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Exam 11: Comparing Two Populations or Treatments

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

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One method of human locomotion that appears to have no function except to express joy is skipping, a skill usually learned in early and middle childhood. In surveys gathered over a 3-year period, University students were asked, "Approximately how long ago did you last spontaneously skip?" The total sample sizes and the numbers who responded "Less than 1 week ago" are given in the table below: ​ a) Construct a 95% confidence interval for the difference in the population proportions of males and females responding "Less than 1 week." Assume that it is reasonable to regard these samples as representative of the corresponding populations. b) Does there appear to be a difference between the proportions of men and women responding "Less than 1 week?" Provide statistical justification for your answer.

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a) The distribution of differences in th...

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Body fat and lean body mass can be estimated in living animals by measuring the total body electrical conductivity (TOBEC). This technique is useful when attempting to determine if "diet drugs" are working on laboratory rats over time--the researchers need not sacrifice the animals to measure the amount of adipose body fat after periods of drug usage. However, the procedure requires that the animal be totally inside the measurement chamber, which is fairly small. Some rats are so big that their tails must be tucked under their bodies before putting them into the measurement chamber. This is causing concern among the researchers, because there is a possibility the tail position might alter the measurements. To see if the TOBEC measurement is altered by the tail position of the rats, an experiment was run on 16 rats ranging in weight from 210 to 505 g. The rats were randomly assigned the order of measurement, with 8 rats measured in the tail-under position first, and 8 rats in the tail-extended position. The data for the body fat measures, and the differences, are presented in the table below. ​ a) Using graphical display(s) of your choice show that the assumptions necessary for the paired t-test are plausible. b) Test the hypothesis that there is no difference between the population mean TOBEC body fat measurements of rats in the tail-under vs. the tail-extended positions. For purposes of the statistics you may assume that these rats are a random sample of laboratory rats. c) Write a short paragraph based on your analysis above, explaining your results for laboratory technicians. Your paragraph should advise them whether or not it is necessary to make sure the tail positions of the rats are the same when replicating the lean mass body measurements during the experiment.

One of the most striking human accomplishments is the achievement of intelligible speech by age five. Deficits in speech acquisition often occur when a child has hearing difficulty or does is slow to develop the motor functions associated with speech. Other deficits in speech acquisition are of unknown origin. It has been hypothesized that a family history of speech communication disorder might be a factor in speech delay. The theory is that if members of the family have difficulty communicating, this may increase the probability of a speech deficit in a young family member. A random sample of healthy two-year-olds was followed over the course of a year. At age 3, each child was classified as having a speech delay or not. These data, cross-classified by a family history of speech communication disorder, is given in the table below: ​ a) At the .05 significance level, do the data provide evidence of the hypothesized developmental differences? Provide statistical evidence for your conclusion. b) Is this an observational or experimental study? Justify your response in a few sentences.

In many animal societies, a role assumed by the adult male is a sentry, warning adult females and juveniles of approaching predators. Some students of animal behavior believe that in human society this role is active also. An interesting variation of this theory is that the human female accepts this role. In one 4-month study in Seattle, an investigator observed people crossing the street at an intersection with no traffic control. The observed behavior was the scanning behavior ("looking both ways") of adult males and females. The investigator reasoned that if the adult female accepted the protection of the adult male, the scanning rate when alone would be greater than when accompanied by an adult female. Data from this study is given below. Do these data support the investigator's hypothesis that the female scanning rate would be less when accompanied by a male? Assume that it is reasonable to regard these samples as representative of the corresponding populations. Provide statistical justification for your answer.

Assume you obtained a MINITAB output and a boxplot of differences between the number of fish at a depth of 5 meters and 10 meters. ​ ​ ​ Choose the correct statement. ​

When testing hypotheses about and constructing confidence intervals for μ₁ − μ₂, we utilize the statistic, . What are the mean and variance of the sampling distribution of ?

Suppose that you wish to compare two treatments for preventing baldness in men. (The measure of baldness you will use is the mean number of healthy hair follicles per square cm.) In some circumstances, you may be able to randomly assign the treatments to the men, and in some circumstances, you may not be able to randomly assign the treatments to the men. Respond to the following questions in a few sentences. a) If you are testing the hypothesis that the means are equal for the population, how would your statistical procedures differ, if at all, depending on whether you randomly assigned subjects to treatments or whether you merely observed differences in men who chose one treatment or the other? b) How would your interpretation of the results differ, if at all, depending on whether you randomly assigned subjects to treatments or whether you merely observed differences in men who chose one treatment or the other?

