Suppose you plan to take a random sample of 1,000 Hispanics in the United States. Assume that the distribution of origin has not changed since 2010.

a. What proportion of Hispanics in the United States in 2010 were of Mexican origin? What is the expected number of Hispanics in your sample who will be of Mexican origin?

b. What proportion of Hispanics in the United States in 2010 were of Puerto Rican origin? What is the expected number of Hispanics in your sample who will be of Puerto Rican origin?

c. Now suppose that you take a random sample of 1,000 Hispanics from your state and find 521 Hispanics of Mexican origin. A friend takes a second random sample of 1,500 Hispanics from your state and finds 847 of Mexican origin. What is the best estimate of the proportion of Hispanics in your state who are of Mexican origin?

d. Suppose you plan to take yet another random sample from your state, this time of 500 Hispanics. Based on your work in Part c, what is the best estimate of the expected number of Hispanics of Mexican origin you will find in this sample?

Tonsils, at the back of the throat, are organs of the lymphatic system. They may help the immune system fight disease. Sometimes, when they are chronically inflamed, doctors remove them, with a surgical procedure called a tonsillectomy. Sweden keeps a register of surgery done on people under the age of 20. During a 15-year period, 27,284 young people had a tonsillectomy. For each of them, five young people the same age, sex, and county of residence were randomly selected to serve as controls. The control group ended up with 136,401 young people. After about 23 years, 47 of the tonsillectomy group and 169 of the control group had had a premature heart attack.

a. What is the explanatory variable in this study? What is the response variable? Was this study an experiment? Explain why or why not.

b. Make a table of observed frequencies.

c. What is the absolute risk for each group? Why is the incidence of heart attack so low in both groups?

d. Compute the absolute risk reduction of a heart attack. Use this in a sentence for parents.

e. Compute the relative risk of a heart attack. Use this in a sentence for parents.

f. Assume that you can consider the two groups equivalent to randomly selected samples from all young people with and without tonsillectomies. Compute $\chi^2$.

g. Is the difference in the proportions of people in the two groups who have heart attacks statistically significant? Explain how you know.

h. Write a short article for your school newspaper about this study. Use and explain the terms absolute risk, relative risk, and statistical significance.

Alzheimer's disease (AD) causes progressive cognitive impairment. A diagnostic blood test for AD recently was proposed. An evaluation of this blood test involved 50 known AD patients and 40 controls who did not have AD. Of those who had AD , 48 tested positive on the blood test and 2 tested negative. Of those without AD , 3 tested positive on the blood test and 37 tested negative.

a. Make a table that summarizes the information above.

b. How many people who did not have AD tested positive on the blood test? What is the name for this kind of test result?

c. How many people who did have AD tested negative on the blood test? What is the name for this kind of test result?

d. With a serious disease, it is important that a high proportion of those who have the disease be detected by the test. Was that the case here? What is the name for the proportion of cases that are detected by a test?

e. It also is important that a high proportion of those who do not have the disease should test negative. Was that the case here? What is the name for this proportion?

f. What is the positive predictive value (PPV)? Use this value in a sentence explaining to a person what his or her positive test suggests.

g. What is the negative predictive value (NPV)? Use this value in a sentence explaining to a person what his or her negative test suggests.

h. All in all, do you think this is a good test? Explain.

In some situations, knowing which category a person falls into on one variable helps you better predict which category they fall into on a second variable. For example, suppose you want to estimate the probability that a teen has long hair. If you are told that the teen is female, then that fact helps you-the probability is higher than if you had been told the teen is male. The variables gender and hair length are associated (or, dependent). On the other hand, suppose that you want to estimate the probability that a teen is carrying a cell phone. Then knowing whether the teen is male or female probably does not help you at all-males and females are equally likely to be carrying cell phones. The variables of gender and cell phone possession are independent.

a. In the situation, does knowing whether one of these dolphins is young or adult help you predict whether it is male? That is, are the variables independent or associated? Explain your answer.

b. If the table had been like the one below, would the variables be independent or associated? Explain your answer.

Conduct a chi-square test for the data using technology.

a. What are the degrees of freedom?

b. What is the P-value? Is the P-value larger than 0.05 ?

c. Is the value of $\chi^2$ statistically significant?

