Comparing Two Populations or Treatments

Intervals for difference of proportions

Tests for difference of proportions

Intervals for difference of means

Tests for difference of means

Random Questions

100

In a large Midwestern university (with the class of entering freshmen being on the order of 6000 or more students), an SRS of 100 entering freshmen in 1993 found that 20 finished in the bottom third of their high school class. Admission standards at the university were tightened in 1995. In 1997, an SRS of 100 entering freshmen found that 10 finished in the bottom third of their high school class. Let p₁ be the proportion of all entering freshmen in 1993 who graduated in the bottom third of their high school class, and let p₂ be the proportion of all entering freshmen in 1997 who graduated in the bottom third of their high school class. What is the 99% confidence interval for p₁ - p₂?

What is (-.0288, 0.2288)?

100

An SRS of 100 flights by Nite-flite Airlines showed that 64 were on time. An SRS of 100 flights by Waxwing Airlines showed that 80 were on time. Let p_N be the proportion of on-time flights for all Nite-flite Airline flights, and let p_W be the proportion of all on-time flights for all Waxwing Airlines flights. Is there a difference in the on-time rate for the two airlines? When you test this, you discover a P-value of 0.0117. Interpret this P-value.

What is "If the on-time rates for two airplanes are equal, the probability of getting samples with a difference as far or farther from zero as these samples is 0.0117."

100

Popular wisdom is that eating presweetened cereal tends to increase the number of dental caries (cavities) in children. A sample of children was (with parental consent) entered into a study and followed for several years. Each child was classified as a sweetened-cereal lover or a unsweetened cereal lover. At the end of the study, the amount of tooth damage was measured. Here are the summary data:
Cereal preference n Mean Std. Dev.
Sweetened 10 6.41 5.0
Unsweetened 15 5.20 15.0
Assuming the necessary conditions for inference are met, what is an approximate 95% confidence interval for the difference in the mean tooth damage?

What is (-7.57, 9.9896)?

100

A sociologist studying fertility in the United States and Canada wanted to compare how many babies, on average, women in each country have. The sociologist obtained a random sample of 88 women from each country. Here is a summary of the number of babies for the women in each sample:

Number of babies in US (mean = 1.97, SD = 1.76)

Number of babies in Canada (1.76, 1.41)

The alternate hypothesis is that the populations are different. Assume that all conditions have been met.

What is the P-value associated with these sample results?

What is .1849?

100

According to Humane Society data, 39% of households in the United States have at least one dog. In the United Kingdom, 23% of households have at least one dog. Suppose you select an SRS of 75 households in the U.S. and 80 households in the U.K., and calculate p-hat₁= the proportion of households in the U.S. sample that have a dog, and p-hat₂= the proportion of households in the U.K. sample that have a dog. Describe the sampling distribution of proportions for p-hat₁ - p-hat₂ .

What is Mean = .16, Standard Deviation = 0.073, Approx. Normal

200

To meet the independence condition for an interval, we need to know the samples would be less than 10% of the population.

200

The movie A Civil Action tells the story of a major legal battle that took place in the small town of Woburn, Massachusetts. A town well that supplied water to eastern Woburn residents was contaminated by industrial chemicals. During the period that residents drank water from this well, 16 of the 414 babies born had birth defects. On the west side of Woburn, 3 of the 228 babies born during the same time period had birth defects. What conditions for inference not met in this situation?

What are random and normal?

200

The heights of young men follow a Normal distribution with mean 69.3 inches and standard deviation 2.8 inches. The heights of young women follow a Normal distribution with mean 64.5 inches and standard deviation 2.5 inches. Let M = height of a randomly selected young man and W = the height of a randomly selected young woman. Describe the shape, center, and spread of the distribution of M - W.

What are Normal with mean = 4.8 inches and standard deviation = 3.75 inches.

200

A economist studying income in Argentina and Brazil wanted to compare the average annual income between the two countries. The economist obtained incomes for a random sample of people from each country to test if the average income in Argentina was higher. Here are the results of the test (incomes are shown in thousands of international dollars):

Argentina: n = 100, m = 17.3, SD = 9.2

Brazil: n = 100, m = 15.4, SD = 9.8

At the 1% significance level, is there sufficient evidence to conclude that the average income is higher in Argentina?

What is "no"? The p-value is greater than 1%.

200

Do people smoke less when cigarettes cost more? A random sample of 500 smokers was selected. The number of cigarettes each person smoked per day was recorded over a one-month period before a 30% cigarette tax was imposed and again for one month after the tax was imposed. This is the appropriate inference procedure to use to answer this question:

What is Paired t test for the mean difference?

300

A manufacturer receives parts independently from two suppliers. An SRS of 400 parts from supplier 1 finds 20 that are defective. An SRS of 100 parts from supplier 2 finds 10 that are defective. Let p₁ and p₂ be the proportions of all parts from suppliers 1 and 2, respectively, that are defective. What is the 95% confidence interval for p₁ - p₂?

What is (-0.112559, 0.012559)?

300

A driving school wants to find out which of its two instructors is more effective at preparing students to pass the state's driver's license exam. An incoming class of 100 students is randomly assigned to two groups, each of size 50. One group is taught by Instructor A; the other is taught by Instructor B. At the end of the course, 30 of Instructor A's students and 22 of Instructor B's students pass the state exam. Do these results give convincing evidence that Instructor A is more effective?

What is "Since P-value > 0.05, we fail to reject H_o. There is not convincing evidence that Instructor A's pass rate is higher than Instructor B's."?

