Statistics Chapter 10

The​ chi-square distribution provides a good approximation to the sampling distribution of the​ chi-square statistic if the expected count in each cell is​ _____.

5 or higher

In order to conduct a hypothesis test for homogeneity or​ independence, what must the expected count in each cell​ be?

5 or more

A poll asked people in a certain country what they considered to be the most important problem in their country today. The people were also classified by race. If we wanted to test whether there was an association between response to the question and race of the​ respondent, should we do a test of independence or of​ homogeneity?

A test of independence because there is only one sample.

Color is a (blank) variable and Stop is a (blank) variable.

Color is a categorical variable and Stop is a categorical variable.

What are controlled​ experiments?

Experiments in which the experimenters determine how subjects are assigned to treatment groups.

What are the two tests that can be used to determine if there is an association between categorical​ variables?

Homogeneity and independence

What do large values of the​ chi-square statistic​ indicate?

One should be suspicious of the null hypothesis.

Are tests of means used for numerical or categorical​ data?

Tests of means are used for numerical data.

Are tests of proportions used for categorical or numerical​ data?

Tests of proportions are used for categorical data.

The​ chi-squared statistic measures which of the​ following?

The amount by which the expected counts differ from the observed counts.

The table shows the country of origin and the percentage of​ foreign-born people in a certain country in 2000 and 2007 for the four countries of origin with the highest percentages. Why should a​ chi-square test not be used with these​ data? Country of Origin A B C D 2000 29.5 4.9 4.4 3.3 2007 30.8 5.1 4.5 3.9

The data are of the entire population​ (not a​ sample), and therefore there is no need for inference. The data are given as rates​ (percentages), not frequencies​ (counts), and there is not enough information for us to convert these percentages to counts.

Which of the following is not a characteristic of the​ chi-square distribution?

The distribution is skewed left

In January​ 2012, a study reported that​ "moderate consumption of coffee is associated with a lower​ risk" of stroke. Is this conclusion likely to be the result of an observational study or a controlled​ experiment? Is it saying that coffee lowers the risk of​ stroke?

The phrase​ "is associated​ with" means that it is not possible to infer​ causality, so it should be suspected that this is based on observational studies. The statement is not saying that coffee lowers the risk of stroke.

What is one drawback with​ chi-square tests?

The tests can reveal whether two variables are associated but not how they are associated.

The table shown to the right summarizes the outcomes of a study that students carried out to determine whether humanities students had a higher mean GPA than science students. Identify both of the​ variables, and state whether they are numerical or categorical. If​ numerical, state whether they are continuous or discrete.

The variable Field of Study is categorical. The variable Mean GPA is numerical and continuous.

There is a theory that relative finger length depends on testosterone level. The table shows a summary of the outcomes of an observational study that one of the authors carried out to determine whether men or women were more likely to have a ring finger that appeared longer than their index finger. Identify both of the​ variables, and state whether they are numerical or categorical. If​ numerical, state whether they are continuous or discrete.

The variable gender is categorical The variable ring finger longer or not is categorical

Fill in the blank below. ​______ is a short paragraph at the beginning of a research article that describes the basic findings.

an abstract

What are journal articles that were read by several knowledgeable and experienced researchers in the same fields before​ publication?

peer reviewed

A study was done to see whether participants would ignore a sign that​ read, "Elevator may stick between floors. Use the​ stairs." The people who used the stairs were classified as​ compliant, those who used the elevator as noncompliant. There were three different​ situations, two of which involved a person who was secretly working with the experimenter.​ (This person is called a​ confederate.) In the first​ situation, there was no other person using the stairs or elevator. In the​ second, there was a compliant confederate. In the​ third, there was a noncompliant confederate. Suppose that the people who arrived to use the elevator were randomly assigned to the three groups. There were significant differences between groups. Complete parts a and b below. (a) Can we generalize widely to a large​ group? Why or why​ not? (b) Can we infer​ causality? Why or why​ not?

(a) No, we cannot generalize because this was not a random sample. (b) Yes, we can infer causality because there was random assignment.

In January​ 2012, a health magazine reported that British researchers compared a​ 12-week yoga program with the usual care provided by​ Britain's National Health Service. It shows that yoga was more effective in reducing back pain than the usual care. (a) What do you need to know to decide whether this was an observational study or a controlled​ experiment? (b) Why do controlled experiments with randomization allow us to draw conclusions implying cause and​ effect?

(a) Whether the subjects were assigned randomly to the groups or chose their own treatment. (b) The random assignment minimizes or eliminates the effects of confounding factors.

