Making conclusions about the difference of proportions

Jillian is an analyst for a ride sharing app that connects users with drivers. She wonders if drivers in Dallas are more or less likely to cancel rides than drivers in Houston. She takes a random sample of \[1000\] rides from Dallas and finds that \[30\] were cancelled. A random sample of \[1000\] rides from Houston shows \[24\] cancelled rides. She used these results to test \[H_0: p_\text{D}=p_\text{H}\] versus \[H_\text{a}: p_\text{D} \neq p_\text{H}\]. The test statistic was \[z=0.83\] and the P-value was approximately \[0.41\]. Assume that all conditions for inference were met. At the \[\alpha=0.01\] level of significance, is there sufficient evidence to conclude that the proportion of cancelled rides is different between the two cities?

No, since P is greater than 0.01

Jenny heard that women are more likely than men to choose an even number when asked what their favorite number is. She took a random sample of people and asked them each what their favorite number was. She used the results to test \[H_0: p_\text{W}-p_\text{M}=0\] versus \[H_\text{a}: p_\text{W}-p_\text{M}>0\], where \[p_\text{W}\] represents the proportion of women who would choose an even number and \[p_\text{M}\] represents the proportion of men who would choose an even number. Her sample data produced a test statistic of \[z=0.22\] and P-value of approximately \[0.41\]. Assume that all conditions for inference were met. At the \[\alpha=0.05\] level of significance, is there sufficient evidence to conclude that the proportion of people who would choose an even number is higher for women than it is for men?

No, since P is greater than 0.05

Roberto was comparing how often trains were late in France and Germany. He observed \[10\] late trains in a random sample of \[200\] scheduled stops in France, and \[12\] late trains in a random sample of \[250\] scheduled stops in Germany. He used these results to test \[H_0: p_\text{F}=p_\text{G}\] versus \[H_\text{a}: p_\text{F}>p_\text{G}\]. The test statistic was \[z=0.098\] and the P-value was approximately \[0.46\]. Assume that all conditions for inference were met. At the \[\alpha=0.10\] level of significance, is there sufficient evidence to conclude that the proportion of late trains is higher in France than it is in Germany?

No, since P is greater than 0.1

Ana wonders if adults living in the northern part of her town are more or less likely to support a new school tax than adults in the southern part of town. She takes a random sample of \[100\] adults from each part of town. She finds that \[79\] support the tax in the northern sample, while \[69\] support the tax in the southern sample. Ana used these samples to build a \[95\%\] confidence interval to estimate \[p_\text{N}-p_\text{S}\]. The resulting interval was \[0.1 \pm 0.12\]. She wants to use this interval to test \[H_0:p_\text{N}=p_\text{S}\] versus \[H_\text{a}:p_\text{N} \neq p_\text{S}\]. Assume that all conditions for inference have been met. Based on the interval, what do we know about the corresponding P-value and conclusion at the \[\alpha=0.05\] level of significance?

P is greater than 0.05, Ana cannot conclude that there is a difference between the proportions.

Sanjay is researching if female students are more or less likely than male students to have received extra credit at a large university. He takes a random sample of \[300\] students. Here are his results: Received extra credit?Total\[\hat p\]Female\[23\]\[n_\text{F}=156\]\[\hat p_\text{F}=0.147\]Male\[14\]\[n_\text{M}=144\]\[\hat p_\text{M}=0.097\] Sanjay used this sample to build a \[95\%\] confidence interval to estimate \[p_\text{F}-p_\text{M}\]. The resulting interval was \[0.05 \pm 0.07\]. He wants to use this interval to test \[H_0:p_\text{F}=p_\text{M}\] versus \[H_\text{a}:p_\text{F} \neq p_\text{M}\]. Assume that all conditions for inference have been met. Based on the interval, what do we know about the corresponding P-value and conclusion at the \[\alpha=0.05\] level of significance?

P is greater than 0.05, Sanjay cannot conclude that there is a difference between the proportions.

A university offers a certain course that students can take in-person or in an online setting. Teachers of the course were curious if there was a difference in the passing rate between the two settings. Data from a recent semester showed that \[80\%\] of students passed the in-person setting, and \[75\%\] of students passed the online setting. The teachers used those results to make a \[95\%\] confidence interval to estimate the difference between the proportion of students who pass in each setting of the course \[(p_\text{in-person}-p_\text{online})\]. The resulting interval was approximately \[(-0.04,0.14)\]. They want to use this interval to test \[H_0: p_\text{in-person}=p_\text{online}\] versus \[H_\text{a}: p_\text{in-person} \neq p_\text{online}\]. Based on the interval, what do we know about the corresponding P-value and conclusion at the \[\alpha=0.05\] level of significance?

