Math 121 Chapter 1 Questions fully solved & verified for accuracy(A+graded)

The study of statistics rests on what two major​ concepts?

Variation and data

The circles shown to the right are​ similar, but not exactly the same. This is an example of​ _______.

variation.

Data are more than just​ numbers, because data have​ _____.

context.

Two sections of statistics are​ offered, the first at 8 a.m. and the second at 10 a.m. The 8 a.m. section has 25​ women, and the 10 a.m. section has 15 women. A student claims this is evidence that women prefer earlier statistics classes than men do. What information is missing that might contradict this​ claim?

The percentage of female students in the two classes is unknown. There may be more females in the 8 a.m. because there are more students in the 8 a.m. class than the 10 a.m. class. This claim could be true only if the classes were the same size.

In​ 2008, a highway safety administration reported that the number of pedestrian fatalities in City A was 65 and that the number in City B was 45. Can we conclude that pedestrians are safer in City​ B? Why or why​ not?

No, in order to compare the fatalities the statistics must include the number of fatalities per pedestrian. There may be fewer pedestrians in City B causing the difference.

A student watched picnickers with a large cooler of soft drinks to see whether teenagers were less likely than adults to choose diet soft drinks over regular soft drinks.

The study is

an observational study.

Records of patients who have had broken ankles are examined to see whether those who had physical therapy achieved more ankle mobility than those who did not.

This is an observational study. Since the researchers did not assign subjects to the control or treatment group​ beforehand, they did not satisfy a key feature of controlled experiments.

A group of boys is randomly divided into two groups. One group watches violent cartoons for one​ hour, and the other group watches cartoons without violence for one hour. The boys are then observed to see how many violent actions they take in the next two​ hours, and the two groups are compared.

The study is

a controlled experiment.

A local public school​ encourages, but does not​ require, students to wear uniforms. The principal of the school compares the grade point averages of students at this school who wear uniforms with the GPAs of those who do not wear uniforms to determine whether those wearing uniforms tend to have higher GPAs.

This is an observational study. The principal does not randomly assign students to either wear or not wear uniforms. Random assignment is essential to conducting a controlled experiment.

A group of educators want to determine how effective tutoring is in raising​ students' grades in a math​ class, so they arrange free tutoring for those who want it. Then they compare final exam grades for the group that took advantage of the tutoring and the group that did not. Suppose the group participating in the tutoring tended to receive higher grades on the exam. Does that show that the tutoring​ worked? If​ not, explain why not and suggest a confounding variable.

Because this was an observational​ study, it only shows an​ association; it does not show that the tutoring worked. It could be that more motivated students attended the tutoring and that was what caused the higher grades.

a. The doctor is concerned that if his most severely depressed patients do not receive the​ antidepressants, they will get much worse. He therefore decides that the most severe patients will be assigned to receive the antidepressants. Explain why this will affect his ability to determine which approach works best.

If the doctor decides on the​ treatment, this could introduce bias.

b. What advice would you give the doctor to improve his​ study?

The doctor should randomly assign the patients to the different treatments.

c. The doctor asks you whether it is acceptable for him to know which treatment each patient receives and to evaluate them himself at the end of the study to rate their improvement. Explain why this practice will affect his ability to determine which approach works best.

If the doctor is aware of the treatment each patient​ receives, that might influence his opinion about the effectiveness of the treatment.

d. What improvements to the plan in part​ (c) would you​ recommend?

To prevent​ bias, the experiment should be​ double-blind. Neither the patients nor the doctor evaluating the patients should know whether each patient received medication.

a. Was the study a controlled experiment or an observational​ study? Explain how you know.

The study was a controlled experiment because the children were randomly assigned to either surgery or constant watch. This is essential to conducting a controlled experiment.

b. Assuming that the study was properly​ conducted, can we conclude that the early surgery caused the​ improvements? Explain.

We can conclude that the early surgery caused the improvements because it was a randomized controlled experiment.

Some people believe that wearing copper bracelets is a good treatment for arthritis of the hand. To test this​ belief, suppose you recruit 100 people and supply them all with copper bracelets. After the patients wear the bracelets for a​ month, you ask them whether or not their pain is less than it was before they began wearing the bracelets. Explain how to improve this study.

To improve the​ study, the patients should be randomly divided into two​ groups; one group will be given the copper​ bracelets, and the other group will be given​ non-copper bracelets. After a​ month, the patients will be surveyed on the levels of their pain.

A study concludes that the use of pesticides is associated with the development of​ Parkinson's disease, a neurological disease that causes people to shake. The study reported that exposure to bug killers and weed killers is​ "associated with" an increase of​ 33% to​ 80% in the chances of getting​ Parkinson's. Does this study show that pesticides cause​ Parkinson's disease? Why or why​ not?

The study does not show that pesticides cause​ Parkinson's disease. This was an observational study because researchers could not have deliberately exposed people to pesticides. Observational studies cannot conclude causation.

A study was done to see whether a smaller dose of flu vaccine could be used successfully. In this​ study, the usual amount of vaccine was injected into half the​ patients, and the other half of the patients had only a small amount of vaccine injected. The response was measured by looking at the production of antibodies. In the​ end, the lower dose of vaccine was just as effective as a higher dose for those under 65 years old. What more do we need to know to be able to conclude that the lower dose of vaccine was equally effective at preventing the flu for those under​ 65?

The patients need to be randomly assigned the full or lower dose. Without randomization there could be​ bias, however, with randomization we can infer causation.

A study was conducted to see whether participants would ignore a sign that​ said, "Elevator may stick between floors. Use the​ stairs." Those who used the stairs were said to be​ compliant, and those who used the elevator were said to be noncompliant. There were three possible​ situations, two of which involved confederates. A confederate is a person who is secretly working with the experimenter. In the first​ situation, there was no confederate. In the second​ situation, there was a compliant confederate​ (one who used the​ stairs), and in the third​ situation, there was a noncompliant confederate​ (one who used the​ elevator). The subjects tended to imitate the confederates. What more do you need to know about the study to determine whether the presence or absence of a confederate causes a change in the compliance of​ subjects?

Identify whether there was random assignment to groups. Without random assignment there is the possibility of​ bias, so we cannot infer causation.

Identify the sample size of the study. Without enough participants to observe the full range of variability in subjects we cannot control for other relevant​ factors, so we cannot infer causation.

A study reported on the effects of vitamin C in breast milk for​ breast-feeding mothers. The children whose mothers had chosen to take high doses of vitamin C had a​ 30% lower risk of developing allergies. Can you conclude that the use of vitamin C caused the reduction in​ allergies? Why or why​ not?

You cannot conclude that the use of vitamin C caused the reduction in allergies because the researchers did not randomly assign mothers to treatment and control groups. This step is necessary for identifying causation.

A college magazine suggested that overeating reduces brain function. Is this likely to be a conclusion from observational studies or randomized​ experiments? Can we conclude that overeating causes a reduction in brain​ function? Why or why​ not?

This is likely to be from observational studies. It would not be ethical to assign people to overeat. We cannot conclude causation from observational studies because of the possibility of confounding factors.