Title: Inference for Two Proportions
Course: Question Banks

Q: The May 10, 2007 issue of The New England Journal of Medicine describes two large randomized, placebo-controlled trials of a vaccine for HPV-related cancer. HPV is human papillomavirus, the most common form of sexually transmitted infection. For the group of people who received the vaccine, the HPV-related cancer rate was much lower than for those who received a placebo.

Which best describes the goal of this study?
- Test a claim or make an estimate about a population proportion (You are correct that we are working with proportions. But this scenario involves two groups.)
- Compare proportions from two populations (You are right that you could interpret this scenario as a comparison of two populations (those who had the HPV vaccine and those who did not). Then compare the proportion with HPV-related cancer in each population. But there is a better description of the goal of this experiment.)
- Establish a cause-and-effect relationship between two categorical variables (This is an experiment. The goal of an experiment is to establish a cause-and-effect relationship. In this case, the researchers want to show that the vaccine reduces HPV-related cancer.)
A: Establish a cause-and-effect relationship between two categorical variables

Q: The following excerpts are from a Gallup article titled "In US, Water Pollution Worries Highest Since 2001," published March 21, 2017

Which of the following excerpts does NOT describe a comparison that is appropriate for testing a hypothesis about the difference in two population proportions?
- "While four in five nonwhites (80%) are worried a great deal about pollution of drinking water, concern is far less prevalent among whites, at 56%" (In the excerpt, we could compare white and non-white people on the issue of drinking water polution. So this scenario would be appropriate for testing a hypothesis about the difference in two population proportions.)
- "Across income groups, 75% of those who earn less than $30,000 annually are concerned a great deal about pollution of drinking water. This compares with smaller majorities among those in middle-income (64%) ." (In the excerpt, we could compare low-income to middle-income proportion concerned with drinking water pollution. So this scenario would be appropriate for testing a hypothesis about the difference in two population proportions.)
- "The latest percentages of Americans who are worried "a great deal" about the pollution of drinking water (63%) and of rivers, lakes and reservoirs (57%) have inched past " (The excerpt does not contain two populations. There is a single population, vaguely defined as "people." The two percentages describe responses to different options of a single question. So these are two values for the same categorical variable. Thus, this scenario would not be appropriate for testing a hypothesis about the difference in two population proportions.)
A: "The latest percentages of Americans who are worried "a great deal" about the pollution of drinking water (63%) and of rivers, lakes and reservoirs (57%) have inched past "

Q: In the article Carbonated Beverage Consumption and Bone Mineral Density among Older Women: The Rancho Bernardo Study (Am J Public Health, 1997; 97:276-279), 1000 women from Rancho Bernardo, California participated in a self-administered questionnaire about their medical history, lifestyle, and lifetime consumption of carbonated beverages. Ever drinkers were defined as “those who had ever consumed any carbonated beverage daily for a year or longer.” The researchers reported

“ In these women, bone mineral density did not differ by ever or current drinker status.“

Which best describes the type of study?
- An experiment comparing two treatments (Researchers did not randomly assign soda drinking behavior to patients, therefore this is not an experiment.)
- An observational study comparing two populations (Since the categories ever/current drinkers were not randomly assigned, this is an observational study comparing two populations.)
A: An observational study comparing two populations

Q: As part of the Gallup-Healthways Well-Being Index, a random sample of 484,350 adults aged 18 and older, living in all 50 U.S. states and the District of Columbia were polled through phone interviews conducted Jan. 2, 2014-Dec. 30, 2016. In an April 6, 2017 article Obesity Quadruples Diabetes Risk for Most US Adults , Gallup reports

“...the odds of having been diagnosed with diabetes are substantially higher among those who are obese than among those who are overweight or normal weight, and remain elevated between the ages of 25 and 64. The peak years of elevated risk are between ages 35 and 39. At this stage in life, obese individuals are over five times more likely than their normal weight counterparts to have been diagnosed with diabetes.”

