Homework 3 (Deadline: March 6, 2022)
1. While campaigning for higher political office during a recent election, a certain candidate claimed that “At least 75% of voters want the country to end all foreign aid to all countries.” When a polling organization conducted a survey to investigate this claim, they found that 242 out of a sample of n = 346 expressed agreement with the statement. At a level of α = 0.02, test H0 : p ≥ 0.75 against H1 : p < 0.75. What is the p-value?
2. A statistics instructor is concerned that after her students perform well on the first of two major examinations in the introductory-level class, their performance appears to drop off on the second. Since this pattern appears to repeat itself across many sections of the same statistics class at her university, she wants to confirm that the downward trend in performance on the two 100-point examinations is real. To this end, she collects the examination results for a random sample of n = 12 students from the previous academic year. The scores on Exam 1 are: 79, 92, 81, 80, 79, 80, 78, 88, 86, 88, 77, and 93. On Exam 2, they are: 80, 75, 67, 82, 76, 71, 78, 78, 80, 77, 78, and 75. Create a data frame that organizes this data into two variables and 12 observations and use R to answer questions (b) and (c).
(a) Are these data independent or paired? Why?
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(b) What is the point estimate of the difference between the two population means, μ1 − μ2?
(c) Formulate a hypothesis and test it (α = 0.05).
3. This question uses the mtcars data set that is installed in R. (Remember that to see all the installed data sets, simply enter data() at the R prompt in the Console; to view the mtcars data set itself, enter mtcars at the R prompt; to learn more about the data set, including the variables and observations, enter ?mtcars at the prompt and wait fro the R Help page to open.) In this case, we are interested in the relationship between an automobile’s quarter-mile time (qsec) and gross horsepower (hp).
(a) Create a scatter plot of the two variables. What does the pattern of points suggest about the relationship (if any) between the variables?
(b) Letting the quarter-mile time be the dependent variable, estimate the regression equation. Write out the regression equation, yˆ = b0 + b1x.
(c) What is the value of multiple r-squared?
(d) What is the p-value of the coefficient?
4. The file BostonHousing.csv contains information collected by the US Bureau of the Census concerning housing in the area of Boston, Massachusetts. The dataset includes information on 506 census housing tracts in the Boston area. The goal is to predict the median house price in new tracts based on information such as crime rate, pollution, and number of rooms. The dataset contains 13 predictors, and the response is the median house price (MEDV). The following table describe each of the predictors and the response.
Questions:
a. Fit a multiple linear regression model to the median house price (MEDV) as a function of CRIM, CHAS, and RM. Write the equation for predicting the median house price from the predictors in the model. b. Using the estimated regression model, what median house price is predicted for a tract in the Boston
area that does not bound the , has a crime rate of 0.1, and where the average number of rooms per house is 6?
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