# Statistical Inference about Slope

Site: | Saylor Academy |

Course: | MA121: Introduction to Statistics |

Book: | Statistical Inference about Slope |

Printed by: | Guest user |

Date: | Friday, July 19, 2024, 7:35 PM |

## Description

This section further details two types of inferences on the slope parameter, considering both confidence intervals and hypothesis testing.

## Statistical Inferences About β1

#### Learning Objectives

- To learn how to construct a confidence interval for , the slope of the population regression line.
- To learn how to test hypotheses regarding .

#### Confidence Intervals for

where and the number of degrees of freedom is .

#### Definition

**sample standard deviation of errors**. It estimates the standard deviation σ of the errors in the population of y-values for each fixed value of (see Figure 10.5 "The Simple Linear Model Concept" in Section 10.3 "Modelling Linear Relationships with Randomness Present").

#### Example 6

Construct the 95% confidence interval for the slope of the population regression line based on the five-point sample data set

##### Solution:

The point estimate of was computed in Note 10.18 "Example 2" in Section 10.4 "The Least Squares Regression Line" as . In the same example was found to be . The sum of the squared errors was computed in Note 10.23 "Example 4" in Section 10.4 "The Least Squares Regression Line" as . Thus

#### Example 7

Using the sample data in Table 10.3 "Data on Age and Value of Used Automobiles of a Specific Make and Model" construct a 90% confidence interval for the slope of the population regression line relating age and value of the automobiles of Note 10.19 "Example 3" in Section 10.4 "The Least Squares Regression Line". Interpret the result in the context of the problem.##### Solution:

The point estimate if was computed in Note 10.19 "Example 3", as was . Their values are and . The sum of the squared errors was computed in Note 10.24 "Example 5" in Section 10.4 "The Least Squares Regression Line" as . Thus

Confidence level 90% means so . From the row labeled in Figure 12.3 "Critical Values of " we obtain . Therefore

#### Testing Hypotheses About

The test statistic has Student's t-distribution with degrees of freedom.

#### Example 8

Test, at the 2% level of significance, whether the variable x is useful for predicting y based on the information in the five-point data set

##### Solution:

We will perform the test using the critical value approach.

- Step 1. Since is useful for prediction of precisely when the slope of the population regression line is nonzero, the relevant test is

vs. @

- Step 2. The test statistic is

and has Student's t-distribution with degrees of freedom.

- Step 3. From Note 10.18 "Example 2", and . From Note 10.30 "Example 6", . The value of the test statistic is therefore

- Step 4. Since the symbol in is "≠" this is a two-tailed test, so there are two critical values . Reading from the line in Figure 12.3 "Critical Values of " labeled , . The rejection region is .

- Step 5. As shown in Figure 10.9 "Rejection Region and Test Statistic for " the test statistic falls in the rejection region. The decision is to reject . In the context of the problem our conclusion is:

The data provide sufficient evidence, at the 2% level of significance, to conclude that the slope of the population regression line is nonzero, so that is useful as a predictor of .

*Figure 10.9 Rejection Region and Test Statistic for Note 10.33 "Example 8"*

#### Example 9

A car salesman claims that automobiles between two and six years old of the make and model discussed in Note 10.19 "Example 3" in Section 10.4 "The Least Squares Regression Line" lose more than $1,100 in value each year. Test this claim at the 5% level of significance.

##### Solution:

We will perform the test using the critical value approach.

- Step 1. In terms of the variables and , the salesman's claim is that if is increased by 1 unit (one additional year in age), then y decreases by more than 1.1 units (more than $1,100). Thus his assertion is that the slope of the population regression line is negative, and that it is more negative than −1.1. In symbols, . Since it contains an inequality, this has to be the alternative hypotheses. The null hypothesis has to be an equality and have the same number on the right hand side, so the relevant test is

vs. @

- Step 2. The test statistic is

- Step 3. From Note 10.19 "Example 3", and . From Note 10.31 "Example 7", . The value of the test statistic is therefore

- Step 4. Since the symbol in is "<" this is a left-tailed test, so there is a single critical value . Reading from the line in Figure 12.3 "Critical Values of " labeled , . The rejection region is (−∞,−1.860].