Which best describes the type of study?
- An experiment comparing two treatments (Researchers could not randomly assign diabetes to patients, therefore this is not an experiment.)
- An observational study comparing two populations (Since the (obese and not obese) sample was not randomly assigned to diabetes and non-diabetes groups, this is an observational study comparing two populations.)
A: An observational study comparing two populations

Q: In the article Randomised clinical trial: yoga vs written self-care advice for ulcerative colitis. (Aliment Pharmacol Ther., April 2017), 77 patients with ulcerative colitis were randomly assigned to yoga or written self-care advice.

“The yoga group had significantly higher disease-specific quality of life compared to the self-care group after 12 weeks. “

Which best describes the type of study?
- An observational study comparing two populations (Researchers randomly assigned patients to treatment groups (yoga or self-care advice) therefore this is an experiment.)
- An experiment comparing two treatments (Researchers randomly assigned patients to treatment groups (yoga or self-care advice) therefore this is an experiment.)
A: An experiment comparing two treatments

Q: A researcher is interested in the issue of cheating on college campuses. She conducts an anonymous survey about cheating among college freshmen. One survey group included freshmen who live on campus, and the other survey group included freshmen who live off-campus.

The standard error formula for the difference between sample proportions is

Calculate the standard error for a survey comparing proportions of cheating of freshmen living on campus to freshmen living off campus, where p1 = 0.20, n1= 130, p2=0.15, n2=117.
- -0.048 (The appropriate calculation is sqrt(0.2*0.8/130 + 0.15*0.85/117).)
- 0.0023 (The appropriate calculation is sqrt(0.2*0.8/130 + 0.15*0.85/117).)
- -0.0023 (The appropriate calculation is sqrt(0.2*0.8/130 + 0.15*0.85/117).)
- 0.048 (The appropriate calculation is sqrt(0.2*0.8/130 + 0.15*0.85/117).)
A: 0.048

Q: The following is an excerpt from a press release on the AFL-CIO website published in October of 2003.

Wal-Mart exemplifies the harmful trend among America’s large employers to shirk health insurance responsibilities at the cost of their workers and community.... With reduced coverage and increased workers’ premium fees, Wal-Mart—the largest private employer in the U.S.—sets a troubling standard. Fewer than half of Wal-Mart workers are insured under the company plan—just 46 percent. This rate is dramatically lower than the 66 percent of workers at large private firms who are insured under their companies’ plans, according to a new Commonwealth Fund study released today which documents the growing trend among large employers to drop health insurance for their workers.

Suppose we want to see if this difference reflects insurance coverage for workers in our community. We select a random sample of 50 Wal-Mart employees and 50 employees from other large private firms in our community. Suppose that 20 of the Wal-Mart employees and 35 of the other employees have insurance through their employer.

Does the difference in insurance coverage reported in the press release reflect what we see in our community?
- No, the press release does not reflect our community. In our community 30% fewer Walmart workers have insurance. The press release reports a 20% difference. (You are correct that 30% fewer Walmart workers in the sample have insurance: (20 / 50 – 35 / 50 = -0.30). But this difference is expected in random sampling from populations that differ by 20%. It is within 2 standard errors of the mean.)
- Yes, the press release reflects our community. The difference in sample proportions from our community is within two standard errors of the difference reported in the press release. (In our community 30% fewer Walmart workers in the sample have insurance: (20 / 50 – 35 / 50 = -0.30). But this difference is expected in random sampling from populations that differ by 20% since the standard error is about 10%.)
A: Yes, the press release reflects our community. The difference in sample proportions from our community is within two standard errors of the difference reported in the press release.

Q: Imagine that you are a researcher investigating steroid use by members of a certain gym. You conduct one survey that includes members from the gym, and another survey that includes no members from the gym.

The graph below represents the sampling distribution of differences between the proportion of gym members who have used steroids and the proportion of non-gym members who have used steroids. The difference in proportions is p_1 - p_2 = 0.05 (gym members minus non-gym members).

