Question 8: Testing for Statistical Significance in R

To solve Question 8, you will need to test whether the effect of tenure (or the combination of tenure and exper since they both increase by one year) is statistically significantly different from zero at a 7% significance level. Here are the steps and the corresponding code in R to conduct this test:

Step-by-Step Explanation:

1. Null Hypothesis:
   You want to check if the combined effect of an additional year of tenure and experience is statistically different from zero. The null hypothesis is that the coefficient of tenure (or combined with exper) is zero.

   $H_0: \beta_{tenure} = 0$

   The alternative hypothesis is:

   $H_1: \beta_{tenure} \neq 0$

2. Load the necessary R packages:
   You will need the lm() function from the stats package for linear regression, which comes pre-installed with R. You also may use broom for cleaner output, though it's optional.

3. Run the regression model:
   Regress ln(wage) on educ, exper, and tenure using the lm() function.

4. Obtain the p-value:
   You can extract the p-value for the coefficient of tenure (which also includes the effect of exper since it increases simultaneously) to determine if it's statistically significant at the 7% significance level.

5. Decision rule:
   If the p-value is less than 0.07 (7% significance level), you reject the null hypothesis.

R Code Implementation:

rCopy code
# Load necessary packages (install broom if not already installed)
# install.packages("broom") # Uncomment if broom is not installed
library(broom)

# Assuming your data is stored in a data frame called `data`
# Example data frame: data <- data.frame(ln_wage, educ, exper, tenure)

# Perform the regression
model <- lm(ln_wage ~ educ + exper + tenure, data = data)

# Summary of the model
summary(model)

# Get the p-value for the tenure coefficient (which represents the combined effect of tenure and experience)
# Use the `broom` package for a clean output (optional)
tidy(model)

# Extract the p-value for the tenure coefficient
p_value_tenure <- tidy(model)$p.value[which(tidy(model)$term == "tenure")]

# Compare the p-value with the 7% significance level
if (p_value_tenure < 0.07) {
  print("Reject the null hypothesis: The effect of tenure is statistically significant.")
} else {
  print("Fail to reject the null hypothesis: The effect of tenure is not statistically significant.")
}

Explanation of Decision:

• After running the regression, focus on the p-value associated with the tenure coefficient.
• If the p-value is smaller than 0.07, this means there is evidence to suggest that tenure has a statistically significant effect on wages, and you should reject the null hypothesis.
• If the p-value is greater than 0.07, you fail to reject the null hypothesis, implying that the effect of tenure on wages is not statistically significant at the 7% significance level.

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Would you like more details or have any questions on how to proceed?

5 Related Questions:

1. How do I interpret the regression coefficients in the summary output?
2. What does it mean if the p-value is very close to 0.07?
3. How do I calculate confidence intervals for the tenure effect in R?
4. Can I test for statistical significance using a different significance level?
5. How can I visualize the relationship between tenure and wage using R?

Tip:

When performing hypothesis tests, always ensure that the assumptions of the regression model (like homoscedasticity and normality of residuals) are met to ensure valid results.