STA 106 Spring 2021: Homework 1
Instructor: Vaidotas Characiejus April 6, 2021
The assignment has to be submitted online on Gradescope until the deadline of April 12, 2021 at 11:59pm. Late homework submissions will be accepted for 24 hours after the deadline with a 30% penalty. No submissions will be accepted by email.
It is not a problem to discuss what kind of an approach or idea other students use. However, you are not allowed to share your solutions with other students. Everyone has to write their own solutions.
Suspected misconduct will be reported to the Office of Student Support and Judicial Affairs and, if established, will result in disciplinary sanctions up through Dismissal from the University and a grade penalty up to a grade of “F” for the course.
Please read the statement on the course materials in the syllabus.
1. (Problem 15.1) In an experiment to study the effect of the location of a product display in drugstores of a chain, the manager of one of the drugstores rearranged the displays of other products so as to increase the traffic flow at the experimental display. Does this action potentially lead to selection bias or measurement bias? Discuss.
2. (Problem 15.7) In a study to evaluate the quality of three alternative recipes for salsa, six containers of salsa – two from each of the three recipes – were randomly assigned to six taste panels. Each taste panel consisted of a team of four trained taste-testers. Each panel reached a consensus score for the assigned recipe. What is the experimental unit in this study? Why?
3. (Problem 15.10) In a study to investigate the effect of color of paper (blue, green, orange) on response rates for questionnaires distributed by the “windshield method” in supermarket parking lots, four supermarket parking lots were chosen in a metropolitan area and 10 questionnaires of each color were assigned at random to cars in the parking lots.
(a) Is this study experimental, observational, of mixed? Why?
(b) Identify all factors, factor levels, and factor-level combinations.
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(c) What type of study design is being implemented here? (d) What is the basic unit of study?
4. (Problem 15.11) A rehabilitation center researcher was interested in examining the relationship between physical fitness prior to surgery of persons undergoing corrective knee surgery and the time required in physical therapy until successful rehabilitation. Data on the number of days required for successful completion of physical therapy and the prior physical fitness status (below average, average, above average) were collected.
(a) Is this study experimental, observational or mixed? Why?
(b) Identify all factors, factor levels, and factor-level combinations.
(c) What type of study design is being implemented here? (d) What is the basic unit of study?
5. The Kenton Food Company wished to test four different package designs for a new breakfast cereal. Twenty stores, with approximately equal sales volumes, were selected as the experimental units. Each store was randomly assigned one of the package designs, with each package design assigned to five stores. A fire occurred in one store during the study period, so this store had to be dropped from the study. Hence, one of the designs was tested in only four stores. The stores were chosen to be comparable in location and sales volume. Other relevant conditions that could affect sales, such as price, amount and location of shelf space, and special promotional efforts, were kept the same for all of the stores in the experiment. Sales, in number of cases, were observed for the study period, and the results are recorded in Table 1. This study is a completely randomized design with package design as the single, four-level factor.
Package design Store (j) Total Mean 12345
i Yi1 Yi2 Yi3 Yi4 Yi5 Yi· Y ̄i· 1 1117161415 73 14.6 2 1210151911 67 13.4 3 23201817 78 19.5 4 2733222628 136 27.2
Number of stores
ni
5
5
4
5
nT = 19
designs–Kenton Food Company example.
Suppose we are interested in testing whether there is any difference in sales for
package designs 3 and 4. Suppose also that σ = 10. (a) Write down the null and alternative hypotheses.
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All designs Y·· = 354 Y ̄·· = 18.36
Table 1: Number of cases sold by stores for each of four package
(b) Write down the test statistic. Compute the p-value of the test. (c) What is your conclusion (assume α = 0.01 level of significance)?
6. Suppose that X1, . . . , Xn are iid random variables and let X ̄ = n−1 ni=1 Xi. (a) Show that the sum of residuals always equal to zero, i.e., show that
(b) Show that
n
( X i − X ̄ ) = 0 .
i=1
nn
( X i − X ̄ ) 2 = X i 2 − n X ̄ 2 .
i=1 i=1
(c) Answerthefollowingquestionsandprovideabriefexplanationtoyouranswers. What is the degree of freedom of (X1,…,Xn)?
What is the degree of freedom of (X1,…,Xn,X ̄)?
What is the degree of freedom of the residuals (X1 −X ̄,…,Xn −X ̄)?
What is the degree of freedom of (X1 −X ̄,··· ,Xn −X ̄,X ̄) ?
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