Consider simulating this data with 5000 samples from each of the gym member group and the non-gym member group. Based on the sampling distribution of difference in proportions, which of the following results for p 1- p 2 would be most unusual?
- 0.0 (The difference in proportions is p1 - p2 = 0.05, so a difference in sample proportions of 0.0 is not unusual.)
- -0.06 (The difference in proportions is p1 - p2 = 0.05, so a difference in sample proportions of -0.06 is not the farthest choice from 0.05.)
- -0.1 (The difference in proportions is p1 - p2 = 0.05, so a difference in sample proportions of -0.1 is not the farthest choice from 0.05.)
- 0.29 (The difference in proportions is p1 - p2 = 0.05, so a difference in sample proportions of 0.29 is the farthest choice from 0.05.)
A: 0.29

Q: Imagine that you are a researcher investigating steroid use by members of a certain gym. You conduct one survey that includes members from the gym, and another survey that includes no members from the gym.

We can use a simulation model to estimate the true difference between the proportion of gym members who have used steroids and the proportion of non-gym members who have used steroids. The graph below represents the sampling distribution of differences.

The difference in proportions is p1 - p2 = 0.05 (gym members minus non-gym members), and the standard error is 0.08. The z-score for this difference in sample proportions is 0.885.

Which of the following is the most appropriate conclusion?
- There is a statistically significant difference between the proportion of gym members who have used steroids and the proportion of non-gym members who have used steroids. (The z-score is less than 2, which means the difference is within 2 standard deviations, so there is not a statistically significant difference in proportions.)
- We do not have enough information to make a conclusion about the difference between the proportion of gym members who have used steroids and the proportion of non-gym members who have used steroids. (The z-score is less than 2, which means the difference is within 2 standard deviations. There is enough evidence to conclude there is not a statistically significant difference in proportions.)
- There is not a statistically significant difference between the proportion of gym members who have used steroids and the proportion of non-gym members who have used steroids. (The z-score is less than 2, which means the difference is within 2 standard deviations, so there is not a statistically significant difference in proportions.)
A: There is not a statistically significant difference between the proportion of gym members who have used steroids and the proportion of non-gym members who have used steroids.

Q: The 95% confidence interval estimating the gender difference in teen depression rates (“female” minus “male”) in the U.S. is 0.032 and 0.085.

Decide whether the following interpretation of this confidence interval is valid or invalid:

Based on a 95% confidence interval and the 6% difference observed in this study, we estimate that the gender difference in U.S. teen depression rates is between 3.2% and 8.5%, with girls having higher rates. 95% of the time this method produces an interval that contains the true difference.
- invalid (Both sentences are accurate. The difference is defined “females minus males,” so a positive difference says girls have higher rates. Also, the second sentence uses a good general description of the phrase 95% confident because it makes a probability statement about the method.)
- valid (The difference is defined “females minus males,” so a positive difference says girls have higher rates. Also, the second sentence uses a good general description of the phrase 95% confident because it makes a probability statement about the method.)
A: valid

Q: The 95% confidence interval estimating the gender difference in teen depression rates (“female” minus “male”) in the U.S. is 0.032 and 0.085.

Decide whether the following interpretation of this confidence interval is valid or invalid:

We are 95% confident that the percentage of female teens who are depressed is between 3.2% to 8.5% higher than the percentage of male teens who are depressed.
- invalid (We use the phrase “95% confident” when we describe a specific confidence interval. The difference is defined “females minus males,” so a positive difference says girls have higher rates.)
- valid (We use the phrase “95% confident” when we describe a specific confidence interval. The difference is defined “females minus males,” so a positive difference says girls have higher rates.)
A: valid

Q: The 95% confidence interval estimating the gender difference in teen depression rates (“female” minus “male”) in the U.S. is 0.032 and 0.085.

Decide whether the following interpretation of this confidence interval is valid or invalid:

We are 95% confident that similar studies will show a gender difference in the samples that is between 3.2% and 8.5%.
- valid (This statement says we are 95% confident that similar studies will give the same confidence interval. This is incorrect because we expect other studies to give different sample differences. So the interval will change, too.)
- invalid (This statement says we are 95% confident that similar studies will give the same confidence interval. This is incorrect because we expect other studies to give different sample differences. So the interval will change, too.)
A: invalid

Q: The 95% confidence interval estimating the gender difference in teen depression rates (“female” minus “male”) in the U.S. is 0.032 and 0.085.

