Proportions Based on Samples

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 (Game – Concert) in population proportions is (−0.154, 0.138). Which of the following gives the correct outcome of the researcher’s test of the claim?

• 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 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.

• Because the interval includes 0, the researcher can conclude that the proportion of iPod owners at the two venues is the same.

• Because the interval includes −0.008, the researcher cannot conclude that the proportion of iPod owners at the two venues is different.

• Because the interval includes 0, the researcher cannot conclude that the proportion of iPod owners at the two venues is different.

Answer explanation

The fact that 0 is included in a confidence interval for 𝑝̂ 1−𝑝̂ 2 means that we don’t have convincing evidence of a difference between the true proportions.

A telephone poll of an SRS of 1234 adults found that 62% are generally satisfied with their lives. The announced margin of error for the poll was 3%. Does the margin of error account for the fact that some adults do not have telephones?

• Yes; taking an SRS eliminates any possible bias in estimating the population proportion.

• No; the margin of error only accounts for sampling variability.

• Yes; the margin of error accounts for undercoverage but not nonresponse.

• No; the margin of error accounts for nonresponse but not undercoverage.

• Yes; the margin of error accounts for all sources of error in the poll.

Answer explanation

An SRS eliminates bias in the choice process but does not help with bias like undercoverage. The margin of error does not account for any bias that is part of the data collection process.

At a baseball game, 42 of 65 randomly selected people own an iPad. At a rock concert occurring at the same time across town, 34 of 52 randomly selected people own an iPad. A researcher wants to test the claim that the proportion of iPad owners at the two venues is different. A 90% confidence interval for the difference (Game – Concert) in population proportions is (−0.154, 0.138). Which of the following gives the correct outcome of the researcher’s test of the claim?

• Because the interval includes 0, the researcher can conclude that the proportion of iPad 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 iPads than at the baseball game.

• Because the interval includes 0, the researcher cannot conclude that the proportion of iPad owners at the two venues is different.

• Because the interval includes −0.008, the researcher cannot conclude that the proportion of iPad owners at the two venues is different.

• Because the interval includes more negative than positive values, the researcher can conclude that a higher proportion of people at the rock concert own iPads than at the baseball game.

The Gallup Poll interviews 1600 people. Of these,18% say that they jog regularly. The news report adds: “The poll had a margin of error of plus or minus three percentage points at a 95% confidence level.” You can safely conclude that

• 95% of all Gallup Poll samples like this one give answers within ±3% of the true population value.

• if Gallup took many samples, 95% of them would find that 18% of the people in the sample jog.

• the percent of the population who jog is certain to be between 15% and 21%.

• 95% of the population jog between 15% and 21% of the time.

• we can be 95% confident that the sample proportion is captured by the confidence interval.

A confidence interval for a difference in proportions is −0.077 to 0.013. What are the point estimate and the margin of error for this interval?

• –0.032, 0.045

• –0.032, 0.090

• –0.032, 0.180

• –0.045, 0.032

• –0.045, 0.090

In a random sample of 100 students from a large high school, 37 regularly bring a reusable water bottle from home. Which of the following gives the correct value and interpretation of the standard error of the sample proportion?

• In samples of size 100 from this school, the sample proportion of students who bring a reusable water bottle from home will be at most 0.095 from the true proportion.

• In samples of size 100 from this school, the sample proportion of students who bring a reusable water bottle from home will be at most 0.048 from the true proportion.

• In samples of size 100 from this school, the sample proportion of students who bring a reusable water bottle from home typically varies by about 0.095 from the true proportion.

• In samples of size 100 from this school, the sample proportion of students who bring a reusable water bottle from home typically varies by about 0.048 from the true proportion.

• There is not enough information to calculate the standard error.

Many television viewers express doubts about the validity of certain commercials. In an attempt to answer their critics, Timex Group USA wishes to estimate the true proportion p of all consumers who believe what is shown in Timex television commercials. What is the smallest number of consumers that Timex can survey to guarantee a margin of error of 0.05 or less at a 99% confidence level? Round up to the nearest whole number.

What is the critical value for calculating a 94% confidence interval for a population proportion? Round to three decimal places.

A radio talk show host with a large audience is interested in the proportion p of adults in his listening area who think the drinking age should be lowered to 18. To find this out, he poses the following question to his listeners: “Do you think that the drinking age should be reduced to 18 in light of the fact that 18-year-olds are eligible for military service?” He asks listeners to go to his website and vote “Yes” if they agree the drinking age should be lowered and “No” if not. Of the 100 people who voted, 70 answered “Yes.” Which of the following conditions are violated?

I. Random

II. 10%

III. Large Counts

• I only

• II only

• III only

• I and II

• I, II, and III

A marketing assistant for a technology firm plans to randomly select 1000 customers to estimate the proportion who are satisfied with the firm’s performance. Based on the results of the survey, the assistant will construct a 95% confidence interval for the proportion of all customers who are satisfied. The marketing manager, however, says that the firm can afford to survey only 250 customers. How will this decrease in sample size affect the margin of error?

• The margin of error will be about 4 times larger.

• The margin of error will be about 2 times larger.

• The margin of error will be about the same size.

• The margin of error will be about half as large.

• The margin of error will be about one-fourth as large.