300

A quiz question gives random samples of n = 10 observations from each of two Normally distributed populations. Tom uses a table of t distribution critical values and 9 degrees of freedom to calculate a 95% confidence interval for the difference of two population means. Janelle uses her calculator's two-sample t interval with 16.87 degrees of freedom to compute the 95% confidence interval. Assume that both students calculate the intervals correctly. Who's interval is wider?

What is "Tom's"?

300

A study of road rage asked separate random samples of 596 men and 523 women about their behavior while driving. Based on their answers, each respondent was assigned a road rage score on a scale of 10 to 20. Are the conditions for performing a two-sample t test satisfied?

What is "Yes, we have two independent random samples with large sample sizes."?

300

Consumers Union wants to see which of two brands of calculator is easier to use. They recruit 100 volunteers and randomly assign them to two equal-sized groups. The people in one group use calculator A and those in the other group use calculator B. Researchers record the time required for each volunteer to carry out the same series of simple calculations on the assigned calculator. This would be the appropriate inference procedure for this study.

What is Two-sample t interval for the difference between two means.

400

A manufacturer receives parts independently from two suppliers. An SRS of 400 parts from supplier 1 finds 20 that are defective. An SRS of 100 parts from supplier 2 finds 10 that are defective. Let p₁ and p₂ be the proportions of all parts from suppliers 1 and 2, respectively, that are defective. How do you interpret the confidence interval (a,b) for the difference of proportions between the 2 suppliers?

What is "We are 95% confident that the interval from a to b captures the true difference in the proportion of defective parts from the two suppliers"?

400

A large clinical trial of the effect of diet on breast cancer assigned women at random to either a normal diet or a low-fat diet. To check that the random assignment did produce comparable groups, we can compare the two groups at the start of the study. Asked if there is a family history of breast cancer: 3396 of the 19,541 women in the low-fat group and 4929 of the 29,294 women in the control group said "Yes". If the random assignment worked well, there should not be a significant difference in the proportions with a family history of breast cancer. Determine if the observed difference is significant.

What is "We do not have enough evidence to conclude that there is a statistically significant difference in the proportions of women assigned to the two groups who have a family history of breast cancer."?

400

A teacher gives students time in class to study for a test. The teacher assigns each of their 24 students to either study using a set of flashcards the teacher created or to study by creating their own flashcards. They plan to construct a two-sample t interval comparing the test scores of the two groups.

Which of the following are conditions for this type of interval?

A The test scores for each student are independent of each other.
B The teacher assigns the students to the groups randomly.
C There are at least 10 students who pass the test and 10 students who fail the test.

What is A and B? (C is the normality condition for proportions)

400

A researcher wished to compare the average amount of time spent in extracurricular activities by high school students in a suburban school district with that in a school district with that in a school district of a large city. The researcher obtained an SRS of 60 high school students in a large suburban school district and found the mean time spent in extracurricular activities per week to be 6 hours with a standard deviation of 3 hours. The researcher also obtained an independent SRS of 40 high school students in a large city district and found the mean time spent in extracurricular activities to be 5 hours with a standard deviation of 2 hours. Suppose that the researcher decided to carry out a significance test of H_o: mu_suburban = mu_city versus a two-sided alternative. This would be the test statistic:

What is t = 2?

400

At a baseball game, 42 of 65 randomly selected people report owning an iPod. At a rock concert occurring at the same time across town, 34 of 52 randomly selected people report owning an iPod. A researcher wants to test the claim that the proportion of iPod owners at the two venues is different. A 90% confidence interval for the difference in population proportions is (-0.154, 0.138). Do you have convincing evidence that the proportion of iPod owners at the two venues is different?

What is no. Since the confidence interval includes 0, the researcher cannot conclude that the proportion of iPod owners at the two venues is different.

500

An article in the Arizona Daily Star (11-19-08) described an experiment to investigate the effect of gingko on dementia. Researchers randomly assigned 1500 elderly volunteers to receive gingko and 1500 elderly volunteers to receive a placebo. After six years, 277 of the ginkgo group and 246 of the placebo group had dementia. Calculate and interpret a 95% confidence interval for the true difference the proportion of patients who receive gingko that get dementia and the proportion of patients who receive placebo that get dementia.

What is "I am 95% confident that the interval from –0.006 to 0.048 captures the true difference in the proportion of elderly people like the ones in the study who take gingko and get dementia and the proportion who receive placebo and get dementia."

500

In an experiment to learn whether Substance M can help restore memory, the brains of 20 rats were treated to damage their memories. The rats were trained to run a maze. After a day, 10 rats (determined at random) were given M and 7 of them succeeded in the maze. Why should a two-sample z test for "no difference" against "a significantly higher proportion of the M group succeeds" not be used:

What is because the Normal condition is violated?

500

You are constructing a 90% confidence interval for the difference of means from simple random samples from two independent populations. The sample sizes are n₁ = 6 and n₂ = 18. You draw dot plots of the samples to check the normality condition for two-sample t-procedures. Which of the following descriptions of those dot plots would suggest that it is safe to use t-procedures?
I. The dot plot of sample 1 is roughly symmetric, the dot plot of sample 2 is moderately skewed left. There are no outliers.
II. Both dot plots are roughly symmetric, sample 2 has an outlier.
III. Both dot plots are strongly skewed to the right. There are no outliers.

What is I?

500

What is "reject H_o at the significance level of 0.05. There is convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts."?

500

An SRS of size 100 is taken from Population A with proportion of 0.8 successes. An independent SRS of size 400 is taken from Population B with proportion 0.5 of successes. The sampling distribution for the difference (Population A - Population B) in sample proportions has this mean and standard deviation:

What is mean = 0.3 and standard deviation = 0.047?