​Methods: We measured adult height in 944 of 1042 participants ​(90.6​%) in the Childhood Asthma Management​ Program; adult height was determined at a mean ​(±​SD) age of 24.5±2.52.5 years. Starting at the age of 5 to 13​ years, the participants had been randomly assigned to receive 400μg of​ budesonide, 16 mg of​ nedocromil, or placebo daily for 4 to 6 years. ​Results: Mean adult height was 1.7cm lower​ (95% confidence interval​ [CI], −1.9 to −0.2​) in the budesonide group than in the placebo group ​(P=0.001​) and was 0.7 cm lower​ (95% CI, −0.4 to 0.7​) in the nedocromil group than in the placebo group ​(P=0.64​). (a) Identify the treatment variable and the response variable. (b) Was this a controlled experiment or an observational​ study? Explain. (c) Does the first​ interval, ​(−1.9 to −0.2​), capture​ 0? What does that​ show? (d) From the​ interval, ​(−1.9 to −0.2​), can you conclude that the use of budesonide in childhood reduces the heights of the children when they become​ adults? Why or why​ not? (e) Does the second​ interval, ​(−0.4 to 0.7​), capture​ 0? What does that​ show? (f)From the​ interval, ​(−0.4 to 0.7​), can you conclude that the use of nedocromil in childhood reduces the heights of the children when they become​ adults? Why or why​ not?

(a) The treatment variable is the drug​ (budesonide, nedocromil, or​ placebo). The response variable is adult height. (b) This was a controlled experiment because the subjects were randomly assigned to a drug by the researchers. (c) No. It shows that there is a statistically significant difference between the mean heights. (d) Yes, you can conclude that budesonide reduces adult​ height, because the interval shows a real difference and it was a randomized experiment. (e) Yes. It shows that there is no statistically significant difference between the mean heights. (f) No, you cannot conclude that nedocromil reduces adult​ height, because the interval contains zero.

​Methods: In this​ two-center, prospective, randomized​ trial, we enrolled 980​ children, 0 to 36 months of​ age, undergoing surgery with cardiopulmonary bypass. Patients were randomly assigned to either tight glycemic control...targeting a blood glucose level of 80 to 110 mg per deciliter...or standard care in the cardiac intensive care unit. ​Results: A total of 446 of the 490 children assigned to tight glycemic control ​(91​%) received insulin versus 12 of 490 children assigned to standard care ​(2​%)...Tight glycemic control was not associated with a significantly decreased rate of health care associated infections ​(8.8 vs. 10.1 per 1000​ patient-days, P=0.64​). ​Conclusions: Tight glycemic control can be achieved with a low hypoglycemia rate after cardiac surgery in​ children, but it does not significantly change the infection rate...as compared with standard care. (a) What is the research question that these investigators are trying to​ answer? (b) What is their answer to the research​ question? (c) What were the methods they used to collect​ data? (controlled experiment or observational​ study) (d) Is the conclusion appropriate for the methods used to collect​ data? (e) To what population do the conclusions​ apply? (f) Have the results been replicated in other​ articles? Are the results consistent with what other researchers have​ suggested?

(a) They want to know if the tight glycemic control reduces the rate of infections in children who have heart surgery. (b) No, tight glycemic control does not reduce the risk of infections. (c) This was a controlled experiment because the patients were randomly assigned to a group by the researchers. (d) Yes. Mentioning that the tight glycemic control does not lower the infection rate was appropriate. (e) Because there was no random​ sampling, the conclusion cannot be generalized​ widely, and the results apply only to these patients. (f) The summary did not mention any other studies.

In the special case in which both categorical variables have only two​ categories, the test of homogeneity is identical to which of the​ following?

A​ two-tailed z-test of two proportions

In order to reach causal conclusions and conclude that the entire population would be affected​ similarly, a research study must use which of the​ following?

Both A and B

A recent study was conducted to determine whether a drug was effective at helping​ east-bound jet passengers adjust to jet lag. Subjects were randomly assigned either to one of three different doses of the drug​ (low, medium,​ high) or to a​ placebo, flown east in a plane in which they could not drink alcohol or coffee or take sleeping​ pills, and then examined in a lab where their state of wakefulness was measured and classified into categories​ (low, normal,​ alert). If we test whether treatments for jet lag are associated with​ wakefulness, are we doing a test of independence or of​ homogeneity? Explain.

It would be a test of homogeneity because the passengers were randomly assigned to different groups.

A large number of surgery patients get infections after​ surgery, which can sometimes be quite serious. Researchers randomly assigned some surgery patients to receive a simple antibiotic ointment after​ surgery, others to receive a​ placebo, and others to receive just cleansing with soap. If we wanted to test the association between treatment and whether or not patients get an infection after​ surgery, would this be a test of homogeneity or of​ independence? Explain.

