Hypothesis Testing for Regression: 1980 vs. 1999 Per Capita Income

Math Problem Statement

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Topic 7 Homework (Nonadaptive) Question 8 of 16 (1 point)|Question Attempt: 1 of Unlimited

Dennis

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Question 8 For the 50 U.S. states, the least-squares regression equation relating the 1980 per capita income and the 1999 per capita income is =y+7.971.97x, with x denoting 1980 per capita income and y denoting 1999 per capita income. The standard error of the slope of this least-squares regression line is approximately 1.64. (Source: U.S. Bureau of Economic Analysis, Survey of Current Business, May 2000.) Based on this information, test for a significant linear relationship between the variables x and y by doing a hypothesis test regarding the population slope β1. (Assume that the variable y follows a normal distribution for each value of x and that the other regression assumptions are satisfied.) Use the 0.10 level of significance, and perform a two-tailed test. Then complete the parts below.

(If necessary, consult a list of formulas.)

(a) State the null hypothesis H0 and the alternative hypothesis H1. H0: H1: (b) Determine the type of test statistic to use. ▼(Choose one) (c) Find the value of the test statistic. (Round to three or more decimal places.)

(d) Find the two critical values at the 0.10 level of significance. (Round to three or more decimal places.) and (e) Based on the data, can we conclude (using the 0.10 level) that there is a significant linear relationship between 1980 per capita income and 1999 per capita income for U.S. states? Yes No

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Solution

Let's break this question down step-by-step. You're asked to test for a significant linear relationship between the 1980 per capita income $x$ and the 1999 per capita income $y$ using a hypothesis test for the population slope $\beta_1$ . The given regression equation is:

$y = 7.97 + 1.97x$

The standard error of the slope ( $SE$ ) is 1.64, and we need to perform a hypothesis test at a 0.10 significance level.

(a) State the hypotheses

Null Hypothesis $(H_0)$ : The slope of the regression line is 0 (no linear relationship between $x$ and $y$ ). $H_0: \beta_1 = 0$
Alternative Hypothesis $(H_1)$ : The slope of the regression line is not 0 (there is a significant linear relationship between $x$ and $y$ ). $H_1: \beta_1 \neq 0$

(b) Determine the type of test statistic

Since we're testing the slope of the regression line, the appropriate test statistic to use is a t-test.

(c) Find the value of the test statistic

The formula for the t-statistic in a regression is:

$t = \frac{\hat{\beta_1} - 0}{SE}$

Where:

$\hat{\beta_1}$ is the estimated slope of the regression line (1.97 in this case),
$SE$ is the standard error of the slope (1.64).

Substitute the values into the formula:

$t = \frac{1.97 - 0}{1.64} = \frac{1.97}{1.64} \approx 1.201$

So, the t-statistic is approximately 1.201.

(d) Find the critical values at the 0.10 level of significance

For a two-tailed test at a 0.10 significance level, we need to find the critical t-values. Assuming the sample size is large (degrees of freedom ( \approx 50 -

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Math Problem Analysis

Mathematical Concepts

Regression Analysis
Hypothesis Testing
Linear Relationships
Significance Testing

Formulas

Least-squares regression equation: y = β0 + β1x
Test statistic for the slope: t = (β1 - 0) / SE
Critical t-values for two-tailed test

Theorems

Hypothesis testing for regression slopes
t-distribution for testing statistical significance

Suitable Grade Level

Undergraduate/College Level

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