Homework Assignment 4 - Simple Linear Regression
ITDSIU24002 - Vũ Đức Thiên Ân
 Based on the table, we observe that:
 Regression Equation
o Intercept (𝛽0 ): 0.6445
o Slope Coefficient (𝛽1 ): 0.6061
o Model Equation ( 𝑦̂): 0.6445 + 0.6061x
=> A 1% increase in depression is associated with a 0.61% rise in anxiety. However,
because the P-value (0.146) exceeds 0.05, this relationship is not statistically significant
and lacks predictive reliability.
 Hypothesis Test for 𝜷𝟏
o
Statistical Decision: Our analysis yielded a P-value of 0.146. Since this
value is considerably higher than the conventional alpha level of 0.05, we
do not have sufficient evidence to reject the null hypothesis.
=> Conclusion: This implies that the observed correlation might be due to random chance
rather than a consistent linear trend within the studied timeframe for Vietnam.
 R-Squared
o R-squared (𝑅2 ): 0.074 (7.4%)
o Interpretation: The model shows a weak fit with an 𝑅2 of only 7.4%,
meaning depression rates explain very little of the variance in anxiety. The
remaining 92.6% of the variability is likely driven by external factors not
captured in this bivariate analysis.
 Confidence Intervals - CI
o CI for Intercept (𝛽0 ) at 95%: [-1.464, 2.753]
o CI for Slope ( β1 ) at 95%: [-0.225, 1.437]
o The 95% confidence interval for the slope (β1 ) ranges from -0.225 to 1.437.
=> Since the interval includes zero, the effect is statistically insignificant. This matches
the high P-value, suggesting the true impact may be zero or negative.
 Residual Diagnostics

Average of Residuals (Zero Mean)
o
𝐻0 : The mean of the residuals equals 0.
o
𝐻𝑎 : The mean of the residuals is significantly different from 0.
o
Statistical Result: With a P-value ≈ 1.0 (from the 1-sample t-test), we
fail to reject 𝐻0 .
 Conclusion: The model perfectly satisfies the assumption that the residuals are
centered around zero, ensuring no systematic bias in the estimations.

Consistency of Variance (Homoscedasticity)
o
𝐻0 : Residual variance is constant (Homoscedasticity).
o
𝐻𝑎 : Residual variance is non-constant (Heteroskedasticity).
o
Breusch-Pagan Test: P-value = 0.0021
o
White Test: P-value = 0.0008
 Conclusion: Since both P-values are far below 0.05, we reject 𝐻0 . This indicates a
violation of the constant variance assumption. The model’s error magnitude
fluctuates, suggesting that the relationship between depression and anxiety might
be more volatile at different levels of prevalence.

Normality of Error Distribution
o
𝐻0 : Residuals are normally distributed.
o
𝐻𝑎 : Residuals do not follow a normal distribution.
o
Shapiro-Wilk Test: P-value = 0.0013
o
Normaltest (D'Agostino): P-value = 0.0333
 Conclusion: The majority of tests yield P-values < 0.05, leading us to reject the
normality assumption. Although the Jarque-Bera test (p = 0.241) was less
sensitive, the overall evidence points toward distributional anomalies.
 The scatter plot shows a slight upward trend, but the data points are widely
dispersed from the regression line. This visual evidence confirms a weak linear
relationship and supports the low 𝑅2 (7.4%) and non-significant P-value (0.146).
 The residuals deviate significantly from the diagonal reference line in the Q-Q
plot. This confirms that the model’s errors are not normally distributed, violating a
fundamental OLS assumption and indicating that a simple linear model may not be
the best fit for this data.
Data: Mental Health (EDA + Prediction)