It would be a test of homogeneity because the patients were randomly assigned to different groups.

A December 2009 poll asked a random sample of people whether they thought their country was heading in the right direction. Each respondent answered​ "Right Direction" or​ "Wrong Direction" and was classified as​ Republican, Democrat, or Independent. If we were interested in testing whether party affiliation was associated with the answer to the​ question, would this be a test of homogeneity or of​ independence? Explain.

It would be a test of independence because there was only one sample.

A health magazine reported on a​ women's health study. Experts used a questionnaire to collect data from​ 39,000 women​ 55-69 years old concering their dietary supplements. After 19 years they looked at death rates. Many of the supplements were associated with higher risk of death. ​ However, the consumption of calcium was associated with a lower death rate. Does this study show that consumption of calcium causes lower death​ rates?

No. The study was probably an observational​ study, and causality cannot be inferred from an observational study.

​"Publication bias" refers to the tendency of scientific and medical journals to prefer to publish which of the​ following?

articles with "positive" findings

An outcome of an experiment or study that is large enough to have real effect on​ people's health or lifestyle is said to have which of the​ following?

clinical significance

What is the practice of stating a hypothesis after first looking at the​ data?

data dredging

For two or more samples and one categorical response​ variable, to determine if there is an association between categorical​ variables, a test of​ _________ is used.

homogeneity

For one sample and two categorical response​ variables, to determine if there is an association between categorical​ variables, a test of​ _______ is used.

independence

In a recent​ study, 611 patients were randomly​ assigned, in a​ 4:4:1 ratio, to receive 5 mg of a drug twice​ daily, 10 mg of the drug twice​ daily, or placebo. The primary end​ point, assessed at month​ 3, was the percentage of patients with at least a​ 20% improvement of symptoms. At month​ 3, a higher percentage of patients in the drug groups than in the placebo groups experienced a​ 20% improvement of symptoms​ (59.8% in the​ 5-mg drug group and​ 65.7% in the​ 10-mg drug group vs.​ 26.7% in the placebo​ group, P<0.001 for both​ comparisons) (a) Identify the treatment variable and the response variable. (b) Was this a controlled experiment or an observational​ study? (c) Do the sample percentages suggest that the drug was effective in achieving a​ 20% reduction in​ symptoms? (d) What does the small​ p-value show? (e) Can you conclude that the use of the drug increases the chances of a​ 20% improvement in​ symptoms? Why or why​ not?

(a) The treatment variable is the drug​ used: 5 mg of the​ drug, 10 mg of the​ drug, or placebo. The response variable is whether the patient experienced a​ 20% improvement of symptoms in three months. (b) This was a controlled experiment because the patients were randomly assigned to a group by the researchers. (c) Yes, both of the two drug groups had a larger percentage of patients that experienced a​ 20% improvement in symptoms than the placebo group did. (d) It shows that the percentage of patients who improved with the drug is significantly higher than the percentage who improved with the​ placebo, and the difference in percentages did not occur by chance. (e) Yes, you can conclude that the drug increases the chances of a​ 20% improvement in symptoms because the study was​ well-designed with random assignment and there was a significant effect.

A recent study randomly assigned 316 patients with chronic liver disease of diverse causes and platelet counts of less than​ 50,000 per cubic millimeter to receive a​ drug, at a dose of 75 mg​ daily, or a placebo for 14 days before a planned elective invasive procedure that was performed within 5 days after the last dose. The primary end point was the avoidance of a platelet transfusion​ before, during, and up to 7 days after the procedure. A platelet transfusion was avoided in 108 of 157 who received the drug ​(69​%) and in 29 of 159 who received a placebo ​(18​%) ​(p-valueless than<​0.001) (a) Identify the treatment variable and the response variable. (b) Was this a controlled experiment or an observational​ study? Explain. (c) Do the sample percentages suggest that the drug was effective in reducing the chance of needing platelet​ transfusions? (d) What does the small​ p-value show? (e) Can you conclude that the use of the drug reduces the chance of needing a platelet​ transfusion? Why or why​ not?

(a) The treatment variable is whether the patient received the drug or the placebo. The response is whether the patient avoided the need for a platelet transfusion. (b) This was a controlled experiment because the patients were randomly assigned to a group by the researchers. (c) Yes, because 6969​% of patients on the drug avoided platelet transfusions and that was better than the 1818​% on the placebo who avoided them. (d) We can reject the hypothesis that the treatments and outcomes are independent. It shows that the drug had a significant effect. (e) Yes, you can conclude that the drug reduces the chance of needing a transfusion because the study was​ randomized, placebo-controlled, and there was a significant effect.

