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

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. Each person in a random sample of female students and a random sample of male students at a large middle school was 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: Number responding "Less than 1 week" Gender Males 71 411 Females 64 253 a) Construct a $95 \%$ confidence interval for the difference in the proportions of males and the proportion of females at this school who would respond "Less than 1 week." b) Referring to your work in part (a), does there appear to be evidence of a difference in the population proportions of males and females at this school who would respond "Less than 1 week?" Provide appropriate statistical justification for your answer.

If the confidence interval for $p _ { 1 } - p _ { 2 }$ does not contain 0 , it is plausible that $p _ { 1 } = p _ { 2 }$ .

If the $P$ -value in a test of $H _ { o } : p _ { 1 } - p _ { 2 } = 0$ is $0.014$ , you would reject the null hypothesis at the $0.01$ significance level.

The pre-historic development of agriculture is an active area of research among anthropologists. The transition from a hunter-gatherer society to an agricultural society is generally assumed to be signaled by an increase in cavities in teeth. Investigators recently studied teeth unearthed in two societies in Colorado: Basketmakers, societies living in small villages, and Post-Basketmakers, societies living in above-ground architectures such as pueblos. The investigators were interested in whether the shift in shelter was associated with a shift in consumption from hunting (meat) to agriculture (maize). Data on the numbers of individuals with cavities are shown in the table below. Cavities tend to occur more among populations subsisting primarily on agricultural products. You may assume that it is reasonable to regard these two samples as representative of teeth from these two populations. Population Cavities No Cavities Basketmaker 52 14 Post-Basketmaker 35 21 a) Construct and interpret a 90% confidence interval for the difference in population proportions of individuals with cavities, Basketmaker - Post-Basketmaker. b) Interpret the 90% confidence level c) Based on your results in parts (a) and (b), would you say that a shift in consumption patterns is a reasonable conclusion on the part of the investigators? Justify your response with statistical evidence from parts (a) and/or (b).

Some psychologists hypothesize 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 delay in a young family member. Random samples of healthy two-year-olds from families with speech communication disorders and from families with no speech communication disorder were followed over the course of a year. At age 3, each child was classified as having a speech delay or not. These data are shown in the table below: Speech delay and gender at 3 years old Family history of Speech communication disorder Speech Delay No Speech Delay Total With 36 135 171 Without 64 404 468 Total 100 539 639 a) Construct a $95 \%$ confidence interval for the difference between the population proportions of healthy two-year-olds from families with speech communication disorders who have a speech delay and this proportion for those from families without speech communication disorders. b) Referring to your work in part (a), does there appear to be a difference between the population proportions of healthy two-year-olds from families with speech communication disorders who have a speech delay and this proportion for those from families without speech communication disorders? Provide statistical justification for your answer. c) Is this an observational study or an experiment? Justify your response in a few sentences.

Careful and constant and hygiene is necessary to reduce the chance of transmitting pathogens by healthcare professionals. Despite the importance of hand hygiene, getting good compliance is challenging. One complicating factor is the potential for observer bias. That is, compliance monitors might be fellow workers who might report higher compliance than neutral observers. A study designed to assess this possibility was conducted in a hospital in a large city. Trained observers working "under cover" - monitored hand hygiene compliance of a randomly selected sample of healthcare professionals over a calendar year. Some observers worked in the same hospital as the professional being observed (Unit-based) while others did not (Non- Unit-based). Data from one of the hospital units are presented in the table below. Do these data provide sufficient evidence of the existence of observer bias? Hand hygiene compliance: Unit- and Non-unit-based monitors Observers Unit based Non Unit based Total Compliance 66 224 290 Non-compliance 25 182 207 Total 91 406 497 a) What are the appropriate null and alternative hypotheses? b) Show the necessary calculations to demonstrate it is reasonable to use the large- sample test for a difference in population proportions for these data. c) Test the hypothesis you indicated in part (a), and interpret the result in the context of the study.

S Suppose that you have estimated the difference between two population proportions, $p _ { 1 } - p _ { 2 }$ , using a confidence interval. What are the appropriate interpretations of the following results? a) Both endpoints of the confidence interval for $p _ { 1 } - p _ { 2 }$ are positive. b) Both endpoints of the confidence interval for $p _ { 1 } - p _ { 2 }$ are negative. c) Zero is included in the confidence interval.

S Suppose that you have estimated the difference between two population proportions, $p _ { 1 } - p _ { 2 }$ , using a confidence interval. What are the appropriate interpretations of the following results? a) Both endpoints of the confidence interval for $p _ { 1 } - p _ { 2 }$ are positive. b) Both endpoints of the confidence interval for $p _ { 1 } - p _ { 2 }$ are negative. c) Zero is included in the confidence interval.