Decide whether the following interpretation of this confidence interval is valid or invalid:

There is a 95% chance that female teens are 6% more likely to be depressed than male teens.
- valid (There are several problems here. The statement uses the sample difference as if it was the population difference. Also, this statement makes a probability statement about a single specific value. There is not a 95% chance that the actual difference in population proportions is equal to 6%. The actual difference in population proportions is some value and not a random variable.)
- invalid (There are several problems here. The statement uses the sample difference as if it was the population difference. Also, this statement makes a probability statement about a single specific value. There is not a 95% chance that the actual difference in population proportions is equal to 6%. The actual difference in population proportions is some value and not a random variable.)
A: invalid

Q: Robert D. Kersey published an article titled “Anabolic-Androgenic Steroid Use among California Community College Student Athletes” (Journal of Athletic Trainers 31(3):237—41, 1996) comparing various aspects of users and nonusers. The study used an advanced random sampling technique to select 10 representative community colleges in California and then to select a random sample of student athletes from the 10 colleges. The group of 1,185 male and female student-athletes completed an anonymous questionnaire. Of the sample, 4.2% of the males and 1.2% of the females admitted to using steroids.

The researcher did not conduct an inference procedure in the study even though he went to a lot of trouble to select a random sample. The gender difference in steroid use is 3% in the sample. If he used this 3% difference to calculate a confidence interval, what is he estimating?

Pick the best description of the population to complete the statement below.

p_1 - p_2 represents the percentage that male steroid use is larger (or smaller) than female steroid use for what population?
- students attending California community colleges in 1996 (This population is too broad to be represented by the sample. The sample came from a subgroup of the students in CA community colleges in 1996.)
- the athletes in the sample of 1,185 (This is a description of the sample, not the population.)
- community college student athletes in 1996 (This population is too broad to be represented by the sample. The sample came from a subgroup of community college student athletes in 1996.)
- student athletes in California community colleges in 1996 (The sample is a random sample of this population.)
A: student athletes in California community colleges in 1996

Q: Robert D. Kersey published an article titled “Anabolic-Androgenic Steroid Use among California Community College Student Athletes” (Journal of Athletic Trainers 31(3):237—41, 1996) comparing various aspects of users and nonusers. The study used an advanced random sampling technique to select 10 representative community colleges in California and then to select a random sample of student athletes from the 10 colleges. The group of 1,185 male and female student-athletes completed an anonymous questionnaire. Of the sample, 4.2% of the males and 1.2% of the females admitted to using steroids.

The researcher did not conduct an inference procedure in the study even though he went to a lot of trouble to select a random sample. The gender difference in steroid use is 3% in the sample. If he used this 3% difference to calculate a confidence interval, what is he estimating?

The sample contained 1,185 community college athletes. But the researcher did not report the number of males and females in the sample. Since 1.2% of the females reported steroid use, how many female athletes would the sample have to contain in order for us to calculate a confidence interval?
- 10 (Are you thinking that the count of successes has to be 10? This is correct. But you need to determine how many female athletes we need to achieve this count of successes.)
- 593 (Are you thinking that half of the sample has to be female? We do not have to have equal numbers in each sample to calculate a confidence interval for the difference in population proportions.)
- It is impossible to determine this from the information given. (Since p-hat is small, only 0.012, we will need a lot of females in the sample to get a count of at least 10 steroid users. For those who like formulas, we need to solve n(0.012) = 10.)
- 834 (Since p-hat is small, only 0.012, we will need a lot of females in the sample to get a count of at least 10 steroid users. For those who like formulas, we need to solve n(0.012) = 10. So we solve 10 / 0.012 to find the number of females we need in the sample. We do not need to worry about the number of non-steroid users since this is 0.988 of the sample.)
A: 834

Q: HPV is a group of more than 150 types of related viruses. About ⅓ of HPV viruses are sexually transmitted. Only a few cause cancer. A recent pair of studies claim that about half of adult men and one quarter of adult women are infected with HPV. (Sources: HPV Infection Common Among American Females, ScienceDaily, Feb. 28, 2007 and How Many Men Have HPV?, Population Reference Bureau, May 2011).

