Deck 13: Asking and Answering Questions About the Difference Between Two Means

Question 1

In an introductory marketing class students were presented with 6 items they couldbid on in an auction. They were asked to bid privately and also estimate the "typical"bid for each item by their classmates. The items were randomly selected from a largelist of items that students might purchase. An initial analysis of the data establishedthe plausibility that the distribution of differences (estimated - actual) isapproximately normal.
a) Construct a 95% confidence intervalfor the mean difference between theactual bid and the estimated"typical" bid for the population ofitems. b) Do the data indicate that the meandiffers for the actual and estimated"typical" bids?
Provide anappropriate statistical justificationusing your response in part (a).

Answer 1

a)
b) There is sufficient evidence of a difference in the mean actual and estimated bids; 0 is not in the confidence interval.

Question 2

The number of degrees of freedom associated with the two-sample t test isthe same as the number of degrees of freedom associated with the paired ttest statistic.

Answer 2

False

Question 3

When wildlife biologists study small animals, the animals are trapped and thenanesthetized to prevent discomfort to the animal. A study of the effect of theanesthetic, Isoflurane, on eastern gray squirrels (Sciurus carolinensis) resulted in thepulse data presented below. The biologists were interested in comparing the effectsof Isoflurane in two different seasons, winter and summer.
An initial analysis of the data revealed that it was reasonable to assume thedistribution of pulses for each season is approximately normal. It was also judged tobe reasonable to regard these samples as representative of the eastern gray squirrelpopulation.
a) Test the hypothesis of no difference in eastern gray squirrel mean pulse for winterand summer.
b) Do the data indicate that the mean pulse differs for the two seasons?
Provide anappropriate statistical justification using your response in part (a).

Answer 3

a)
The problem states that the approximate normality of pulses is plausible; the t-procedure is appropriate.
b) Since the P-value is < .05 we have sufficient evidence for a difference in summer and winter mean pulse in eastern gray squirrels.

Question 4

Question 5

Male and female Downy Woodpeckers (Picoides pubescens)
The data in the table at right are the bill lengths of 12 male and12 female randomly selected Downy Woodpeckers caught andreleased in a banding survey. The investigators would like toknow whether these data provide evidence that the male andfemale Downy Woodpeckers differ in mean bill length. Aninitial analysis of the data established the plausibility that thedistributions of bill lengths are approximately normal.
a) Construct a 95% confidence interval for the difference in mean bill length formales and for females.
b) Do the data indicate that the mean bill length differs for males and females?
Provide an appropriate statistical justification using your response in part (a).

Question 6

A number of butterfly species mate for hours, and if a mating couple is disturbed, oneof the butterflies is responsible for flying, carrying its partner with it. Not only aremating pairs more noticeable to predators, the added weight may hamper the flightduring escape. Random samples of Green-veined White (Pieris napi) butterflieswere the subjects of a Swedish study to investigate the escape flights of singlebutterflies and of mating pairs when exposed to a predator. Data on the initial takeoffangle of escape are presented below. The investigators considered performing ahypothesis test to determine if there was evidence that the mean take off angle wasdifferent for singles and pairs. Using a graphical procedure of your choice,determine if the t-test is appropriate.

Question 7

A number of butterfly species mate for hours, and if a mating couple is disturbed, oneof the butterflies is responsible for flying, carrying its partner with it. Not only aremating pairs more noticeable to predators, but the added weight may hamper theflight during escape. Random samples of Green-veined White (Pieris napi)butterflies were the subjects of a Swedish study to investigate the escape flights ofsingle butterflies and of mating pairs when exposed to a predator. Data on the initialtakeoff escape velocities are presented below. The investigators consideredperforming a hypothesis test to determine if there was evidence that the mean take offescape velocity was different for singles and pairs. Using a graphical procedure ofyour choice, determine if the t-test is appropriate.

Question 8

When analyzing paired data the population distribution of differences, notthe separate population distributions, should be credibly normal.

Question 9

The estimated standard deviation of used in hypothesis tests about is the same as the estimated standard deviation used in calculatingconfidence intervals for

Question 10

In many animal species the males and females differ slightly in structure, coloring,and/or size. The hominid species Australopithecus is thought to have lived about 3.2million years ago. ("Lucy," the famous near-complete skeleton discovered in 1974, isan Australopithecus.) Forensic anthropologists use partial skeletal remains toestimate the mass of an individual. The data below are estimates of masses frompartial skeletal remains of this species found in sub-Saharan Africa. Appropriategraphical displays of the data indicate that it is reasonable to assume that thepopulation distributions of mass are approximately normal for both males andfemales. You may also assume that these samples are representative of the respectivepopulations. Do these data provide convincing evidence that the mean estimated masses differ forAustralopithecus males and females?
Provide appropriate statistical justification foryour conclusion.

