Problem 1: Hypothesis Testing with Multiple Samples Suppose a researcher wants to test whether there is a significant difference in the average productivity of employees across three different departments in a company. The productivity scores are measured over a week and given in the table below. Department A Department B Department C 85 80 90 88 85 92 82 84 95 90 86 89 87 89 91 State the null and alternative hypotheses for testing the difference in means across the departments. Perform an ANOVA test to evaluate if there is a significant difference in average productivity across the three departments. Use a significance level of 0.05. Calculate the F-statistic and conclude whether the null hypothesis is rejected. Problem 3: Logistic Regression Analysis A hospital collects data on patient survival outcomes based on age and cholesterol levels. The goal is to model the probability of survival (1 = survived, 0 = did not survive) using logistic regression. The data is summarized in the table below: Age (years) Cholesterol (mg/dL) Survival Outcome (1 = survived, 0 = not survived) 45 210 1 54 250 0 60 300 0 37 180 1 50 220 1 63 270 0 Fit a logistic regression model where the probability of survival depends on age and cholesterol level. Interpret the coefficients for age and cholesterol in the logistic model. Calculate the predicted probability of survival for a 50-year-old patient with a cholesterol level of 240 mg/dL. Problem 4: Bayesian Inference with Prior Distributions Suppose the number of defects in a batch of products follows a Poisson distribution with parameter λ\lambdaλ. The prior distribution for λ\lambdaλ is Gamma distributed with shape parameter α=2\alpha = 2α=2 and rate parameter β=3\beta = 3β=3. You observe 8 defects in a randomly selected batch of products. Write down the likelihood function based on the Poisson distribution and the prior distribution of λ\lambdaλ. Derive the posterior distribution of λ Find the posterior mean and variance of λ\lambdaλ.
Problem 1: Hypothesis Testing with Multiple Samples
Suppose a researcher wants to test whether there is a significant difference in the average productivity of employees across three different departments in a company. The productivity scores are measured over a week and given in the table below.
| Department A | Department B | Department C |
|---|---|---|
| 85 | 80 | 90 |
| 88 | 85 | 92 |
| 82 | 84 | 95 |
| 90 | 86 | 89 |
| 87 | 89 | 91 |
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State the null and alternative hypotheses for testing the difference in means across the departments.
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Perform an ANOVA test to evaluate if there is a significant difference in average productivity across the three departments. Use a significance level of 0.05.
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Calculate the F-statistic and conclude whether the null hypothesis is rejected.
Problem 3: Logistic Regression Analysis
A hospital collects data on patient survival outcomes based on age and cholesterol levels. The goal is to model the probability of survival (1 = survived, 0 = did not survive) using logistic regression. The data is summarized in the table below:
| Age (years) | Cholesterol (mg/dL) | Survival Outcome (1 = survived, 0 = not survived) |
|---|---|---|
| 45 | 210 | 1 |
| 54 | 250 | 0 |
| 60 | 300 | 0 |
| 37 | 180 | 1 |
| 50 | 220 | 1 |
| 63 | 270 | 0 |
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Fit a logistic regression model where the probability of survival depends on age and cholesterol level.
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Interpret the coefficients for age and cholesterol in the logistic model.
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Calculate the predicted probability of survival for a 50-year-old patient with a cholesterol level of 240 mg/dL.
Problem 4: Bayesian Inference with Prior Distributions
Suppose the number of defects in a batch of products follows a Poisson distribution with parameter λ\lambdaλ. The prior distribution for λ\lambdaλ is Gamma distributed with shape parameter α=2\alpha = 2α=2 and rate parameter β=3\beta = 3β=3. You observe 8 defects in a randomly selected batch of products.
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Write down the likelihood function based on the Poisson distribution and the prior distribution of λ\lambdaλ.
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Derive the posterior distribution of λ
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Find the posterior mean and variance of λ\lambdaλ.
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