Proportions and Critical Values

A preliminary study suggests that 17% of people agree on an issue. How large must your sample size be so you could be within ± 5% with 95% confidence?

• 152

• 30

• 2

• 217

Which confidence would result in a wider interval (assuming nothing else changed), 91% or 97%?

• 97%, z* is larger for higher confidence intervals

• 97%, the standard error is larger for higher confidence intervals

• 91%, z* is larger for lower confidence intervals

• 91%, the standard error is smaller for lower confidence intervals

What is the critical value for calculating a 94% confidence interval for a population proportion? Enter the critical value to 2 decimal places.

Answer explanation

In a 94% confidence interval, the tail area is (1 – 0.94)/2 = 0.03. Use invNorm or a standard normal table to determine the critical value for this tail area.

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

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

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

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

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

Answer explanation

95% of confidence intervals will capture the true proportion, not any particular sample proportion.

A confidence interval for a difference in proportions is −0.077 to 0.013. What are the point estimate rounded to 3 decimal places.

Answer explanation

The point estimate is the midpoint of the interval.

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?

• 550

• 600

• 650

• 700

• 750

Thirty-five people from a random sample of 125 workers from Company A admitted to using sick leave when they weren’t really ill. Seventeen employees from a random sample of 68 workers from Company B admitted that they had used sick leave when they weren’t ill. Which of the following is a 95% confidence interval for the difference in the proportions of workers at the two companies who would admit to using sick leave when they weren’t ill?

Answer explanation

Formula for a confidence interval for the difference in two proportions is:

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.

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 0.048 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.

• 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.

Answer explanation

Because 𝑛 = 100 < 10% of all students at this large high school the standard error of 𝑝̂ is approximately √(0.37(1−0.37)/100) = 0.048.

Assume that 32% of the population is nearsighted, what is the standard deviation for the sample proportion for a group of 100 new students?

• 0.0022

• 0.0032

• 0.0466

• 0.0566