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Let’s work through each part of the problem.

### Given Information:
The least-squares regression equation is given as:
\[
\hat{y} = 7.97 + 1.97x
\]
- **x** represents the 1980 per capita income.
- **y** represents the 1999 per capita income.
- The standard error of the slope (\(b_1\)) is approximately 0.75.
- We are testing for a significant linear relationship between **x** and **y** at a 0.05 significance level.

### Part (a): State the Null and Alternative Hypotheses
The hypotheses for testing the slope of the regression line are as follows:

- **Null Hypothesis (\(H_0\))**: There is no linear relationship between \(x\) and \(y\). Thus, the population slope \(\beta_1\) is zero.
  \[
  H_0: \beta_1 = 0
  \]

- **Alternative Hypothesis (\(H_1\))**: There is a significant linear relationship between \(x\) and \(y\). Thus, the population slope \(\beta_1\) is not zero.
  \[
  H_1: \beta_1 \neq 0
  \]

### Part (b): Determine the Type of Test Statistic to Use
Since we are testing the significance of the slope in a regression, we use the **t-test** for the slope.

### Part (c): Calculate the Test Statistic
The test statistic for the slope in a regression is calculated as:
\[
t = \frac{b_1}{\text{SE}(b_1)}
\]
where:
- \(b_1 = 1.97\) (the estimated slope from the regression equation).
- \(\text{SE}(b_1) = 0.75\).

Substitute these values:
\[
t = \frac{1.97}{0.75} \approx 2.627
\]

### Part (d): Find the p-value
To find the p-value, we need the degrees of freedom, which is typically \(n - 2\) for a regression with one predictor. Since we are working with data from 50 U.S. states, \(n = 50\), so the degrees of freedom is \(50 - 2 = 48\).

Using a t-distribution table or calculator with 48 degrees of freedom:
- For a two-tailed test, and a \(t\)-value of approximately 2.627, the p-value is around **0.011**.

### Part (e): Conclusion
Since the p-value (0.011) is less than the significance level (0.05), we **reject the null hypothesis**. There is sufficient evidence to conclude that there is a significant linear relationship between 1980 per capita income and 1999 per capita income for U.S. states.

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Do you have any questions, or would you like more detailed explanations for any of these parts?

### Related Questions:
1. How does changing the significance level affect the decision to reject or fail to reject the null hypothesis?
2. What would the hypotheses look like if we tested for a positive relationship rather than any linear relationship?
3. Why is it important to ensure the regression assumptions are satisfied before conducting this test?
4. How would the interpretation change if the p-value were greater than 0.05?
5. Can we infer causation from this regression analysis?

#### Tip:
In hypothesis testing, always ensure that the assumptions of the test (like normality of residuals in regression) are met before drawing conclusions.

For the 50 U.S. states, the least-squares regression equation relating the 1980 per capita income and the 1999 per capita income is ŷ = 7.97 + 1.97x, 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 0.75. 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. Use the 0.05 level of significance, and perform a two-tailed test.

Learn how to conduct a hypothesis test for the slope in a linear regression model comparing 1980 and 1999 per capita incomes for U.S. states. This problem involves calculating the test statistic, determining the p-value, and interpreting the results at a 0.05 significance level.

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