Proportions

Tatiana wonders if the same proportion of teens and adults check social media at least once per day. She wants to obtain a random sample of people from each group to test if there is a significant difference between the proportion of teens and adults that check social media at least once per day.

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Kiley has a dime and a nickel, and she wonders if they have the same likelihood of showing heads when they are flipped. She flips each coin 100 times to test if there is a significant difference in the proportion of flips that they each land showing heads.

Which of the following is an appropriate set of hypotheses for Kiley's significance test?

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Caroline thinks she can flip a coin so it lands showing heads more often with her right hand. She flipped a coin 40 times with each hand. She wants to test whether she gets more heads with her right hand than her left hand. (Assume all conditions have been met.)

Which of the following would be an appropriate test statistic for their test?

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An economist was comparing unemployment rates between states. A random sample of 1000 adults in Alaska showed that 70 were unemployed, and a random sample of 1000 adults in Minnesota showed that 30 were unemployed.

The economist want to use these results to whether the unemployment rate in Alaska is higher than that of Minnesota. Assume that all conditions have been met.

Which of the following would be an appropriate test statistic for their test?

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A large school district was considering changing the start times to its schools. The change would involve high schools starting later and elementary schools starting earlier. The school district was curious if parents supported this change, so they surveyed separate random samples of parents from both types of school in the district.

The school district wants to determine if the proportion of parents who support the change is significantly different for each type of school. Assume that all conditions have been met.

Which of the following would be an appropriate test statistic for the district's test?

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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.

What can the sociologist conclude?

• There is sufficient evidence to show that there is a difference between proportions.

• There is insufficient evidence to show that there is a difference between proportions.

• There is sufficient evidence to show that there is no difference between proportions.

• There is insufficient evidence to show that there is no difference between 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 whether there is a difference in the proportion of cancelled rides between the two cities. The test statistic was z=0.83 and the p-value was approximately 0.41.

At the α=0.01 level of significance, is there sufficient evidence to conclude that the proportion of cancelled rides is different between the two cities?

• Yes, since the p-value is greater than 0.01.

• Yes, since the test statistic is greater than 0.01.

• No, since the p-value is greater than 0.01.

• No, since the test statistic is greater than 0.01.

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.

Sanjay used this sample to build a 95% confidence interval to estimate the difference between the proportion of females and males receiving extra credit. The resulting interval was 0.05±0.07.

Based on the interval, what do we know about the corresponding P-value and conclusion at the α=0.05 level of significance?

• The P-value is less than α=0.05, and he cannot conclude that there is a difference between the proportions.

• The P-value is less than α=0.05, and he should conclude that there is a difference between the proportions.

• The P-value is greater than α=0.05, and he cannot conclude that there is a difference between the proportions.

• The P-value is greater than α=0.05, and he should conclude that there is a difference between the proportions.

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 results of the study produced a test statistic of z=-2.90 and P-value of approximately 0.002. Assume that all conditions for inference were met.

At the α=0.01 level of significance, is there sufficient evidence to conclude that the proportion of unpaid tickets is lower when both payment options are offered?

• Yes, since the P-value is less than 0.01.

• Yes, since the test statistic is less than 0.01.

• No, since the P-value is less than 0.01.

• No, since the test statistic is less than 0.01.

At a baseball game, 42 of 65 randomly selected people own an iPod. At a rock concert occurring at the same time across town, 34 of 52 randomly selected people own 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(game − concert) is (−0.154, 0.138). Which of the following gives the correct outcome of the researcher’s test of the claim?

• Because the confidence interval includes 0, the researcher can conclude that the proportion of iPod owners at the two venues is the same.
• Because the center of the interval is –0.008, the researcher can conclude that a higher proportion of people at the rock concert own iPods than at the baseball game.
• Because the confidence interval includes 0, the researcher cannot conclude that the proportion of iPod owners at the two venues is different.
• Because the confidence interval includes more negative than positive values, the researcher can conclude that a higher proportion of people at the rock concert own iPods than at the baseball game.