A shoe manufacturer compared material A and material B for the soles of shoes. Twelve volunteers each got two shoes. The left wa
s made with material A and the right with material B. On both shoes, the material was 1 inch thick. Volunteers used the shoes normally for two months and returned them to the manufacturer. Technicians then measured the thickness of the soles and recorded the amount of wear (in microns):
Volunteer 1 2 3 4 5 6 7 8 9 10 11 12
Sole A 379 378 328 372 325 304 356 309 354 318 355 392
Sole B 372 376 328 368 283 252 369 321 379 303 328 411
They wish to test
H0: µA −µB = 0
against
HA: µA −µB 6= 0,
where µA and µB are the unknown population mean amounts of sole wear, using α = 0.05.
(a) Are the two samples paired or independent? Explain your answer.
(b) Make a normal QQ plot of the differences within each pair. Is it reasonable to assume a normal population of differences?
(c) Choose a test appropriate for the hypotheses above and justify your choice based on your answers to parts (a) and (b). Perform the test by computing a p-value, make a test decision, and state your conclusion in the context of the problem
Volunteer 1 2 3 4 5 6 7 8 9 10 11 12
Sole A 379 378 328 372 325 304 356 309 354 318 355 392
Sole B 372 376 328 368 283 252 369 321 379 303 328 411
They wish to test
H0: µA −µB = 0
against
HA: µA −µB 6= 0,
where µA and µB are the unknown population mean amounts of sole wear, using α = 0.05.
(a) Are the two samples paired or independent? Explain your answer.
(b) Make a normal QQ plot of the differences within each pair. Is it reasonable to assume a normal population of differences?
(c) Choose a test appropriate for the hypotheses above and justify your choice based on your answers to parts (a) and (b). Perform the test by computing a p-value, make a test decision, and state your conclusion in the context of the problem
1 answer:
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Complete Question
The complete question appears in the first uploaded image
Answer:
a) Yes, selecting 15 corresponds to a binomial experiment
b)
c)
d)
Regarding question a:
For an experiment to qualify as binomial
the trials have to be independent
each trial must yield one of two possible outcomes
Given that the selection of 15 individuals is random, we ascertain that the trials are independent and the outcomes are “either the individual saves for retirement or does not save for retirement.”
Therefore, we conclude that the selection of 15 people at random is indeed a binomial experiment.
In question b:
The probability that all selected adults do not save for retirement is mathematically modeled as
r = 15 implies all selected adults
n refers to the population size equating to 15
From the problem, p = 0.20
and q can be calculated as
=>
=>
Thus
Regarding question c:
The probability that exactly five of the selected adults do not save for retirement is mathematically modeled as
In relation to question d:
The probability that at least one of the selected adults opts not to save for retirement can be mathematically expressed as