Suppose that we use the data one of these studies to calculate the 90%, 95%, and 99% confidence intervals for the difference in male and female infection rates. Which interval is the 90% confidence interval?
- 0.192 to 0.308 (This is the 95% confidence interval. Notice that it is the 2nd widest interval. The 95% confidence interval will have the second largest margin of error.)
- 0.174 to 0.326 (This is the 99% confidence interval. Notice that it is the widest interval. The 99% confidence interval will have the largest margin of error.)
- 0.201 to 0.299 (The 90% confidence interval is the most narrow because it has the smallest margin of error.)
A: 0.201 to 0.299

Q: A survey was conducted to research the public's views on potholes. When the results were analyzed, researchers found that 65% of adults age 20 to 40 view potholes as a serious problem, but only 47% of adults age 60 and older view potholes as a serious problem. Researchers are interested in the difference in proportions of adults age 60 and over and adults age 20 to 30. The standard error for the difference in proportions is 0.011.

Which of the following is the margin of error for a 95% confidence interval?
- 0.011 (The margin of error is calculated by multiplying the critical value (Z-score) and standard error. It looks like you forgot to multiply by the critical value (Z-score) for a 95% confidence interval.)
- 0.018 (The margin of error is calculated by multiplying the critical value (Z-score) and standard error. For a 95% confidence interval, the critical value is Z = 1.96. It looks like you used the wrong critical value in your calculation.)
- 0.028 (The margin of error is calculated by multiplying the critical value (Z-score) and standard error. For a 95% confidence interval, the critical value is Z = 1.96. It looks like you used the wrong critical value in your calculation.)
- 0.021 (You chose the right critical value Z = 1.96 for a 95% confidence interval. The margin of error is calculated by multiplying the critical value (Z-score) and standard error.)
A: 0.021

Q: A survey was conducted to research the public's views on potholes. When the results were analyzed, researchers found that 65% of adults age 20 to 40 view potholes as a serious problem, but only 47% of adults age 60 and older view potholes as a serious problem. Researchers are interested in the difference in proportions of adults age 60 and over and adults age 20 to 30. The standard error for the difference in proportions is 0.011.

Researchers want to decrease the margin of error by adjusting the confidence level. Which confidence interval will have the smallest margin of error (MOE)?
- 95% confidence interval (The margin of error (MOE) is calculated by multiplying the critical value Z by the standard error. For 95% confidence, the critical value is Z = 1.96. This will be a larger margin of error than for 90% confidence.)
- 99% confidence interval (The margin of error (MOE) is calculated by multiplying the critical value Z by the standard error. For 99% confidence, the critical value is Z = 2.576. This will be a larger margin of error than for 90% confidence.)
- 90% confidence interval (The margin of error (MOE) is calculated by multiplying the critical value Z by the standard error. For 90% confidence, the critical value is Z = 1.645. This is a smaller margin of error than for 95% or 99% confidence.)
A: 90% confidence interval

Q: The Women’s Health Initiative conducted an experiment in 2002 to study the side-effects of hormone replacement therapy for women with menopausal symptoms. The experiment randomly assigned over 16,000 U.S. women to receive a hormone treatment or a placebo. The experiment was double blind. After 5 years a larger proportion of the hormone group had breast cancer and heart disease. Researchers were so alarmed by the results that the experiment was ended early to prevent further harm to the health of the women participating in the hormone group.

As a result of this experiment the use of hormone replacement therapy fell by 66%.

Yet the British Menopause Society and the International Menopause Society questioned this reaction. The Women’s Health Concern, a British non-profit group that provides independent and unbiased information about women’s health wrote, “It must be remembered that the WHI data on which the concerns were raised related to overweight North American women in their mid-sixties and not to the women that are treated with HRT for their menopausal symptoms in the United Kingdom, who are usually around the age of menopause, namely 45-55 years.”

The objections raised by the Women’s Health Concern are an example of what?
- Inference based on non-random data (the experiment randomly assigned women to a hormone treatment or a placebo. So inference is appropriate.)
- Inappropriate cause-and-effect conclusions (the experiment was well-designed with random assignment, use of a placebo, and double blind design. So cause-and-effect conclusions are valid.)
- Overgeneralization (The objection is that the original group of women in the experiment was not randomly selected (though the sample was randomly assigned to two treatments.) The women were all from the U.S. and tended to be overweight and older. So conclusions will not generalize to all women.)
A: Overgeneralization

Q: Suppose that you are a researcher, and you're interested in the possible effects of calcium on joint health. You conduct a study that measures joint health and calcium levels in a random sample of 675 people who died in good health. You find that 8.5% of the 82 people with low calcium levels had poor joint health when they died. However, only 1% of the 593 people with regular calcium levels had poor joint health.

