Master Advanced Statistical Methods: Key Concepts and Practice
School
Capital Community College, Hartford**We aren't endorsed by this school
Course
PHYS 8011
Subject
Statistics
Date
Dec 12, 2024
Pages
4
Uploaded by HighnessFoxPerson1237
Exam Name: Advanced Statistical Methods AssessmentExam Time: 120 minutesTotal Score: 100 pointsInstructions:1. Please write your answers on the designated answer sheet.2. Show all your work for calculation questions.3. All questions carry equal marks unless otherwise specified.4. Marks will be deducted for incorrect answers in multiple-choice questions.---1. The probability density function of the standard normal distribution is given by:\[ f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \](a) Calculate the probability that a randomly selected value from this distribution is less than 1.5. (10 points)(b) Find the 95th percentile of this distribution. (10 points)---2. A binomial experiment involves 10 trials, and the probability of success on each trial is 0.2.(a) Calculate the probability of getting exactly 2 successes. (5 points)(b) What is the mean of the distribution? (5 points)---3. A Poisson process has a rate parameter of 5 events per hour. Calculate the probability of observing exactly 3 events in a 2-hour interval. (10 points)---4. Given the following joint probability mass function for two discrete random variables X and Y:\[ P(X=x, Y=y) = \frac{2x + 3y}{72}, \quad \text{for} \quad x=1,2,3, \quad \text{and} \quad y=1,2 \](a) Calculate the marginal probability mass function of X. (10 points)(b) Are X and Y independent? Justify your answer. (10 points)
---5. Suppose you have a regression model with the following equation:\[ y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon \]You perform a hypothesis test to determine if \(\beta_1\) is equal to zero. You obtain a p-value of 0.05. Interpret the result in the context of the model. (10 points)---6. A random sample of size 50 is drawn from a normal population with an unknown mean \(\mu\) and a known variance of 9. The sample mean is found to be 22.(a) Construct a 95% confidence interval for the population mean. (10 points)(b) Test the hypothesis that \(\mu = 20\) at a 5% significance level. (10 points)---7. A completely randomized design is conducted with 4 treatments and 3 replicates. The sum of squares for treatments is 50, the sum of squares for error is 10, and the total sum of squares is 80.(a) Calculate the degrees of freedom for treatments. (5 points)(b) What is the mean square for treatments? (5 points)---8. Given the following data:\[ \text{X: } 2, 3, 5, 7, 11 \]\[ \text{Y: } 4, 6, 10, 14, 22 \](a) Calculate the covariance between X and Y. (10 points)(b) Compute the Pearson correlation coefficient. (10 points)---9. Consider a bivariate normal distribution with parameters \(\mu_X = 10\), \(\mu_Y = 20\), \(\sigma_X^2 = 25\), \(\sigma_Y^2 = 36\), and \(\rho_{XY} = 0.6\).(a) Calculate the probability that \(X \leq 8\) and \(Y \leq 25\). (10 points)(b) Find the conditional probability that \(X \leq 8\) given \(Y = 20\). (10 points)
---10. Suppose you have a time series data set and you fit an AR(2) model. The estimated parameters are:\[ \phi_1 = 0.5, \quad \phi_2 = -0.3 \](a) Write down the AR(2) model for this data. (5 points)(b) What does the estimated model imply about the dependence of the current value on past values? (5 points)---11. In a nonparametric test for two independent samples, you obtain a rank sum test statistic of 100 and a p-value of 0.01. Interpret the result in the context of the test. (10 points)---12. A Bayesian model for a binary outcome is given by:\[ P(\theta | y) \propto P(y | \theta) P(\theta) \]where \(\theta\) is the probability of success, \(y\) is the observed number of successes in \(n\) trials, and \(P(\theta)\) is the prior distribution of \(\theta\).(a) Derive the posterior distribution if \(P(\theta)\) is a beta distribution with parameters\(\alpha\) and \(\beta\). (10 points)(b) Calculate the expected value of \(\theta\) given the observed data. (10 points)---13. You are given a data set with 1000 observations, and you are tasked with predicting the number of clicks on a website advertisement. Which of the following techniques would be most appropriate for this task?(a) Logistic regression(b) Linear regression(c) Time series analysis(d) Naive Bayes classifierExplain your choice. (10 points)
---14. A company is interested in analyzing the effectiveness of a new drug. They conduct a randomized controlled trial and collect data on the drug's effectiveness. Which statistical method should they use to analyze the data?(a) ANOVA(b) Chi-squared test(c) t-test(d) Pearson correlationJustify your answer. (10 points)---15. A researcher wants to analyze the relationship between three variables: income, education level, and life satisfaction. Which statistical technique should the researcher use to analyze the relationship among these variables?(a) Factor analysis(b) Cluster analysis(c) Multiple regression analysis(d) Discriminant analysisExplain your choice. (10 points)---END OF EXAM