P is greater than 0.05, they cannot conclude that there is a difference between the proportions.

A sociologist took a random sample of \[1200\] drivers and found that \[59\] of the \[610\] men in the sample had received a speeding ticket, while \[28\] of the \[590\] women in the sample had received a speeding ticket. The sociologist used those results to make a \[99\%\] confidence interval to estimate the difference between the proportion of male and female drivers who have received a speeding ticket \[(p_\text{M}-p_\text{W})\]. The resulting interval was \[(0.011, 0.087)\]. They want to use this interval to test \[H_0: p_\text{M}=p_\text{W}\] versus \[H_\text{a}: p_\text{M} \neq p_\text{W}\] at the \[\alpha=0.01\] significance level. Assume that all conditions for inference have been met. Based on the interval, what do we know about the corresponding P-value and conclusion at the \[\alpha=0.01\] level of significance?

P is less than 0.01, they should conclude that there *is* a difference between the proportions.

A market researcher obtained separate random samples of \[500\] car owners from East coast of the United States and \[500\] car owners from the West coast. They found that \[58\] people in the West coast sample owned an electric car compared to \[35\] in the East coast sample. The researcher used those results to make a \[95\%\] confidence interval to estimate the difference between the proportion of car owners in each region who own an electric car \[(p_\text{W}-p_\text{E})\]. The resulting interval was \[(0.010, 0.082)\]. They want to use this interval to test \[H_0: p_\text{W}=p_\text{E}\] versus \[H_\text{a}: p_\text{W} \neq p_\text{E}\] at the \[\alpha=0.05\] significance level. Assume that all conditions for inference have been met. Based on the interval, what do we know about the corresponding P-value and conclusion at the \[\alpha=0.05\] level of significance?

P is less than 0.05, they should conclude that there is a difference between the proportions.

A large school district found that \[88\] of \[100\] randomly selected students at East High School had internet access at home. A separate random sample from West High showed \[79\] of \[100\] students had internet access at home. Administrators used those results to make a \[90\%\] confidence interval to estimate the difference between the proportion of students at each school who have internet access at home \[(p_\text{E}-p_\text{W})\]. The resulting interval was approximately \[0.09\pm0.086\]. They want to use this interval to test \[H_0: p_\text{E}=p_\text{W}\] versus \[H_\text{a}: p_\text{E} \neq p_\text{W}\] at the \[\alpha=0.10\] significance level. Assume that all conditions for inference have been met. Based on the interval, what do we know about the corresponding P-value and conclusion at the \[\alpha=0.10\] level of significance?

P is less than 0.1, they should conclude that there is a difference between the proportions.

A veterinarian is studying a certain disease that seems to be affecting male cats more often than female cats. They obtain a random sample of records from \[500\] cats. They find \[24\] of \[259\] male cats have the disease, while \[14\] of \[241\] female cats have the disease. The veterinarian used these results to test \[H_0: p_\text{M}=p_\text{F}\] versus \[H_\text{a}: p_\text{M} > p_\text{F}\]. The test statistic was \[z=1.46\] and the P-value was approximately \[0.07\]. Assume that all conditions for inference were met. At the \[\alpha=0.10\] level of significance, is there sufficient evidence to conclude that a larger proportion of male cats have the disease?

Yes, since P is less than 0.01

A biologist is studying a certain disease affecting oak trees in a forest. They are curious if there's a difference in the proportion of trees that are infected in the north and south sections of the forest. They take separate random samples of \[100\] trees from each section. They find \[32\] infected in the North sample and \[18\] infected in the South sample. The biologist used these results to test \[H_0: p_\text{N}=p_\text{S}\] versus \[H_\text{a}: p_\text{N} \neq p_\text{S}\]. The test statistic was \[z=2.29\] and the P-value was approximately \[0.022\]. Assume that all conditions for inference were met. At the \[\alpha=0.05\] level of significance, is there sufficient evidence to conclude that the proportion of infected trees is different in the north and south sections of the forest?

Yes, since P is less than 0.05

A city used to require mailed payments for parking tickets. City officials piloted a system that allowed people to choose between paying by mail or paying online. They were curious if giving people both payment options would result in fewer unpaid parking tickets. To test the new system, each parking ticket one month was printed with either a "mail only" payment option or both payment options (mail and online). Officers flipped a coin to determine which message was printed on each ticket. The data from their study was used to test \[H_0 : p_\text{B} = p_\text{M}\] versus \[H_\text{a} : p_\text{B} < p_\text{M}\], where \[p_\text{B}\] is the proportion of unpaid tickets with both payment options and \[p_\text{M}\] is the proportion of unpaid tickets with the mail only option. The results of the study produced a test statistic of \[z=-2.90\] and P-value of approximately \[0.002\].

Yes, since P is less than 0.1