- Step 5. As shown in Figure 10.10 "Rejection Region and Test Statistic for " the test statistic falls in the rejection region. The decision is to reject . In the context of the problem our conclusion is:

The data provide sufficient evidence, at the 5% level of significance, to conclude that vehicles of this make and model and in this age range lose more than $1,100 per year in value, on average.

*Figure 10.10 Rejection Region and Test Statistic for Note 10.34 "Example 9"*

This text was adapted by Saylor Academy under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License without attribution as requested by the work's original creator or licensor.

## Key Takeaways

- The parameter , the slope of the population regression line, is of primary interest because it describes the average change in with respect to unit increase in .
- The statistic , the slope of the least squares regression line, is a point estimate of . Confidence intervals for can be computed using a formula.
- Hypotheses regarding are tested using the same five-step procedures introduced in Chapter 8 "Testing Hypotheses".

## Exercises

#### Basic

For the Basic and Application exercises in this section use the computations that were done for the exercises with the same number in Section 10.2 "The Linear Correlation Coefficient" and Section 10.4 "The Least Squares Regression Line".

3. Construct the 90% confidence interval for the slope of the population regression line based on the sample data set of Exercise 3 of Section 10.2 "The Linear Correlation Coefficient".

5. For the data in Exercise 5 of Section 10.2 "The Linear Correlation Coefficient" test, at the 10% level of significance, whether is useful for predicting (that is, whether ).

7. Construct the 90% confidence interval for the slope of the population regression line based on the sample data set of Exercise 7 of Section 10.2 "The Linear Correlation Coefficient".

9. For the data in Exercise 9 of Section 10.2 "The Linear Correlation Coefficient" test, at the 1% level of significance, whether x is useful for predicting (that is, whether ).

#### Applications

11. For the data in Exercise 11 of Section 10.2 "The Linear Correlation Coefficient" construct a 90% confidence interval for the mean number of new words acquired per month by children between 13 and 18 months of age.

13. For the data in Exercise 13 of Section 10.2 "The Linear Correlation Coefficient" test, at the 10% level of significance, whether age is useful for predicting resting heart rate.

15. For the situation described in Exercise 15 of Section 10.2 "The Linear Correlation Coefficient"

- Construct the 95% confidence interval for the mean increase in revenue per additional thousand dollars spent on advertising.
- An advertising agency tells the business owner that for every additional thousand dollars spent on advertising, revenue will increase by over $25,000. Test this claim (which is the alternative hypothesis) at the 5% level of significance.
- Perform the test of part (b) at the 10% level of significance.
- Based on the results in (b) and (c), how believable is the ad agency’s claim? (This is a subjective judgement.)

17. For the data in Exercise 17 of Section 10.2 "The Linear Correlation Coefficient" test, at the 10% level of significance, whether course average before the final exam is useful for predicting the final exam grade.

19. For the data in Exercise 19 of Section 10.2 "The Linear Correlation Coefficient" test, at the 1/10th of 1% level of significance, whether, ignoring all other facts such as age and body mass, the amount of the medication consumed is a useful predictor of blood concentration of the active ingredient.

21. For the data in Exercise 21 of Section 10.2 "The Linear Correlation Coefficient"

- Construct the 95% confidence interval for the mean increase in strength at 28 days for each additional hundred psi increase in strength at 3 days.
- Test, at the 1/10th of 1% level of significance, whether the 3-day strength is useful for predicting 28-day strength.

#### Large Data Set Exercises

23. Large Data Set 1 lists the SAT scores and GPAs of 1,000 students.

http://www.gone.2012books.lardbucket.org/sites/all/files/data1.xls

- Compute the 90% confidence interval for the slope of the population regression line with SAT score as the independent variable and GPA as the dependent variable .
- Test, at the 10% level of significance, the hypothesis that the slope of the population regression line is greater than 0.001, against the null hypothesis that it is exactly 0.001.

25. Large Data Set 13 records the number of bidders and sales price of a particular type of antique grandfather clock at 60 auctions.

http://www.gone.2012books.lardbucket.org/sites/all/files/data13.xls

- Compute the 95% confidence interval for the slope of the population regression line with the number of bidders present at the auction as the independent variable and sales price as the dependent variable .
- Test, at the 10% level of significance, the hypothesis that the average sales price increases by more than $90 for each additional bidder at an auction, against the default that it increases by exactly $90.