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Assignment sample solution of STAT8002 - Advanced Statistical Modelling

Q.1 A pharmaceutical company conducted a clinical trial to test the effectiveness of a new drug. The data contains two groups: treatment and control, with their respective blood pressure readings after the trial. Perform an independent two-sample t-test at a 1% significance level to determine whether the drug has a significant effect on blood pressure reduction. State the null hypothesis, alternative hypothesis, and conclusion.

Q.2 You are tasked with analyzing the relationship between customer satisfaction (dependent variable) and three independent variables: product quality, price, and delivery time. Use a multiple linear regression model to determine the significant predictors of customer satisfaction. Discuss the model assumptions and validate the model.

Q.3 A dataset contains information about monthly temperatures in Melbourne over the last 20 years. Use a time series analysis approach to identify seasonal patterns and predict temperatures for the next 12 months. Apply the ARIMA model and explain your process.

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Statistics Assignment Sample

Q1:

Answer :

Q.1 Ans. : Null Hypothesis (H₀): There is no difference in mean blood pressure between the treatment and control groups (μ1=μ2\mu_1 = \mu_2μ1​=μ2​).
Alternative Hypothesis (H₁): There is a difference in mean blood pressure between the groups (μ1≠μ2\mu_1 \neq \mu_2μ1​=μ2​).

  • Test Statistic Formula: t=xˉ1−xˉ2s12n1+s22n2t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}t=n1​s12​​+n2​s22​​​xˉ1​−xˉ2​​.
  • Calculation: Calculate the means, variances, and sample sizes of both groups. Compute the test statistic and compare it to the critical t-value for a 1% significance level.
  • Conclusion: If the p-value is less than 0.01, reject H0H₀H0​ and conclude that the drug has a significant effect.

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Answer :

Q.2 Ans.: Model Formula: Y=β0+β1X1+β2X2+β3X3+ϵY = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_3 + \epsilonY=β0​+β1​X1​+β2​X2​+β3​X3​+ϵ, where YYY is customer satisfaction, and X1,X2,X3X_1, X_2, X_3X1​,X2​,X3​ are the independent variables.
Model Fitting: Use statistical software (e.g., R, Python, or SPSS) to fit the regression model.

  • Assumption Checks:
    1. Linearity: Scatterplots between YYY and predictors.
    2. Independence: Durbin-Watson test.
    3. Homoscedasticity: Residual vs. fitted values plot.
    4. Normality: Q-Q plot of residuals.
  • Validation: Use cross-validation techniques or split the data into training and test sets.
  • Conclusion: Report the significant predictors and interpret their coefficients (e.g., “a 1-unit increase in product quality leads to a 5-point increase in customer satisfaction”).

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Answer :

Q.3 Ans.  Model Formula: Y=β0+β1X1+β2X2+β3X3+ϵY = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_3 + \epsilonY=β0​+β1​X1​+β2​X2​+β3​X3​+ϵ, where YYY is customer satisfaction, and X1,X2,X3X_1, X_2, X_3X1​,X2​,X3​ are the independent variables.

  • Model Fitting: Use statistical software (e.g., R, Python, or SPSS) to fit the regression model.
  • Assumption Checks:
    1. Linearity: Scatterplots between YYY and predictors.
    2. Independence: Durbin-Watson test.
    3. Homoscedasticity: Residual vs. fitted values plot.
    4. Normality: Q-Q plot of residuals.
  • Validation: Use cross-validation techniques or split the data into training and test sets.
  • Conclusion: Report the significant predictors and interpret their coefficients (e.g., “a 1-unit increase in product quality leads to a 5-point increase in customer satisfaction”).