If the hypothesis test has P-value = 0.03, can you conclude that low calcium levels cause poor joint health?
- Yes (A causal conclusion is not appropriate for this study because the study is not an experiment. In this observational study you are not able to control for the other factors that contribute to poor joint health. At best, with a single observational study, you can establish a correlation between low calcium levels and poor joint health, but correlation does not imply causation. The best evidence that an association is due to causation comes from a controlled experiment.)
- No (A causal conclusion is not appropriate for this study because the study is not an experiment. In this observational study you are not able to control for the other factors that contribute to poor joint health. At best, with a single observational study, you can establish a correlation between low calcium levels and poor joint health, but correlation does not imply causation. The best evidence that an association is due to causation comes from a controlled experiment.)
A: No

Q: Suppose that you are a researcher, and you're interested in the possible effects of calcium on joint health. You conduct a study that measures joint health and calcium levels in a random sample of 675 people who died in good health. You find that 8.5% of the 82 people with low calcium levels had poor joint health when they died. However, only 1% of the 593 people with regular calcium levels had poor joint health.

Which of the following statistical designs best describes measuring joint health and calcium levels in a random sample from 675 people?
- Experiment (An experiment deliberately imposes some treatment on individuals in order to observe the response. This study measures variables of interest (joint health and calcium levels) from a random sample of people who died. In this case you could not influence the responses through random assignment to treatments because the people are already dead.)
- Randomized comparison (A randomized comparison is not a type of statistical design for a study. This study measures variables of interest (joint health and calcium levels) from a random sample of death records.)
- Observational study (An observational study observes individuals and measures variables of interest, but does not attempt to influence the responses. In this case, you examined a random sample of death records, so there was no random assignment to treatment groups.)
A: Observational study

Q: A study published in Alcoholism: Clinical & Experimental Research investigated a potential link between energy drink consumption and alcohol dependence. Let p_1 and p_2 represent the proportions of people who develop alcohol dependence among consumers and non-consumers of energy drinks, respectively. What are the appropriate null and alternative hypotheses?
- H_0: p_1 − p_2 = 0 and H_a: p_1 − p_2 < 0. (The first proportion represents consumers of energy drinks. If energy drink consumption is linked to alcohol dependence, this proportion is larger than the other.)
- H_0: p_1 − p_2 = 0 and H_a: p_1 − p_2 ≠ 0. (The first proportion represents consumers of energy drinks. If energy drink consumption is linked to alcohol dependence, this proportion is larger than the other.)
- H_0: p_1 − p_2 = 0 and H_a: p_1 − p_2 > 0. (The first proportion represents consumers of energy drinks. If energy drink consumption is linked to alcohol dependence, then the first proportion is greater than the second proportion (p_1 > p_2). This is equivalent to the claim that p_1 − p_2 > 0.)
A: H_0: p_1 − p_2 = 0 and H_a: p_1 − p_2 > 0.

Q: Recall the (make-believe) matched-pairs investigation of the effect of drinking two beers on reaction time in a driving simulator. We set a significance level of 5% and tested the following hypotheses.

- H_0: Drinking 2 beers has no effect on reaction time.
- H_a: Drinking 2 beers slows reaction time.

We defined µ as the mean of the difference in reaction time (before minus after) for all students at this college after drinking two beers. We rewrote the hypotheses in terms of µ.

