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Consider the accompanying data on [tex]\( x \)[/tex] = research and development expenditure (millions of dollars) and [tex]\( y \)[/tex] = growth rate (% per year) for eight different industries.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & 2.023 & 5.037 & 0.904 & 3.571 & 1.157 & 0.327 & 0.378 & 0.191 \\
\hline
[tex]$y$[/tex] & 1.90 & 3.96 & 2.44 & 0.88 & 0.37 & -0.90 & 0.49 & 1.01 \\
\hline
\end{tabular}

(a) Would a simple linear regression model provide useful information for predicting growth rate from research and development expenditure? Test the appropriate hypotheses using a 0.05 significance level.

Calculate the test statistic. (Round your answer to two decimal places.)
[tex]\[ t = \square \][/tex]

Use technology to find the [tex]\( P \)[/tex]-value for this test. (Round your answer to four decimal places.)
[tex]\[ P \text{-value} = \square \][/tex]

What can you conclude?

A. Fail to reject [tex]\( H_0 \)[/tex]. We have convincing evidence of a useful linear relationship between growth rate and research and development expenditure.

B. Reject [tex]\( H_0 \)[/tex]. We do not have convincing evidence of a useful linear relationship between growth rate and research and development expenditure.

C. Fail to reject [tex]\( H_0 \)[/tex]. We do not have convincing evidence of a useful linear relationship between growth rate and research and development expenditure.

D. Reject [tex]\( H_0 \)[/tex]. We have convincing evidence of a useful linear relationship between growth rate and research and development expenditure.

(b) Use a [tex]\( 90\% \)[/tex] confidence interval to estimate the average change in growth rate associated with a [tex]\( \$ 1,000,000 \)[/tex] increase in expenditure. (Use technology to find the critical value. Round your answers to four decimal places.)
[tex]\[ \square \% \text{ per year} \][/tex]


Sagot :

To determine whether a simple linear regression model provides useful information for predicting growth rate from research and development expenditure, we will follow these steps:

### Part (a): Hypothesis Testing

Step 1: State the hypotheses

- Null Hypothesis ([tex]\(H_0\)[/tex]): There is no significant linear relationship between growth rate and research and development expenditure ([tex]\( \beta = 0 \)[/tex]).
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): There is a significant linear relationship between growth rate and research and development expenditure ([tex]\( \beta \neq 0 \)[/tex]).

Step 2: Calculate the test statistic

From linear regression analysis, we have the slope [tex]\( \beta \)[/tex], standard error of the slope [tex]\( SE_{\beta} \)[/tex], and the t-statistic.

Given:
- Test statistic [tex]\( t = 2.28 \)[/tex]

Step 3: Determine the p-value

We use the t-statistic to find the corresponding p-value.

Given:
- [tex]\( P \text{-value} = 0.0625 \)[/tex]

Since the p-value (0.0625) is greater than the significance level of 0.05, we compare it with the significant level as follows:

Step 4: Draw a conclusion

Since the p-value (0.0625) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Conclusion:
Fail to reject [tex]\( H_0 \)[/tex]. We do not have convincing evidence of a useful linear relationship between growth rate and research and development expenditure.

### Part (b): Confidence Interval for Slope

Step 1: Calculate the confidence interval for the slope

We want to construct a 90% confidence interval for the average change in growth rate associated with a [tex]$1,000,000 increase in expenditure. Given: - slope (\( \beta \)) = 0.5751273930806419 - standard error of the slope (\( SE_{\beta} \)) = 0.25184706198095234 The critical value for a 90% confidence interval with 6 degrees of freedom (\( df = n - 2 \)) is approximately 1.943. Step 2: Compute the confidence interval The formula for the confidence interval is: \[ \text{Slope CI} = \beta \pm \text{critical t} \times SE_{\beta} \] Given: - Lower bound = 0.08576954016490956 - Upper bound = 1.0644852459963743 Hence, the confidence interval is: \[ 0.086 \leq \beta \leq 1.064\] ### Conclusion: (a) - Test Statistic: \( t = 2.28 \) - P-value: \( P = 0.0625 \) - Conclusion: Fail to reject \( H_0 \). We do not have convincing evidence of a useful linear relationship between growth rate and research and development expenditure. (b) - The 90% confidence interval for the average change in growth rate associated with a $[/tex]1,000,000 increase in expenditure is approximately [tex]\([0.086, 1.064]\)[/tex] percent per year.