Sometimes you will see risk given in the form " 1 in n. ." For example the risk of breast cancer for a woman was given as 1 in 8.

a. Refer to the two-way frequency table analyzed in the Think About This Situation. Write the risk of getting lung cancer for the boys who smoke in the form " 1 in n. "

b. Write the risk of getting lung cancer for the boys who do not smoke in the form " 1 in n."

c. Write a procedure that can be used to convert numbers like those in the table to the form " 1 in n."

d. In the Civil War, 77 of 425 Confederate generals were killed in action; 47 of 583 Union generals were killed.

Look back at your work. Does your analysis of these data convince you that loud music, especially played through headphones and MP3 players, is damaging the hearing of teens? Explain.

In this unit, when you used technology to determine the value of $\chi^2$, you may have noticed that you were also given the degrees of freedom df and a P-value p. The P-value is the probability of getting a value of $\chi^2$ as large or larger than the one you computed for your sample if it had been taken from a population where the two variables are independent. If the P-value is smaller than 0.05, the value of $\chi^2$ is statistically significant.

a. Conduct a chi-square test for the data in Extensions Task 16 using technology. What are the degrees of freedom?

b. What is the P-value? Is the P-value smaller than 0.05 ?

c. Is the value of $\chi^2$ statistically significant?

SAT Critical Reading score is a quantitative variable.

a. How were the data organized to permit SAT Critical Reading scores to be analyzed as a categorical variable using a frequency table?

b. Identify for which a similar technique was used.

Write a short summary of what the research summarized below tells you about the placebo effect.

Remifentanil is a very quick acting and effective pain-reliever. However, its effect doesn't last very long, with half of its usefulness gone in 3 to 4 minutes. In a test of the placebo effect, twenty-two healthy volunteer subjects were hooked up to an IV so that the Remifentanil could be administered. In the first step, before being given Remifentanil, the subjects were subjected to moderate pain (70 on a scale of 0 to 100) caused by a heat source attached to their calf and their reaction measured. In the second step, the subjects were given Remifentanil, but were led to believe that it hadn't been turned on yet. Their rating of their pain was significantly lower. In the third step, the subjects were told that the drug had been started, and their pain rating dropped about the same amount again as it had in the second step. In the final step, the subjects were told that the drug had been stopped, even though it hadn't. The subjects reported a considerable increase in pain intensity, almost to as high a level as in the first step when they were receiving no Remifentanil at all. The subject's brains were scanned during these tests and the activity in the areas of the brain that are involved with pain was consistent with the subjects' reports of their pain.

A random sample of college students were asked to identify their favorite season of the year and whether they preferred to spend most of their time outside, inside, or did not care. The table below is based on their responses.

How many times would you have to flip a coin before convincing someone that it was fair? The plot shows the number of flips of a fair quarter on the horizontal axis and the proportion of times, so far, that the coin came up heads.

a. Was the first flip a head or a tail?

b. Was the 50 th flip a head or a tail?

c. How many heads were in the longest run of consecutive heads?

d. Describe what is happening as the number of flips increases.

e. About how many flips did it take until the proportion of heads stays pretty close to 0.5 ?

f. Predict what would happen to the pattern in the plot if the person continued flipping the quarter. Explain why this should be the case.

When absolute risk is low, it is often reported as the number of cases per 1,000 people, per 10,000 people, or per 100,000 people. For example, says that 54 people out of every 100,000 who get the HPV vaccine have an adverse event.

Read this statement about collisions of autos and deer in Pennsylvania: "The highest rate for 2009 was Fulton County, west of Adams County along the Maryland border, with 16.83 per 10,000 people." In 2009, Fulton County had 25 deer-related accidents. About how many people lived in Fulton County?

Sometimes one or both of the categorical variables has more than two categories. In other words, the table is larger than 2 rows and 2 columns. For example, the following table gives information about a random sample of traffic crashes in Virginia in 2011.

a. Compute a table of expected frequencies, in the usual way, for a chi-square test of independence.

b. Compute the value of $\chi^2$ in the usual way.

c. The larger the table, the larger the critical value will be. To find the critical value in such cases, you compute the degrees of freedom (df): Multiply the number R of rows minus 1 times the number C of columns minus 1 .

$$d f=(R-1)(C-1)$$

Then look up the critical value in the following table. Note that the critical values are the same as in the table in Lesson 2. However, this table uses Degrees of Freedom rather than Number of Categories.

Is the value of $\chi^2$ from Part b statistically significant?

d. Where does the largest difference between the observed and expected frequency occur? What can you conclude from this test?