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9th Edition

David R. Anderson + 4 others

Publisher: South-Western College Pub

ISBN: 9780357045435

Chapter 14.6, Problem 32E

Textbook Problem

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The data from exercise 1 follow.

x_{i} |
1 | 2 | 3 | 4 | 5 |

y_{i} |
3 | 7 | 5 | 11 | 14 |

- a. Use equation (14.23) to estimate the standard deviation of
*ŷ** when*x*= 4. - b. Use expression (14.24) to develop a 95% confidence interval for the expected value of
*y*when*x*= 4. - c. Use equation (14.26) to estimate the standard deviation of an individual value of
*y*when*x*= 4. - d. Use expression (14.27) to develop a 95% prediction interval for
*y*when*x*= 4.

**a.**

To determine

Estimate the standard deviation of

**Calculation:**

The estimate of standard deviation of

Where *x*, *y* when *x* for sample of size *n*.

Simple linear regression model can be expressed as

*Least squares criterion:*

The least square criterion can be obtained as by minimizing sum of squares of difference between observed and predictor variable, that is,

Where *i*–th observation and *i*–th observation.

*Slope and y–intercept of the estimated regression equation:*

The slope can be obtained as,

And the *y*–intercept is,

where *i*-th observation.

*i*-th observations.

*n* is the total number of observations.

It is known for a sample size *n* that mean of a random variable *x* can be obtained as,

Thus, mean of the random variable *x* is,

Thus, mean of the random variable *y* is,

The value of

1 | –2 | 3 | –3.67 | 7.34 | 4 |

2 | –1 | 7 | 0.33 | –0.33 | 1 |

3 | 0 | 5 | –1.67 | 0 | 0 |

4 | 1 | 11 | 4.33 | 4.33 | 1 |

5 | 2 | 14 | 7.33 | 14.66 | 4 |

1 | –2 | 3 | –3.67 | 7.34 | 4 |

Total | 26 | 10 |

Thus, using the table the slope of the estimated regression equation is,

Thus, using the value of slope estimate the *y*–intercept of the estimated regression equation is,

Hence, the estimated regression equation is

*SSE:*

The Sum of squares due to error (SSE) is obtained as *i*-th observation and *i*-th observation

**b.**

To determine

Obtain a 95% confidence interval for the expected value of *y* when

**c.**

To determine

Estimate the standard deviation of an individual value of *y* for

**d.**

To determine

Obtain a 95% prediction interval for *y* when

Essentials Of Statistics For Business & Economics

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