When testing hypotheses about and constructing confidence intervals for μ₁ − μ₂, we utilize the statistic, . What are the mean and standard deviation of the sampling distribution of ?

Viruses are infectious agents that often cause diseases in plants. Different viruses have different potency levels, and this fact can be used to detect whether a new virus is infecting plants in the field. In a potency comparison experiment, two viruses were placed on a tobacco leaf of 10 randomly selected tobacco plants in a field. The viruses were randomly assigned to one-half of each of the leaves. The table below presents the potency of the viruses, as measured by the number of lesions appearing on the leaf half. Test the hypothesis that there is no difference in mean number of lesions for Virus X and Virus Y.

Suppose you need to estimate a difference in mean systolic blood pressures between men and women. You've constructed a 95% confidence interval for this test and obtained the lower limit of −25.07 and the upper limit of 6.47. What conclusion can you make? ​

For two independent samples, .

An article proving effectiveness of a new hair growth shampoo was published on a producer's website. The producer states that two samples were compared, both consisted of 35 subjects. One group used the new shampoo, while the other used a shampoo without the components that activate hair growth. Both groups used the products three times a week. After a month, hair growth was measured for both samples. The producer declares that their shampoo significantly increases hair growth, because the P-value was less than 0.001 in a paired t test at a 0.01 significance level. The article also provided the following information on hair growth for the two experimental groups. ​ ​ Is the producer's statement about the effectiveness of the product correct? ​

The teacher noticed that 15 out of 21 girls read more than 90 words per minute by the end of the second grade, while only 8 out of 18 boys showed the same result. The null hypothesis is that the proportions of girls and boys who read more than 90 words per minute are equal. Calculate the two-sided P-value and make the following conclusion at α = 0.05 using the output of the Shiny app "Randomization Test for Two Proportions". ​

The home range of an animal is the average area an animal occupies while foraging for food and defending its territory. It is thought that home ranges of animals usually do not change much, except when an area is under ecological stress. As part of a study of white-tailed deer in Florida in 1991 and 1992, deer were radiocollared and their movements followed over the course of a year. The home range data for a random sample of 18 white-tailed deer are shown below in Table 1. The raw area of the home range for each of the deer is reported in hectares for the years 1991 and 1992. (A hectare is a metric unit of area equal to 2.471 acres.) The investigators are interested in determining whether the home range white-tailed deer change over the course of as little time as a year. ​ a) Using graphical display(s) of your choice show that the assumptions necessary for determining a change in the mean home ranges are plausible. b) Construct a 95% confidence interval for the difference in means of the home ranges from 1991 to 1992. c) Do the data provide evidence of a change in the size of the home ranges between 1991 and 1992? Provide statistical justification for y.

The teacher noticed that 13 out of 20 girls read 90 words per minute by the end of the second grade, while only 9 out of 17 boys showed the same result. ​ Interpret the following output produced by the Shiny app "Bootstrap Confidence Interval for Difference in Two Proportions". ​ ​

Are the given hypotheses essentially the same? People at age 30 have less free time than they had at 20. and 30-year old people have less free time than 20-year old people.

The perception of danger is an important characteristic for survival of animals. In a field experiment in Costa Rica, investigators located and directly approached black iguanas; that is, they walked straight towards them. Two treatments were randomly assigned to the individual iguanas. In one treatment the investigator gazed at the iguana while approaching, "maintaining eye contact." In a second treatment, the investigator did not gaze at the iguana while approaching. The outcome measure was the distance of the investigator from the iguana when it decided to run away. The researchers believe that eye contact is noticed by the iguana, leading to a longer approach distance. Data from this experiment appears in the table below. ​ a) Using a graphical display of your choosing, assess the assumption that the distributions of approach distances are approximately normal. State your conclusion in a few sentences. b) Assuming that it is OK to proceed with a two-sample t procedure, determine if there is sufficient evidence to conclude that there is a shorter mean approach distance for the "Eye contact" group. c) In a few sentences, state any concerns you have about your conclusions in part (b), based on your results from part (a). If you have no concerns, write "No concerns."

When comparing two population means, it is important to know if the investigation was an experiment where the researcher was able to control which subject gets what treatment, or if the study was observational. Suppose that a statistically significant result is obtained in a study. How would the interpretations of the results differ, based on whether the study was observational or the study was an experiment?

When comparing two population means, it is important to make the distinction between independent samples and paired samples. What makes two samples independent?