathrm{bx}$ are

a. | II only | ||

b. | III only | ||

c. | IV only | ||

d. | I only |

where

Correct answer : (4)

6.

Determine the equation of the regression line for the data and predict the value for $y$′ when $x$ = 3.0.

$x$ | 2.1 | 1.7 | 1.1 | 1.5 | 2.7 |

$y$ | 40 | 37 | 35 | 36 | 42 |

a. | $y$′ = 29.438 - 4.704$x$, 15.326 | ||

b. | $y$′ = 4.704 + 29.438$x$, 93.018 | ||

c. | $y$′ = 4.704 + 29.438$x$, 43.55 | ||

d. | $y$′ = 29.438 + 4.704$x$, 43.55 |

Make a table with values for

To find the regression line, we first have to find out

If

[Substitute the values and simplify.]

Therefore,

Regression line equation is

[Substitute the values and simplify.]

Hence, the equation of the regression line

When

[Substitute the values and simplify.]

Correct answer : (4)

7.

A business man wants to know the likely cost of a new contract based on the data collected from the contracts of the previous years. Determine the equation of the regression line for the data and predict the value for $y$′ when $x$ = 30.

Years | 2000 | 2001 | 2002 | 2003 | 2004 |

No.of employees ($x$) (in 1000) | 10 | 20 | 16 | 14 | 24 |

Total cost of contract ($y$) (in lakh $) | 6 | 15 | 12 | 9 | 20 |

a. | $y$′ = - 4.343 + 0.997$x$, 25.567 | ||

b. | $y$′ = 0.997 - 4.343$x$, - 129.293 | ||

c. | $y$′ = 4.343 + 0.997$x$, 34.253 | ||

d. | $y$′ = 4.343 - 0.997$x$, - 25.567 |

Make a table with values for

[Substitute the values and simplify.]

Therefore,

Regression line equation is

[Substitute the values and simplify.]

Therefore, the equation of the regression line

When

[Substitute the values and simplify.]

Correct answer : (1)

8.

Determine the equation of the regression line and plot the line on the scatter plot for the data.

$x$ | 2 | 8 | 7 | 5 | 3 | 6 | 4 | 1 |

$y$ | 6 | 8 | 4 | 1 | 5 | 7 | 2 | 3 |

a. | $y$′ = 2.893 + 0.357$x$ ; Graph 1 | ||

b. | $y$′ = 0.357 + 2.893$x$ ; Graph 3 | ||

c. | $y$′ = 2.893 - 0.357$x$ ; Graph 2 | ||

d. | $y$′ = - 0.357 + 2.893$x$ ; Graph 4 |

Make a table with values for

Regression line equation is

[Substitute the values and simplify.]

Therefore, the equation of the regression line

Draw a scatter plot for the data and plot the line.

Correct answer : (1)

9.

An experiment was conducted to find out the number of times a tennis ball bounces after it is thrown from a fixed height. This was tried for different heights and the number of times the ball bounced was measured. Determine the equation of the regression line for the data and predict the value for $y$′ when $x$ = 100.

Height (in feet) ($x$) | 10 | 20 | 30 | 45 | 60 |

No.of bounces ($y$) | 6 | 10 | 13 | 19 | 25 |

a. | $y$′ = - 2.152 + 0.377$x$, 35.548 | ||

b. | $y$′ = - 0.377 + 2.152$x$, 215 | ||

c. | $y$′ = 0.377 - 2.152$x$, 215 | ||

d. | $y$′ = 0.377 + 2.152$x$, 216 |

Make a table with values for

[Substitute the values and simplify.]

Therefore,

Regression line equation is

Therefore, the equation of the regression line

When

Correct answer : (1)

10.

The university authorities wanted to predict a student's grade on a statistics midterm score based on his/her SAT scores. Determine the correlation coefficient and the equation of the regression line for the data and predict the value of $y$′ when $x$ = 900. Test the hypothesis that there is a significant relationship between the SAT scores and the Statistcs scores at α = 0.05.

SAT scores ($x$) | 1100 | 1300 | 1000 | 1200 | 1100 | 1200 | 1400 | 1000 |

Statistics Midterm scores($y$) | 89 | 92 | 86 | 92 | 90 | 93 | 98 | 88 |

a. | + 0.941, $y$′ = 62.514 + 0.025$x$, yes | ||

b. | + 0.941, $y$′ = 62.514 + 0.025$x$, no | ||

c. | + 0.491, $y$′ = 62.514 + 0.025$x$, yes | ||

d. | + 0.491, $y$′ = 62.514 + 0.025$x$, no |

Make a table with values for

[Substitute the values and simplify.]

Therefore,

Regression line equation is

[Substitute the values and simplify.]

Therefore, the equation of the regression line

When

State the hypothesis:

H

Since α = 0.05 and there are 8 - 2 = 6 degrees of freedom, the critical values obtained from table are ± 2.447.

Compute the test value,

[Substitute the values in the formula and simplify.]

Reject the null hypothesis, since the test value falls in the critical region as shown.

Therefore, there is a significant relationship between the SAT scores and the Statistics Midterm scores.

Correct answer : (1)