Question 1

p₁ − p₂ is a biased estimator of π₁ − π₂.

Accepted Answer

An estimator is biased if its expected value does not equal the true parameter it is estimating. The notation p₁ and p₂ typically represents sample proportions, and π₁ and π₂ represent population proportions. The difference between two sample proportions (p₁ - p₂) is an unbiased estimator of the difference between two population proportions (π₁ - π₂) because the expectation of the difference is equal to the difference of the expectations.

Question 2

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 ?

Accepted Answer

The mean of the sampling distribution of  is  , the standard deviation is  .

Question 3

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?

Accepted Answer

If a study is not such that the treatments are randomly assigned, then no causal conclusions can be made. Therefore, even if two samples or treatments have a statistically significant difference, it will not be possible to decide that the two populations are different or that the two treatments have a different effect.

Question 4

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."

Accepted Answer

The answer of The perception of danger is an important...

Question 5

The number of degrees of freedom of the two-sample t test are the same as the degrees of freedom for the paired t test statistic.

Accepted Answer

The two-sample t test and paired t test have different formulae for calculating the degrees of freedom, and therefore have different degrees of freedom. In the two-sample t test, the degrees of freedom is calculated as the sum of the individual degrees of freedom for each sample, while in the paired t test, the degrees of freedom is calculated as the sample size minus 1.

Question 6

Comparison of means focuses on the difference between two population or treatment means, μ₁ − μ₂. When testing the hypothesis, H₀ : μ₁ − μ₂ = 0, there are three possible alternative hypotheses. a) Using correct statistical notation, list these three alternative hypotheses. b) For each alternative hypothesis in part (a), give an example of a situation from your own experience or your reading where one might use that alternative hypothesis. Be sure to clearly specify a context, define your variables, and state the null and alternative hypothesis.

Accepted Answer

The answer of Comparison of means focuses on the difference...

Question 7

The standard deviation of p₁ − p₂ used in the large sample test of π₁ − π₂ is the same as the standard deviation used in the large sample confidence interval for π₁ − π₂.

Accepted Answer

In hypothesis testing, a pooled standard deviation is used, while in confidence intervals, the standard deviations of the individual samples are used.

Question 8

The P-value of an upper-tail t test is the area to the right of the calculated t value on the appropriate t curve.

Accepted Answer

The P-value of an upper-tail t test is the area to the right of the calculated t value on the appropriate t curve.

Question 9

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

Accepted Answer

The answer of When comparing two population means, it is...

Question 10

The large sample z test for μ₁ − μ₂ can be used as long as at least one of the two sample sizes, n₁ and n₂, is greater than or equal to 30.

Accepted Answer

The large sample z test for the difference between two means (μ1 - μ2) assumes that both sample sizes (n1 and n2) are large (typically n ≥ 30) to ensure the sampling distribution of the difference is approximately normal, not just one of the samples.

Question 11

As part of the iguana risk assessment experiment described in the previous problem, investigators replicated the experiment with one difference--the investigators approached the iguanas at a tangent, passing by the iguana with a closest distance being 2 meters. Summary statistics for the approach distances for the two treatments, Eye Contact and No Eye Contact, are given in the table below:  An initial analysis of the data revealed that it was reasonable to assume the population approach distances at first run were both approximately normal. 
a) Construct a 95% confidence interval for the difference in approach distances for the eye contact and no eye contact treatments. 
b) Do the data indicate that for a tangential approach, the population means differ? Justify your answer statistically.

Accepted Answer

The answer of As part of the iguana risk assessment...

Question 12

For two independent samples,  .

Accepted Answer

The answer of For two independent samples,  ....

Question 13

Researchers have hypothesized that female Downy Woodpeckers avoid the feeding areas of socially dominant males; that is, the males chase them away from prime spots. An alternative opinion is that there are important physical characteristics of males and females that might lead them to choose different foraging locations. One such characteristic could be the bill length of the males and females; it may be that longer bills let one gender or the other drill deeper into a tree and thus get more food per tree. The data in the table below are the bill lengths of 12 male and 12 female randomly selected Downy Woodpeckers caught and released in a banding survey. The investigators would like to know whether these data provide evidence that the males and females differ in bill size.  ​ 
a) Using a graphical display of your choosing, assess the plausibility of the assumption that the distributions of bill lengths 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 difference in mean bill length for males and females. 
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."

