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

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

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.

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).

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

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.

For both large and small samples the estimated standard deviation of

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?

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).

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

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?

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).

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?

"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?

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.

For paired samples are always numerically equal.

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.

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

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.

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

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.