When patients are admitted to​ hospitals, they are sometimes assigned to a single room with one bed and sometimes assigned to a double​ room, with a roommate. A researcher was interested in the effect of the type of room on the length of stay in the hospital. Suppose that upon admission to the​ hospital, the names of patients who would have been assigned a double room were put onto a list and a systematic random sample was taken. For each​ participant, a coin was​ flipped: If it landed heads​ up, they got a double​ room, and if it landed tails​ up, they got a single room. Then the experimenters observed how many days the patients stayed in the hospital and compared the two groups. The experiment ran for two months. Suppose those who stayed in single rooms stayed​ (on average) one less​ day, and suppose the difference was significant. (a) Can you generalize to others from this​ experiment? If​ so, to whom can you generalize and why can you do​ it? (b) Can you infer causality from this​ study? Why or why​ not?

(a) Yes, you can generalize to other people admitted to this hospital who would have been assigned a double room because of the random sampling from that group. (b) Yes, you can infer causality because there was random assignment.

An article in a medical journal claimed that autism was caused by the​ measles, mumps, and rubella​ (MMR) vaccine. This vaccine is typically given to children​ twice, at about the age of 1 and again at about 4 years of age. The article reports a study of 12 children with autism who had all received the vaccines shortly before developing autism. The article was later retracted by the medical journal because the conclusions were not justified by the design of the study. Explain why the medical journal might have felt that the conclusions were not justified by listing potential flaws in the​ study, as described above. (a) no control group (b) the vaccine was given twice (c) not a randomized experiment (d) no mention of unimmunized children who develop autism (e) sample size is too small (f) the topic is controversial

(a) no control group (c) not a randomized experiment (d) no mention of unimmunized children who develop autism (e) sample size is too small

The alumni office wishes to determine whether students who attend a reception with alumni just before graduation are more likely to donate money within the next two years (a) Describe a study based on a sample of students that would allow the alumni office to conclude that attending the reception causes future donations but that it is not possible to generalize this result to all students. (b) Describe a study based on a sample of students that does not allow fundraisers to conclude that attending receptions causes future donations but does allow them to generalize to all students. (c) Describe a study based on a sample of students that allows fundraisers to conclude that attending the reception causes future donations and also allows them to generalize to all students.

(a) non random, randomly assign them to (b) random, offer them a choice (c) random, randomly assign them to

The​ 3-year recidivism rate in Texas is about​ 30%, which means that​ 30% of released Texas prisoners return to prison within 3 years of release. There have been many attempts to reduce the recidivism rate. Suppose you want to determine whether electronic monitoring bracelets that track the location of the released prisoner reduce recidivism. Suppose that offenders released from prison are observed for three years to see whether they go back to prison and that the ones who wear electronic monitoring bracelets wear them for the first year only. (a) Describe a study based on a sample of released offenders that would allow the legal system to conclude that monitoring causes a reduction in recidivism but would not allow it to generalize to all released prisoners. (b) Describe a study based on a sample of released offenders that does not allow the legal system to conclude that monitoring causes a reduction in recidivism but does allow it to generalize to all released offenders. (c) Describe a study based on a sample of released prisoners that allows the legal system to conclude that monitoring causes a reduction in recidivism and also allows it to generalize to all released offenders.

(a) nonrandom, randomly assign them to (b) random, let them choose to (c) random, randomly assign about half to

Suppose that a new nicotine patch to help people quit smoking was developed and tested. Smokers voluntarily entered the study and were randomly assigned either the nicotine patch or a placebo patch. Suppose that a larger percentage of those using the nicotine patch were able to stop smoking. (a) Can we generalize widely to a large​ group? Why or why​ not? (b) Can we infer​ causality? Why or why​ not?

(a) ​No, we cannot generalize because this was not a random sample. (b) Yes, we can infer causality because there was random assignment.

Do​ chi-square tests apply to categorical or numerical​ data?

Chi-square tests apply to categorical data.

In a hypothesis test to determine if there is an association between two categorical​ variables, what can be said about the null​ hypothesis?

The null hypothesis always states that there is no association between the variables.

What are the expected counts in a​ two-way table?

The numbers of observations in each cell if the null hypothesis were true.

If there is no association between the categorical​ variables, then which of the following must be​ true?

The observed counts in the​ two-way table should be close to the expected counts.

The shape of the​ chi-square distribution is​ ______.

skewed right

In a​ chi-square test of​ independence, the number of degrees of freedom is equal to the product of​ (number of row−​1) and which of the​ following?

​(Number of columns−​1)