- H_0: μ = 0
- H_a: μ < 0

Identify the type I error.
- Researchers conclude that drinking 2 beers slows reaction time, and this is true. (There is no error here. The researchers’ conclusion reflects the real truth about the effect of drinking 2 beers on reaction time.)
- Researchers conclude that drinking 2 beers does not significantly slow reaction time, but this is not really true. (The researchers conclude drinking 2 beers does not significantly affect reaction time, so they fail to reject the null hypothesis. But in reality drinking 2 beers does significantly affect reaction time. So the null hypothesis is false and the alternative hypothesis is true. This is a type II error.)
- In reality drinking 2 beers does not have much of an impact on reaction time, but researchers conclude from this experiment that drinking 2 beers significantly slows reaction time. (The researchers conclude that drinking 2 beers slows reaction time. So their data supports the alternative hypothesis. But in reality the drinking 2 beers does not significantly affect reaction time, so the null is true. This is a type I error.)
A: In reality drinking 2 beers does not have much of an impact on reaction time, but researchers conclude from this experiment that drinking 2 beers significantly slows reaction time.

Q: Recall the (make-believe) matched-pairs investigation of the effect of drinking two beers on reaction time in a driving simulator. We set a significance level of 5% and tested the following hypotheses.

- H_0: Drinking 2 beers has no effect on reaction time.
- H_a: Drinking 2 beers slows reaction time.

We defined µ as the mean of the difference in reaction time (before minus after) for all students at this college after drinking two beers. We rewrote the hypotheses in terms of µ.

- H_0: μ = 0
- H_a: μ < 0

The significance level is 5%. The P-value was 0.014. What is the probability of a type I error for this hypothesis test (assuming that the null hypothesis is true)?
- 0.014 (This is the P-value. The probability of a type I error is the significance level.)
- it is impossible to tell from the information given (The probability of a type I error is the significance level.)
- 0.05 (The probability of a type I error is the significance level.)
A: 0.05

Q: Suppose that you are a researcher conducting a double-blind, placebo-controlled experiment to evaluate two different treatments for acne in teenagers. In this experiment you randomly assign teenage participants to one of the two treatments (TREAT1 and TREAT2).

You conduct a hypothesis test to determine if the side effect rates are significantly different for TREAT1 vs TREAT2 at a 5% level of significance. (The normality conditions are met by the data.) The P-value is 0.04.

What does a P-value of 0.04 mean?
- There is a 4% chance there is no difference in side effect rates for these two treatments. (This says that there is a 4% chance that the null hypothesis is true. The P-value is the chance that two random samples will have a difference in proportions more extreme than the data if the null hypothesis is true.)
- If the side effect rates are the same for the two treatments, there is a 4% that the rates will be different in future experiments. (This interpretation of the P-value gets off to a good start with the phrase “if the side effect rates are the same for the two treatments.” But we always expect rates to differ in future experiments. The P-value is the chance that two random samples will have a difference in proportions more extreme than than observed in this experiment if the null hypothesis is true.)
- If the side effect rates are different for the two treatments, there is a 4% chance that future experiments will show the same differences in side effect rates observed in this experiment. (The P-value is a conditional probability where the condition is that the null hypothesis is true. This statement “if the side effect rates are different for the two treatments” makes the condition that the alternative hypothesis is true.)
- If the side effect rates are the same for the two treatments, there is a 4% chance that future experiments will show differences in side effect rates greater than observed in this experiment. (This is an accurate description of the P-value.)
A: If the side effect rates are the same for the two treatments, there is a 4% chance that future experiments will show differences in side effect rates greater than observed in this experiment.

Q: Is it true that a larger proportion of people who consume at least one energy drink per week are alcohol dependent when compared to people who don’t consume energy drinks?

The null hypothesis claims that there is no difference in alcohol dependence rates between these populations. The alternative hypothesis claims that energy drinkers are more likely to become alcohol dependent.

Which of the following is the Type I error?
- People who drink energy drinks are more likely to become alcohol dependent, but the data suggests that this is not true. (The statement “people who drink energy drinks are more likely to become alcohol dependent” says that the alternative hypothesis is true. So this option describes a situation where the alternative hypothesis is true, but the data does not support it. This is a Type II error. With a Type I error, the null hypothesis is true, but we reject it.)
- Alcohol dependence is not related to energy drink consumption, and the data supports this fact. Our conclusion is correct. (The statement “alcohol dependence is not related to energy drink consumption” says that the null hypothesis is true. If the data support this fact, we fail to reject the null. So we have not made an error. The type I error always involves the rejection of a true null hypothesis.)
- Alcohol dependence is not related to energy drink consumption, but the data suggests that the two variables are related. (The statement “alcohol dependence is not related to energy drink consumption” says that the null hypothesis is true. The proportion of alcohol dependence is the same for those who do and those who don’t drink energy drinks. The type I error involves rejecting a true null hypothesis.)
A: Alcohol dependence is not related to energy drink consumption, but the data suggests that the two variables are related.

