Deck 11: Asking and Answering Questions About the Difference Between Two Population Proportions

Question 1

Suppose that you have estimated the difference between two population proportions, , using a confidence interval. What are the appropriate interpretations of thefollowing results?
a) Both endpoints of the confidence interval for are positive.
b) Both endpoints of the confidence interval for are negative.
c) Zero is included in the confidence interval.

Answer 1

a) A difference of 0 in population proportions is not a plausible value, and p₁>p₂ .

b) A difference of 0 in population proportions is not a plausible value, and p₁< p₂.
c) A difference of 0 in population proportions is a plausible value.It is possible that p₁=p₂ .

Question 2

The pre-historic development of agriculture is an active area of research amonganthropologists. The transition from a hunter-gatherer society to an agriculturalsociety is generally assumed to be signaled by an increase in cavities in teeth.Investigators recently studied teeth unearthed in two societies in Colorado:Basketmakers, societies living in small villages, and Post-Basketmakers, societiesliving in above-ground architectures such as pueblos. The investigators wereinterested in whether the shift in shelter was associated with a shift in consumptionfrom hunting (meat) to agriculture (maize). Data on the numbers of individuals withcavities are shown in the table below. Cavities tend to occur more among populationssubsisting primarily on agricultural products. You may assume that it is reasonable toregard these two samples as representative of teeth from these two populations.
a) Construct and interpret a 90% confidence interval for the difference in populationproportions of individuals with cavities, Basketmaker - Post-Basketmaker.
b) Interpret the 90% confidence level
c) Based on your results in parts (a) and (b), would you say that a shift inconsumption patterns is a reasonable conclusion on the part of the investigators?
Justify your response with statistical evidence from parts (a) and/or (b).

Answer 2

a) Assumption check: There are at least 10 "successes" and at least 10 "failures" in both samples.Thus, the large sample procedure (z) is appropriate.
b) We are 90% confident that the true difference in population proportions with cavities of basketmakers and post-basketmakers between --0.2977 and -0.0281.
c) Since 0 is not in the confidence interval, a conclusion that there was a shift in consumption patterns is reasonable.

Question 3

Answer 3

False

Question 4

Question 5

The large sample confidence interval procedure for estimatingpp p₁?p₂ requires the random samples to be selected independently.

Question 6

One method of human locomotion that appears to have no function except to expressjoy is hopping, a skill usually learned in early and middle childhood. Each person ina random sample of female students and a random sample of male students at a largemiddle school was asked, "Approximately how long ago did you last spontaneouslyhop?
" The total sample sizes and the numbers who responded "Less than 1 weekago" are given in the table below:
a) Construct a 95% confidence interval for the difference in the populationproportions of males and the proportion of females at this school who wouldrespond "2-3 weeks."
b) Referring to your work in part (a), does there appear to be a difference betweenthe population proportions of men and women responding "2-3 weeks?
" Providestatistical justification for your answer.

Question 7

The large sample confidence interval procedure for estimating using z can safely be used if the combined samples have at least 20"successes" and 20 "failures."

Question 8

Question 9

Question 10

Question 11

One method of human locomotion that appears to have no function except to expressjoy is skipping, a skill usually learned in early and middle childhood. Each person ina random sample of female students and a random sample of male students at a largemiddle school was asked, "Approximately how long ago did you last spontaneouslyskip?
" The total sample sizes and the numbers who responded "Less than 1 weekago" are given in the table below: a) Construct a 95% confidence interval for the difference in the proportions of malesand the proportion of females at this school who would respond "Less than 1week."b) Referring to your work in part (a), does there appear to be evidence of a differencein the population proportions of males and females at this school who wouldrespond "Less than 1 week?
" Provide appropriate statistical justification for youranswer.

Question 12

Suppose that you have estimated the difference between two population proportions,p₁?p₂ , using a confidence interval. What are the appropriate interpretations of thefollowing results?
a) Both endpoints of the confidence interval for are positive.
b) Both endpoints of the confidence interval for are negative.
c) Zero is included in the confidence interval.

Question 13

The pre-historic development of agriculture is an active area of research amonganthropologists. The transition from a hunter-gatherer society to an agriculturalsociety is generally assumed to be signaled by an increase in cavities in teeth.Investigators recently studied teeth unearthed in two societies in Colorado:Basketmakers, societies living in small villages, and Post-Basketmakers, societiesliving in above-ground architectures such as pueblos. The investigators wereinterested in whether the shift in shelter was associated with a shift in consumptionfrom hunting (meat) to agriculture (maize). Data on the numbers of individuals withinterproximal (between teeth) cavities are shown in the table below. Inter-proximalcavities tend to occur more among populations subsisting primarily on agriculturalproducts. You may assume that it is reasonable to regard these two samples asrepresentative of teeth from these two populations.
a) Construct and interpret a 90% confidence interval for the difference in populationproportions of individuals with interproximal cavities, Basketmaker - Post-Basketmaker.
b) Interpret the 90% confidence level
c) Based on your results in parts (a) and (b), would you say that a shift inconsumption patterns is a reasonable conclusion on the part of the investigators?
Justify your response with statistical evidence from parts (a) and/or (b).

Question 14

The formula for the standard error of used in hypothesis testingand confidence intervals is the same.

Question 15

Question 16