Question 11

Two samples are said to be independent when the selection of theindividuals in one sample has no bearing on the selection of those in theother sample.

Question 12

"Tail-chasing" by dogs is an anxiety disorder characterized by circling behavior withthe dog's attention directed toward its tail. There may be many reasons for tail-chasing behaviors. To investigate the potential for biochemical causes, a study wasperformed at a small animal clinic at a university. Blood samples were taken fromrandom sample of 15 dogs brought to the clinic by owners worried about the tail-chasing behaviors of their dogs. A control group consisting of a random sample of 15dogs brought to the clinic for other reasons contributed blood samples with theowner's permission. The mean lipoprotein density for the tail-chasing group was 12mg/dl, and the standard deviation was 2.24 mg/dl. The corresponding statistics forthe control group were 13 mg/dl and 1.73 mg/dl. There was no indication ofskewness in either sample distribution.a) Is there sufficient evidence to conclude that there is a difference in meanlipoprotein density in these two samples?
Provide statistical justification for yourresponse.b) Irrespective of your response in part (a), consider the design of this study. Woulda statistically significant difference in mean lipoprotein density be sufficient tomake a case that lipoprotein density is the cause of tail-chasing behavior in dogs?
Why or why not?

Question 13

"Tail-chasing" by dogs is an anxiety disorder characterized by circling behavior withthe dog's attention directed toward its tail. There may be many reasons for tail-chasing behaviors. To investigate the potential for biochemical causes, a study wasperformed at a small animal clinic at a university. Blood samples were taken from arandom sample of 15 dogs brought to the clinic by owners worried about the tail-chasing behaviors of their dogs. A control group consisting of a random sample of 15dogs brought to the clinic for other reasons contributed blood samples with theowner's permission. The mean triglyceride level for the tail-chasing group was 68mg/dl, and the standard deviation was 19.36 mg/dl. The corresponding statistics forthe control group were 61 mg/dl and 11.62 mg/dl. Both sample distributions wereapproximately symmetric.
a) Is there convincing evidence of a difference in mean triglyceride level in thesetwo populations?
Provide statistical justification for your response.
b) Irrespective of your response in part (a), consider the design of this study. Woulda statistically significant difference in mean triglyceride levels be sufficient tomake a case that triglyceride level is a cause of tail-chasing behavior in dogs?
Why or why not?

Question 14

Researchers usually test the hypothesis with an alternativehypothesis of 0rather than using a one-sided alternative.

Question 15

The Amazon River basin includes the largest remaining connected area of tropicalforest in the world, and ecologists regularly monitor its health. One aspect of itshealth is the density of different indigenous species. The Amazon basin includes bothareas that are periodically flooded, and areas that are not periodically flooded. If thedensities of species differ in the flooded and unflooded regions, researchers will wantto take stratified samples in their studies. Stratified sampling would be moreexpensive, since boats must be rented to get to any randomly selected flooded sites.The data below are measures of density (individuals per square kilometer) of primatesin random samples of flooded and unflooded areas in the Amazon basin. a) Do these data provide convincing evidence of a difference in the primate densityin flooded and unflooded forests in Amazonia?
Provide appropriate statisticaljustification for your conclusion.b) Based on your results in part (a), would you recommend using a stratified sample?
Justify your recommendation with a statistical argument.

Question 16

Generally speaking, the "pooled" t is preferred if population variances areunequal.

Question 17

When an alternative hypothesis is 0he P-value is found bydoubling the appropriate area under the t curve.

Question 18

For both large and small samples the estimated standard deviation of

Question 19

For paired samples are always numerically equal.

Question 20

The P-value for a hypothesis test is always found by calculating areasunder the z curve.

Question 21

In a study of captive nectar-feeding bats (Leptonycteris sanborni), data were gatheredon nectar intake over a five-minute flight period. The bats were randomly selectedfor the study. Investigators are interested in the differences in weight gain for malesand females. The bats were weighed before and after feeding, and the relative weightgain was calculated for each animal using the formula: Consider constructing a 95% confidence interval for the population difference in meanweight gains for male and female bats.a) Using a graphic display of your choice, show that it is appropriate to use the tprocedure to construct a confidence interval.b) Calculate and interpret the 95% confidence interval in the context of the problem.c) Do you feel these results can be generalized to non-captive bats?
Why or whynot?