Accepted Answer

The answer of Researchers have hypothesized that female Downy Woodpeckers...

Question 14

In 1992 Hurricane Andrew struck Florida causing widespread devastation. The eye of Hurricane Andrew passed directly over a site where scientists were studying the ecology of white-tailed deer. The deer had been radio-collared prior to Andrew's appearance, and the hurricane provided an opportunity to study the effects of an awesome storm on the deer. The investigators felt that the home ranges of the deer would change, but were unsure of the direction of the change. (The home range is the average area an animal occupies while foraging for food and defending its territory.) Home ranges of animals usually do not change much unless an area is under ecological stress. The home range data for a random sample of 18 white-tailed radio-collared 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 pre-hurricane year of 1992 and the post hurricane year of 1993. (A hectare is a metric unit of area equal to 2.471 acres.)  ​ 
a) Using graphical display(s) of your choice show that the assumptions necessary for determining any change in the mean home ranges are plausible. 
b) Construct a 95% confidence interval for the difference in means of the home ranges from before Andrew to after Andrew. 
c) Do the data provide evidence of a change in the size of the home ranges after Hurricane Andrew? Provide statistical justification for your response.

Accepted Answer

The answer of In 1992 Hurricane Andrew struck Florida causing...

Question 15

As part of the foraging behavior assessment described in the previous problem, investigators also measured the bill depths of the male and female Downy Woodpeckers. Summary statistics for these measures are given in the table below:  An initial analysis of the data revealed that it was reasonable to assume the bill depths for both sexes are approximately normal. 
a) Construct a 95% confidence interval for the difference in bill depths of the males and females. 
b) Do the data indicate that the bill depths differ? Justify your answer statistically.

Accepted Answer

The answer of As part of the foraging behavior assessment...

Question 16

The number of degrees of freedom used in the two-sample t test for independent samples are the same as the degrees of freedom used in the construction of a confidence interval for μ₁ − μ₂.

Accepted Answer

This is true. The degrees of freedom used in the two-sample t test for independent samples and in the construction of a confidence interval for μ₁ − μ₂ are both equal to the sum of the degrees of freedom for each sample, which is (n1-1) + (n2-1).

Question 17

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.

Accepted Answer

The answer of Body fat and lean body mass can...

Question 18

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.

Accepted Answer

Independent samples are selected randomly without influence or regard for the individuals selected in the other sample.

Question 19

For both large and small samples the estimated standard deviation of  is calculated the same way.

Accepted Answer

The estimated standard deviation is calculated using the same formula for both small and large samples, which is the square root of the sum of the squared deviations from the mean divided by the sample size minus one. This formula is not affected by the size of the sample.

Question 20

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 ?

Accepted Answer

The answer of When testing hypotheses about and constructing confidence...

Quiz 11: Comparing Two Populations or Treatments

p1 − p2 is a biased estimator of π1 − π2.

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 ?

The number of degrees of freedom of the two-sample t test are the same as the degrees of freedom for the paired t test statistic.

The standard deviation of p1 − p2 used in the large sample test of π1 − π2 is the same as the standard deviation used in the large sample confidence interval for π1 − π2.

The P-value of an upper-tail t test is the area to the right of the calculated t value on the appropriate t curve.

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

The large sample z test for μ1 − μ2 can be used as long as at least one of the two sample sizes, n1 and n2, is greater than or equal to 30.

For two independent samples, .

The number of degrees of freedom used in the two-sample t test for independent samples are the same as the degrees of freedom used in the construction of a confidence interval for μ1 − μ2.

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.

For both large and small samples the estimated standard deviation of is calculated the same way.

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 ?

p₁ − p₂ is a biased estimator of π₁ − π₂.

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 ?

The standard deviation of p₁ − p₂ used in the large sample test of π₁ − π₂ is the same as the standard deviation used in the large sample confidence interval for π₁ − π₂.

The large sample z test for μ₁ − μ₂ can be used as long as at least one of the two sample sizes, n₁ and n₂, is greater than or equal to 30.

The number of degrees of freedom used in the two-sample t test for independent samples are the same as the degrees of freedom used in the construction of a confidence interval for μ₁ − μ₂.

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 ?