Exam 11: Asking and Answering Questions About the Difference Between Two Proportions

An article describes the results of an experiment. Patients with gluten-related disorders were randomly assigned to one of two groups. One group was a control group who received treatment A. Of the 73 patients in this group, 42 reported improvements at the end of the study period. Those in the second group received treatment B. Of the 98 patients in the supplement group, 69 reported improvements at the end of the study period. ​ Is it possible to use the large-sample z test and the large-sample z confidence interval to draw conclusions about a difference in treatment proportions? Why? ​

A study was carried out to see if seniors were more likely than freshmen to favor banning cell phones at school. Which of the following hypotheses should be tested to answer this question? ​

A marketing strategist would like to estimate the difference between the proportion of men who list "Action" as their favorite movie genre and the proportion of women who list "Action" as their favorite genre. The strategist gathers a random sample of 300 men and 200 women. Of the men, 156 indicate that "Action" is their favorite genre; of the women, 87 indicate that "Action" is their favorite. a.Determine if the large sample confidence interval procedures can be used for estimation. b.Create a 95% confidence interval for the difference of the population proportions. c.Interpret the confidence interval. d.Based on the confidence interval, is it possible that the two population proportions are equal?

An article describes an experiment that investigated if car following behavior for motorcyclists in conditions A is less dangerous than in conditions B. Motorcyclists were randomly assigned to one of two groups. The 60 participants in the first group were placed in conditions A while driving in a simulator. The 60 participants in the second group were placed in conditions B while driving in a simulator. The motorcyclists were instructed to pay attention to neighboring traffic, maintain two car lengths from a virtual lead vehicle, and avoid crashes. Of the motorcyclists from the group A, 38 avoided crashes. Of the motorcyclists from the group B, 17 avoided crashes. Does these suggest that car following driving in conditions A is less dangerous than car following driving in conditions B? Carry out large-sample z test for a difference in treatment proportions and interpret the conclusion in context.

A study is conducted at a university to determine if it's test-preparation seminar improves the proportion of students who pass the AP Statistics exam. The following summary data were obtained. Is there evidence to support the claim that the proportion of passing grades is higher for those who attended the seminar? Compute the test statistic for this test.

One method of human locomotion that appears to have no function except to express joy is skipping, a skill usually learned in early and middle childhood. In surveys gathered over a 3-year period, University students were asked, "Approximately how long ago did you last spontaneously skip?" The total sample sizes and the numbers who responded "Less than 1 week ago" are given in the table below: ​ a)Construct a 95% confidence interval for the difference in the population proportions of males and females responding "Less than 1 week." Assume that it is reasonable to regard these samples as representative of the corresponding populations. b)Does there appear to be a difference between the proportions of men and women responding "Less than 1 week?" Provide statistical justification for your answer.

An article describes the results of an experiment. Students of WEB programming courses were randomly assigned to the two groups. One group was a control group of students who were trained by the teacher A. Of the 69 students in this group, 17 passed the final exam. Those in the second group were trained by the teacher B. Of the 131 students in the supplement group, 9 passed the final exam. Based on the given data, what can you learn about the difference in the proportion of students who passed the exam after being trained by the teacher A and those who passed the exam after being trained by the teacher B? Calculate and interpret a 95% bootstrap confidence interval for the difference in proportions.

The hypothesis is equivalent to the hypothesis .

Compare the results of the survey and the experiment. The report describes the results of a survey of 739 automobile owners age 18 to 29 and 564 automobile owners age 30 to 49 in a certain metropolitan area. Survey participants were asked if they own an automobile that is equipped with an electrical engine; 42% of the automobile owners age 18 to 29 and 36% of the automobile owners age 30 to 49 said "yes". ​ An article describes an experiment that investigated if sending or receiving text messages is more distracting than programming navigation devices. Drivers were randomly assigned to one of two groups. The 60 drivers in the cell phone group sent or received messages while driving in a simulator. The 60 drivers in the navigator group programmed navigator devices while driving in a simulator. The drivers were instructed to exit the highway when they came to a rest stop. Of the drivers programming the navigator devices, 38 noticed the rest stop and exited. For the drivers sending and receiving messages, 17 noticed the rest stop and exited. What is the difference between these two types of data?

An article describes the results of an experiment. Students were randomly assigned to the two groups. One group was a control group of students who used source A while preparing for the test. Of the 54 students in this group, 31 passed the test. Those in the second group used source B while preparing for the test. Of the 48 students in the supplement group, 17 passed the test. Construct and interpret a 95% confidence interval for the difference in the proportion of students who passed the test using source A and the proportion of students who passed the test using source B.