Question 22

Inferences about the difference between two means fall into two categories: thesamples are independent, or the samples are paired.
a) What considerations would lead you to use the techniques for independentsamples rather than those for paired samples?
You may use examples toillustrate your ideas, but examples alone are not sufficient.
b) How do the analyses of independent samples and paired samples differ?
In yourresponse, consider the hypotheses, methods, assumptions, and calculations.

Question 23

After an outbreak of a drug-resistant strain of bacteria (Enterococci faecium), hospitalofficials became concerned that their alcohol-based handrub (ABHR) hand hygieneprogram was not sufficient to prevent spreading this bacteria. The officials solicited20 volunteers to assess the effectiveness of ABHR. The volunteers' hands werecontaminated with E. faecium. After gathering baseline data on the amount ofbacteria present they performed the recommended hand hygiene according to theWorld Health Organization protocol. The amount of bacteria present was thenassessed again. Summary measures and boxplots of the baseline sample, the post-ABHR sample, and the reduction in the amount of bacteria ( log10 (bacteria / mL) ) arepresented below. Do these data provide sufficient evidence that the ABHR iseffective against the E. faecium?

Question 24

After an outbreak of a drug-resistant strain of bacteria (Enterococci faecium), hospitalofficials became concerned that their alcohol-based handrub (ABHR) hand hygieneprogram was not sufficient to prevent spreading this bacteria. The officials solicited20 volunteers to assess the effectiveness of ABHR. The volunteers' hands werecontaminated with E. faecium. After gathering baseline data on the amount ofbacteria present they performed the recommended hand hygiene according to theWorld Health Organization protocol. The amount of bacteria present was thenassessed again. Summary measures and boxplots of the baseline sample, the post-ABHR sample, and the reduction in the amount of bacteria ( log10 (bacteria / mL) ) arepresented below. Do these data provide sufficient evidence that the ABHR iseffective against the E. faecium?

Question 25

When wildlife biologists study small animals, the animals are trapped and thenanesthetized to prevent discomfort to the animal. A study of the effect of theanesthetic, Isoflurane, on Allegheny woodrats (Neotoma magister) resulted in theheartbeat data presented below. The biologists were interested in comparing theeffects of Isoflurane in two different seasons, winter and summer.
An initial analysis of the data revealed that it was reasonable to assume thedistributions of heartbeats for both seasons are approximately normal. It was alsojudged to be reasonable to regard these samples as representative of the Alleghenywoodrat population.
a) Test the hypothesis of no difference between woodrat mean heartbeat rates forwinter and summer.
b) Do the data indicate that the mean heartbeat rates differ?
Provide an appropriatestatistical justification using your response in part (a).

Question 26

Inferences about the difference between two means fall into two categories: thesamples are independent, or the samples are paired.
a) What considerations would lead you to use the techniques for independentsamples rather than those for paired samples?
You may use examples toillustrate your ideas, but examples alone are not sufficient.
b) How do the analyses of independent samples and paired samples differ?
In yourresponse, consider the hypotheses, methods, assumptions, and calculations.

Question 27

When stalking gazelles, cheetah frequently have a choice between two gazelles closeto each other while grazing. A biologist thought the choice of prey might be affectedby the "vigilance" behavior of the gazelles. She defined vigilance as the percentageof the time that a gazelle had its head in the air searching for potential predators. Shefilmed cheetah stalks and analyzed 16 incidents where two same-sex gazelles werewithin 5 meters of each other; thus, either could have been chosen as the cheetah's prey. The table below presents the vigilance levels for each of the gazelles and thedifference (gazelle chased - gazelle ignored) for each pair.
a) Using the scales below, construct comparative dotplots to show that it isreasonable to use the t-procedure to construct confidence intervals for thedifference in population means for males and females.
b) Calculate and interpret the 95% confidence interval in the context of the problem.c) The investigator noticed that many more male pairs than female pairs wereactually stalked by cheetah. Two theories have been proposed for this difference.The first theory is that the gazelle females are generally more vigilant than males.The second theory is that females generally graze near the centers of the herds,protecting the young, and are less accessible to predators.i) Is it possible to use investigator's data be used to support or refute the theorythat females are more vigilant than males?
Is so, how?
If not, why not?
ii) Is it possible to use investigator's data be used to support or refute the theorythat females generally graze near the centers of the herds?
Is so, how?
If not,why not?