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CDS533 Assignment 3
(Statistics for Data Science)
The assignment may consist of either two parts—a problem set and an R component— or just one part. Please follow the guidelines provided below for each.
Problem Set:
1. Manually solve each question and clearly demonstrate all necessary steps.
2. You can either type out your answers in MS Word or provide scanned images of your written answers.
3. Ensure that your handwriting is clear and legible, and that the quality of scanned files is sufficient. Unclear submissions will not be accepted.
R Component:
1. Provide screenshots of R console outputs or insert R plots as required by the questions.
2. For questions involving analysis based on R outputs, please provide detailed explanations to showcase your understanding.
On final submission
Personal Information:
Ensure that your full name and student ID (SID) are written at the very beginning of the document.
Final Submission Format:
● Round your final answer in 3 decimals.
● Your final submission, including the answers from the problem set and R component, should be merged into a single Microsoft Word document (.docx).
Submission Deadline:
Please upload your work in Moodle by 1:30pm 18th Nov, 2024.
NOTE: USE P-VALUE METHOD FOR HYPOTHESIS TESING PROBLEMS.
1. A study of MBA graduates for The American Graduate Survey 1999 revealed that MBA graduates have several expectations of prospective employers beyond their base pay. In particular, according to the study 46% expect a performance-related bonus, 46% expect stock options. 42% expect a signing bonus, 28% expect profit sharing, 27% expect extra vacation/personal days, 25% expect tuition reimbursement, 24% expect health benefits, and 19% expect guaranteed annual bonuses. Suppose a study is conducted in an ensuing year to see whether these expectations have changed. If 125 MBA graduates are randomly selected and if 66 expect stock options, does this result provide enough evidence to declare that a significantly higher proportion of MBAs expect stock options? Let α = 0.05. If the proportion really is 0.50, what is the probability of committing a Type II error? And what is the power of this test?
2. Using R to complete this question. An American karate studio plans to advertise but is unsure as to which of three ads to use. The ads are tested on randomly selected consumers and the reactions measured on an ordinal scale that produces the following data:
Red 80,80,78,81,72,85,96,84,71,75,98
White 75,55,98,92,86,78,87,79,88,87,85,94,99
Blue 72,76,70,77,68,82,85,81,65,69
Test the claim that reactions are the same for the three different ads. Perform the Kruskal Wallis Test. If appropriate, follow with Wilcoxon tests to make the decision which ads should be given the contract, and interpret your results.
3. An experiment was conducted to compare the wearing qualities of three types of paint. Ten point specimens were tested for each paint type and the number of hours until visible abrasion was apparent was recorded. Assume that the variances are not significantly different, that the distributions are approximately normal, that the measures are numerical. Is there evidence to indicate a difference in the three plant types? Construct the ANOVA table, show your calculation. Each group has 10 readings with the following statistics:
a) Compute an ANOVA table for these data (using a hand calculator), including all relevant sums of squares, mean squares, and degrees of freedom.
b) State the statistical model underlying the procedures in the analysis of variance as applied to these data. Define symbols used and make clear all distributional assumptions.
c) State in words and symbols the null hypothesis and alternative hypothesis appropriate to this problem. Compute the relevant F-test and find the p-value for the test.
4. Suppose we have a data set with five predictors, X1= GPA, X2= IQ, X3= Level (1 for College and 0 for High School), X4 = Interaction between GPA and IQ, and X5 = Interaction between GPA and Level. The response is starting salary after graduation (in thousands of dollars). Suppose we use least squarestofit the model,
and get β(̂)0 = 50, β(̂)1 = 20,β(̂)2 = 0.07,β(̂)3 = 35, β(̂)4 = 0.01, β(̂)5 = −10 .
a) Write out the regression equation.
b) Which answer is correct, and why?
i. For a fixed value of IQ and GPA, high school graduates earn more, on average, than college graduates.
ii. For a fixed value of IQ and GPA, college graduates earn more, on average, than high school graduates.
iii. For a fixed value of IQ and GPA, high school graduates earn more, on average, than college graduates provided that the GPA is high enough.
iv. For a fixed value of IQ and GPA, college graduates earn more, on average, than high school graduates provided that the GPA is high enough.
c) Predict the salary of a college graduate with IQ of 110 and a GPA of 4.0.
d) True or false: Since the coefficient for the GPA/IQ interaction term is very small, there is very little evidence of an interaction effect. Justify your answer.
5. Using R to complete this question. This question involves the use of simple linear regression on the “Auto” data set.
a) Use the lm() function to perform a simple linear regression with mpg as the response and horsepower as the predictor. Use the summary() function to print the results. Comment on the output.
For example:
i. Is there a relationship between the predictor and the response?
ii. How strong is the relationship between the predictor and the response?
iii. Is the relationship between the predictor and the response positive or negative?
iv. What is the predicted mpg associated with a horsepower of 98? v. What is the associated 95% confidence intervals of β1 ?
b) Plot the response and the predictor. Use the abline() function to display the least squares regression line.
c) Use the plot() function to produce diagnostic plots of the least squares regression fit. Comment on any problems you see with the fit.
6. A substance used in biological and medical research is shipped by airfreight to users in cartons of 1,000 ampules. The data below, involving 10 shipments, were collected on the number of times the carton was transferred from one aircraft to another over the shipment route (X) and the number of ampules found to be broken upon arrival (Y). Assume the first-order regression model Y = β0 + β1x + ε is appropriate.
|
i |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
xi |
1 |
0 |
2 |
0 |
3 |
1 |
0 |
1 |
2 |
0 |
|
|
16 |
9 |
17 |
12 |
22 |
13 |
8 |
15 |
19 |
11 |
a) Obtain the estimated regression function.
b) Obtain the point estimate of the expected number of broken ampules when x = 1 transfer is made.
c) Estimate the increase in the expected number of ampules broken when there are 2 transfers as compared to 1 transfer.
d) Estimate β1 with a 95% confidence interval. Interpret your interval estimate.
e) Conduct a t test to decide whether or not there is a linear associaton between number of times a carton is transferred (x) and the number of broken ampules (Y). Use a level of significance1 of 5%. State the alternatives, decision rule, and conclusion. What is the p-value of the test?
f) A consultant has suggested, on the basis of previous experience, that the mean number of broken ampules should not exceed 9.0 when no transfers are made. Conduct an appropriate test, using α = 0.025. State the alternatives, decision rule, and conclusion. What is the P-value of the test?