One of the most striking human accomplishments is the achievement of intelligible speech by age five. Deficits in speech acquisition often occur when a child has hearing difficulty or is slow to develop the motor functions associated with speech. Other deficits in speech acquisition are of unknown origin. Males generally develop physiologically at a slower rate than females, and in a recent large study of speech development in young children investigators hypothesized that a greater proportion of boys would have speech deficits for this reason. A random sample of healthy two-year-olds was followed over the course of a year. At age 3, each child was classified as having a speech delay or not. The data by gender is given in the table below: ​ a)At the .05 significance level, do the data provide evidence of the hypothesized developmental differences? Provide statistical evidence for your conclusion. b)Is this an observational or experimental study? Justify your response in a few sentences.

The report describes the results of a survey of 739 automobile owners age 18 to 29 and 564 automobile owners age 30 to 49 in a certain metropolitan area. Survey participants were asked if they own an automobile that is equipped with an electrical engine; 42% of the automobile owners age 18 to 29 and 36% of the automobile owners age 30 to 49 said "yes". Suppose you want to decide if there is evidence that the proportion of owners of automobiles equipped with electrical engine is different for age 18 to 29 and age 30 to 49. What is the value of the test statistics and what is the associated P-value? If a significance level of 0.05 is selected for the test, will you reject or fail to reject the null hypothesis?

An article describes an experiment that investigated if car following behavior for motorcyclists in conditions A is less dangerous than in conditions B. Motorcyclists were randomly assigned to one of two groups. The 60 participants in the first group were placed in conditions A while driving in a simulator. The 60 participants in the second group were placed in conditions B while driving in a simulator. The motorcyclists were instructed to pay attention to neighboring traffic, maintain two car lengths from a virtual lead vehicle, and avoid crashes. Of the motorcyclists from the group A, 38 avoided crashes. Of the motorcyclists from the group B, 17 avoided crashes. Use the given information to construct and interpret a 95% confidence interval for the difference in the proportions of motorcyclists who would avoid crashes.

A study determined that in 2009, 38% of those in a random sample of 100 adult Americans had accounts with some type of social web site. In a similar survey conducted in 2012, 53% reported having accounts. Suppose that the 2012 figure was also based on a random sample of size 100. The 95% confidence interval for the difference in population proportions, P₂₀₁₂-P₂₀₀₉ , is: ​

A study seeks to determine whether the proportion of Dutch boys who listen to music at high volume is greater than this proportion for Dutch girls. After checking to make sure that all of the necessary conditions were met, we calculated the value of the z test statistic to be z = 2.999. Which of the following is the P-value associated with this test? ​

A study compared flu vaccine administered by injection and flu vaccine administered as a nasal spray. Each of 4,000 children under the age of 5 who participated in the study received both a nasal spray and an injection, but only one was the real vaccine and the other was salt water. At the end of the flu season, it was determined that of the 2,000 children receiving the real vaccine by nasal spray, 10.1% got the flu. Of the 2,000 children receiving the real vaccine by injection, 15.6% got the flu. The 95% confidence interval for the difference in treatment proportions is: ​

One of the most striking human accomplishments is the achievement of intelligible speech by age five. Deficits in speech acquisition often occur when a child has hearing difficulty or does is slow to develop the motor functions associated with speech. Other deficits in speech acquisition are of unknown origin. It has been hypothesized that a family history of speech communication disorder might be a factor in speech delay. The theory is that if members of the family have difficulty communicating, this may increase the probability of a speech deficit in a young family member. A random sample of healthy two-year-olds was followed over the course of a year. At age 3, each child was classified as having a speech delay or not. These data, cross-classified by a family history of speech communication disorder, is given in the table below: ​ a)At the .05 significance level, do the data provide evidence of the hypothesized developmental differences? Provide statistical evidence for your conclusion. b)Is this an observational or experimental study? Justify your response in a few sentences.

When comparing two population means, it is important to make the distinction between independent samples and paired samples. What makes two samples independent?

Each person in a random sample of 200 students and a random sample of 150 faculty at a large university was asked "Do you take public transportation to campus?" Fifty-eight percent of the students and 46% of the faculty responded "Yes". This data was used to carry out a test of the hypotheses . The value of the test statistic was z = 2.226. For a significance level of 0.01, which of the following is correct? ​

One method of human locomotion that appears to have no function except to express joy is hopping, a skill usually learned in early and middle childhood. In surveys gathered over a 3-year period, University students were asked, "Approximately how long ago did you last spontaneously hop?" The total sample sizes and the numbers who responded "Less than 1 week ago" are given in the table below: ​ a)Construct a 95% confidence interval for the difference in the population proportions of males and females responding "Less than 1 week." Assume that it is reasonable to regard these samples as representative of the corresponding populations. b)Does there appear to be a difference between the proportions of men and women responding "Less than 1 week?" Provide statistical justification for your answer.