Q: Is it true that a larger proportion of people who consume at least one energy drink per week are alcohol dependent when compared to people who don’t consume energy drinks?

The null hypothesis claims that there is no difference in alcohol dependence rates between these populations. The alternative hypothesis claims that energy drinkers are more likely to become alcohol dependent.

Which is NOT a possible consequence of the Type I error in this situation?
- The government bans the advertisement and sale of energy drinks to minors. This causes a loss of profit, tax revenue and jobs. (This is a possible consequence of a Type I error in this situation. With a Type I error we think there is a connection between consumption of energy drinks and alcoholism, but there is not a connection. So we might take action to protect minors from consuming energy drinks when there really is no danger. Here our error leads to reduced sales of energy drinks.)
- Consumer protection organizations waste time and money raising public awareness that consumption of energy drinks is linked to future alcohol dependence. (This is a possible consequence of a Type I error in this situation. With a Type I error we think there is a connection between consumption of energy drinks and alcoholism, but there is really not a connection. So we may spend time and money to warn the public about the dangers of energy drinks, when no danger exists. Here our error leads to a waste of time and money.)
- People who consume energy drinks are not told about the potential problems. So they continue to consume energy drinks without the knowledge that this behavior may lead to alcohol dependence. (This is NOT a possible consequence of a Type I error. This describes a possible consequence of a Type II error. A Type II error in this situation is where we think there is a no connection between energy drink consumption and alcoholism, but we are wrong. If we think there is a no connection, we will not educate people about potential problems with drinking energy drinks.)
A: People who consume energy drinks are not told about the potential problems. So they continue to consume energy drinks without the knowledge that this behavior may lead to alcohol dependence.

Q: Suppose that researchers conduct a clinical trial to test the effectiveness of a new vaccine to prevent HPV-related cervical cancer. They compare HPV-related cancer rates in women who were vaccinated during adolescence to women who were not. The incidence of cancer in vaccinated women is statistically significant at the 1% level. (P=0.006).

Identify the Type I error.
- Researchers conclude that the vaccine is effective in reducing HPV-related cancer, and this is true. In reality the proportion of vaccinated girls who get HPV-related cancer later is life is significantly smaller. (There is no error here. The researchers conclude that the vaccine is effective and it is.)
- Researchers conclude that the vaccine is not effective in reducing HPV-related cancer, but it is. In reality the proportion of vaccinated girls who get HPV-related cancer later is life is significantly smaller. (The researchers conclude that the vaccine is not effective, so they fail to reject the null hypothesis. But in reality the proportion of vaccinated girls who get HPV-related cancer later is life is smaller. So the null hypothesis is false and the alternative hypothesis is true. This is a Type II error.)
- Researchers conclude that the vaccine is effective in reducing HPV-related cancer, but it is not. In reality the same proportion of vaccinated and unvaccinated girls will get HPV-related cancer. (The researchers conclude that the vaccine is effective. So their data supports the alternative hypothesis. But in reality the proportion with cervical cancer is the same in the two groups, so the null is true. This is a Type I error.)
A: Researchers conclude that the vaccine is effective in reducing HPV-related cancer, but it is not. In reality the same proportion of vaccinated and unvaccinated girls will get HPV-related cancer.

Q: Suppose that researchers conduct a clinical trial to test the effectiveness of a new vaccine to prevent HPV-related cervical cancer. They compare HPV-related cancer rates in women who were vaccinated during adolescence to women who were not. The incidence of cancer in vaccinated women is statistically significant at the 1% level. (P=0.006).

What is the probability of a Type I error for this hypothesis test?
- 0.006 (This is the P-value. The probability of a Type I error is the significance level.)
- it is impossible to tell from the information given (The probability of a Type I error is the significance level.)
- 0.01 (The probability of a Type I error is the significance level.)
A: 0.01