ustion 1: Comparing means again (25 points) Based on an episode of the popular TV series \"Futurama\" Similar to assignment 2 question 4. Suppose that you want to assess the performance of a new treatment for tritonian yetiism. The onset of yetiism is measured by the amount of yeti neurotoxin in the patient's blood. The concentration of this neurotoxin may be reasonably assumed to be normally distributed, possibly with a different mean for individuals who have recieved the new treatment and those who have not. The goal of this study is to determine whether the treatment effectively reduces the concentration of yeti neurotoxin in the blood. If the treatment is effective then the average concentration of yeti neurotoxin will be lower in treated subjects than in subjects not treated. The sample contains 10 people with tritonian yetiism, and 20 total measurements; 1 per patient before treatment and 1 per patient after treatment. The concentration of yeti neurotoxin in the blood of each of the 10 individuals before and after treatment is given in the table below: before after 1 2 3 4 5 6 7 8 9 10 1592.0 984.9 1670.5 990.3 1563.9 994.1 1950.6 994.6 1592.1 999.2 2085.4 994.7 1897.6 992.8 1553.8 980.3 1715.4 996.1 1628.4 986.2 1a) Is it more appropriate to use a paired-data test or an unpaired-data test for this problem? Briefly describe why. 1b) Produce a confidence interval for the difference in mean concentrations of yeti neurotoxin before and after treatment. 1c) Devise a one sided hypothesis test for the question of interest (that the treatment is effective). Compute a p-value for your test. Form a conclusion at 5% significance based on your p-value. 1d) Compute a rejection region for your test based on a significance level of 5%. Form a conclusion based on your rejection region. 1e) Compute the power of your test under the assumption that the mean before treatment is really 991 and the mean after treatment is really 1675. 2 Qustion 2: Comparing variances (30 points) Using the data from question 1, we wish to assess whether the variance before treatment is more than the variance after treatment. 2a) Produce a 95% confidence interval for After Before (standard deviations not variances!) 2b) State a null and alternative hypothesis you would use to determine whether the variance before treatment is more than the variance after treatment. 2c) What is your test statistic for the hypothesis test? What is its distribution under the null hypothesis? (don't forget to state the degrees off freedom) 2d) Compute the observed value of the test statistic 2e) Compute a p-value for your test and form a conclusion at 5% significance. 2f) Why was equality of variances irrelevent for question 1? 3 Qustion 3: 3 3 contingency table (20 points) Consider the following contingency table: Handedness Major Right-handed Left-handed Ambidextrous Total by Major Statistics Psychology Engineering Total by Handedness 43 44 20 107 9 4 3 16 10 5 3 18 62 53 26 141 3.a) Formulate a hypothesis to assess whether handedness is independent from university major. State the null and alternative hypotheses mathematically. 3.b) State the formula of a test statistic you may use for this test. What is its approximate distribution under the null hypothesis? (don't forget to state the degrees off freedom) 3.c) Compute the observed value of the test statistic. 3.d) Compute a p-value for your test. Form a conclusion at 1% significance. 4 Qustion 4: Linear Regression (25 points) We will use the fisher iris dataset in R to regress Petal.Length against Petal.Width. model1<-lm(Petal.Length~Petal.Width) summary(model1) 4a) Which variable is the response, and which variable is the predictor? Write down a mathematical expression that represents the model described above. Explain how this model relates the mean of the response to the value of the predictor. 4b) Run the code provided and include the output in your assignment submission. 4c) Provide a 95% confidence interval for the slope. 4d) What hypothesis would you test to determine whether there is any association between Petal.Length and Petal.Width? State the null hypothesis. Identify the p-value for this test from your output in part b). Form a conclusion at 1% significance. 4e) Compute (X , Y , X 2 , Y 2 , XY ) in R. Use these to estimate 0 and 1 and 2 . Are your estimates consistent with part b)? 5