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8.	(8 points) Is your predicted effect in Question 7 statistically significantly different from zero at the seven percent of level? Clearly write down the null hypothesis (3 points). Test your hypothesis using R by loading relevant packages (2 point). Explain your decision whether to reject the hypothesis (3 points).

To address this question using the information provided from the code snippet in the image, we will approach it step by step.

### 1. **Null Hypothesis (H₀):**
   - The null hypothesis typically states that there is no effect, meaning the coefficients (the predicted effects) are equal to zero.
   - **H₀: coef_tenure + coef_exper = 0**  
     In this case, the null hypothesis can be written as:
     \[
     H_0: \text{The combined effect of tenure and experience on the predicted outcome is zero.}
     \]

### 2. **Test Statistic Calculation:**
   In the code:
   - `coef_tenure` and `coef_exper` are the coefficients (effects) of tenure and experience, respectively.
   - The code sums the two coefficients (`sum_coeff <- coef_tenure + coef_exper`), giving the total predicted effect.
   - The standard errors (`standerror_tenure` and `standerror_exper`) are used to calculate the variance and standard deviation (`variance_sum` and `square_coef`), leading to the calculation of the **t-value** (`t_value <- sum_coeff / square_coef`), which in this case is `6.286762618755409`.

### 3. **P-Value Calculation:**
   - The `p-value` is calculated using the cumulative distribution function for the t-distribution (`p-value <- 2 * pt(-abs(t_value), df = 522)`), where `df` is the degrees of freedom (522).
   - The resulting **p-value** is `6.8519610230127e-10`, which is a very small value close to zero.

### 4. **Decision on the Null Hypothesis:**
   - The p-value (`6.8519610230127e-10`) is compared against the significance level (α = 0.07, since the question mentions a 7% level).
   - Since the p-value is significantly smaller than α = 0.07, we reject the null hypothesis. This indicates that the predicted effect (combined effect of tenure and experience) is statistically significantly different from zero at the 7% level.

### Conclusion:
   - **Decision**: We reject the null hypothesis at the 7% significance level.
   - **Explanation**: The p-value is much smaller than 0.07, indicating strong evidence against the null hypothesis. Therefore, the combined effect of tenure and experience on the predicted outcome is statistically significant.

Would you like more details or clarifications? 

### Here are 5 related questions:
1. What is the formula for calculating the t-value in hypothesis testing?
2. How do you interpret the p-value in hypothesis testing?
3. Why is the sum of the coefficients important in this context?
4. How do you calculate degrees of freedom for a t-test?
5. What is the significance of choosing a 7% significance level?

**Tip**: Always ensure that the assumptions of a t-test are met, such as the normality of residuals and equal variance, for accurate results.

Is your predicted effect in Question 7 statistically significantly different from zero at the seven percent level? Clearly write down the null hypothesis. Test your hypothesis using R by loading relevant packages. Explain your decision whether to reject the hypothesis.

This article explains how to test the statistical significance of the predicted effect of tenure and experience using R. It covers hypothesis testing, the t-test, and p-value calculations. Suitable for undergraduate statistics students.

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