One of the most striking human accomplishments is the achievement of intelligible speech by age five. Males generally develop 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. Random samples of healthy two-year-old males and females were followed over the course of a year, and at age 3 each child was classified as having a speech deficit or not. The data are given in the table below: Speech delay and gender at 3 years old Gender Speech deficit No Speech Deficit Total Female 22 280 302 Male 78 259 337 Total 100 539 639 a) Construct a 95% confidence interval for the difference between the population proportions of males and the proportion of females at this school who would be classified as having a speech deficit. b) Referring to your work in part (a), does there appear to be a difference between the population proportions of males and females at this school who would be classified as having a speech deficit? Provide statistical justification for your answer. c) Is this an observational study or an experiment? Justify your response in a few sentences.

When testing the hypothesis $H _ { o } : p _ { 1 } - p _ { 2 } = 0$ , the alternative hypotheses $H _ { a } : p _ { 1 } - p _ { 2 } > 0$ and $H _ { a } : p _ { 2 } - p _ { 1 } > 0$ can be used interchangeably.

The formula for the standard error of $\hat { p } _ { 1 } - \hat { p } _ { 2 }$ used in hypothesis testing and confidence intervals is the same.

Some animal behaviorists study the evolution of behavior as a consequence of natural selection and the development of species. A common research question involves the extent to which our human behavior is grounded in an evolutionary past, as opposed to culturally imposed. For example, among monkey species in Africa, males function primarily as sentries, warning females that predators are in the area. Might it be so that human males and females exhibit this behavior? That is, do human males assume a protective role and human females a protected role? Researchers studied the look-both-ways behavior of a random sample of females alone and an independent random sample of females with males at a dangerous pedestrian crossing. They reasoned that if females assume a protected role, they should look both ways at a lower proportion when they are with males than when alone. The resulting data are shown in the table below. Social grouping Number of crossings Number looking both ways Adult female, alone 80 60 Adult female with adult male 76 37 a) What null and alternative hypotheses should the researchers test? b) It is reasonable to use the large-sample test for a difference in population proportions. Show that this is true by indicating the necessary calculations. c) Test the hypothesis you indicated in part (a), and interpret the result in the context of the study.

The large sample confidence interval procedure for estimating $p _ { 1 } - p _ { 2 }$ using $z$ can safely be used if the combined samples have at least 20 "successes" and 20 "failures."

If the $P$ -value in a test of $H _ { o } : p _ { 1 } - p _ { 2 } = 0$ is $0.578$ , you have strong evidence that the null hypothesis is true.

The large sample confidence interval procedure for estimating $p _ { 1 } - p _ { 2 }$ requires the random samples to be selected independently.

O 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. Each person in a random sample of female students and a random sample of male students at a large middle school was 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: Number responding "2-3 weeks" Gender Males 104 411 Females 71 253 a) Construct a $95 \%$ confidence interval for the difference in the population proportions of males and the proportion of females at this school who would respond "2-3 weeks." b) Referring to your work in part (a), does there appear to be a difference between the population proportions of men and women responding "2-3 weeks?" Provide statistical justification for your answer.

A $95 \%$ confidence interval for $p _ { 1 } - p _ { 2 }$ is wider than a $90 \%$ confidence interval constructed using the same sample data.

When testing the hypothesis $H _ { o } : p _ { 1 } - p _ { 2 } = 0$ , a test statistic near zero indicates strong support for the null hypothesis.

If the $P$ -value in a test of $H _ { o } : p _ { 1 } - p _ { 2 } = 0$ is $0.0005$ , you have strong evidence that the null hypothesis is false.

Some animal behaviorists study the evolution of behavior as a consequence of natural selection and the development of species. A common research question involves the extent to which our human behavior is grounded in an evolutionary past, as opposed to culturally imposed. For example, among monkey species in Africa males function primarily as sentries, warning females that predators are in the area. Might it be so that human males and females exhibit this behavior? That is, do human males assume a protective role and human females a protected role? Researchers studied the look-both-ways behavior of a random sample of males alone and an independent random sample of males with females at a dangerous pedestrian crossing. They reasoned that if males assume a protective role, they should look both ways at a higher proportion when they are with females than when alone. The resulting data are shown in the table below. Social grouping Number of crossings Number looking both ways Adult male, alone 84 56 Adult male with adult female 76 61 a) What are the appropriate null and alternative hypotheses? b) It is reasonable to use the large-sample test for a difference in population proportions. Show that this is true by indicating the necessary calculations. c) Test the hypothesis you indicated in part (a), and interpret your result in the context of the study.