AP Statistics Unit 6pt1 MC

Mrs. Malatlian selects a random sample of 28 seniors from the 300 seniors at her school and finds that 20 of them are planning to participate in the school’s annual capture the flag game. She wants to construct a confidence interval for p = the proportion of all seniors who plan to participate in the game. Which condition has she failed to meet, if any?

• The sample was selected at random.

• np̂≥10

• n(1-p̂)≥10

• The sample size is less than 10% of the population size.

• 1. All conditions are met.

Answer explanation

n=28

p̂=20/28

np̂=20

n(1-p̂)=8<10

1. The report of a sample survey of 1,014 adults says, “With 95% confidence, the true population proportion of all Americans that expect to spend more money on gifts this year than last year is between 0.09 and 0.15.” What does the phrase “95% confidence” mean

• 95% of all Americans will spend between 9% and 15% more than what they spent last year.

• 9% to 15% of all Americans will spend 95% of what they spent last year.

• 1. There is a 95% chance that the percent who expect to spend more is between 9% and 15%.

• 1. The method used to get the stated interval, when used over and over, produces intervals which include the true population proportion about 95% of the time.

• 1. The population parameter will be between 0.09 and 0.15 in 95% of all confidence intervals computed in this same manner. 

Answer explanation

Confidence intervals give you a range of plausible values if you did all sample of the same size it should capture the truth 95% of the time.

A political candidate hopes that a majority of voters will vote for him. He is told by his polling organization that a 90% confidence interval for the true proportion of voters who support his candidacy is 0.45 to 0.53. What are the values of the point estimate and margin of error for this interval?

• point estimate = 0.50; margin of error = 0.08.

• point estimate = 0.50; margin of error = 0.04.

• point estimate = 0.49; margin of error = 0.08

• 1. point estimate = 0.49; margin of error = 0.04.

• point estimate = 0.49; margin of error cannot be determined without knowing the sample size.

A 95% confidence interval for p, the true proportion of all shoppers at a large grocery store who purchase 2% milk, was found to be (0.436, 0.582). Given that all conditions were met, which of the following is the most appropriate conclusion?

• Based upon these data, the majority of shoppers purchase 2% milk.

• Based upon these data, at least 43.6% of shoppers purchase 2% milk.

• Based upon these data, at least 58.2% of shoppers purchase 2% milk.

• Based upon these data, less than 50% of shoppers purchase 2% milk.

• Based upon these data, 95% of shoppers purchase 2% milk. 
  

Answer explanation

This is because 0.436 is the lower bound of the interval. Which means is is plausible that at least 43.6% of shoppers purchased 2% milk. Also at most 58.2% purchased the 2% milk.

A 95% confidence interval for the proportion of all statistics students from a large high school that have a part-time job is (0.225, 0.395). You plan to compute a 99% confidence interval using the same data. Which of the following statements is correct?

• The intervals will have the same width.

• The 99% confidence interval will be narrower.

• The 99% confidence interval will be wider.

• A 99% confidence interval will always be exactly twice as wide as a 95% confidence interval.

• 1. A 99% confidence interval will always be exactly 1/2 as wide as a 95% confidence interval.

Answer explanation

When you increase the confidence you increase the z* so z*: 1.96→2.58

1. You are told that the proportion p̂ of those who answered "yes" to a poll about internet use is   p̂= 0.70, and the standard error of the proportion is 0.0458. What was the sample size? 

• 50

• 98

• 100

• 200

• The sample size cannot be determined from the information given.

After a college’s football team once again lost a football game to the college’s arch rival, the alumni association conducted a survey to see if alumni were in favor of firing the coach. An SRS of 100 alumni from the population of all living alumni was selected. Sixty-four of the alumni in the sample were in favor of firing the coach. Let p = the true proportion of all living alumni who favor firing the coach. Which of the following gives the correct 95% confidence interval for p?

• 0.64±0.009

• 0.64±0.079

• 0.64±0.094

• 0.64±0.124

• 0.64±0.048

1. A buyer for a grocery chain inspects large truckloads of apples to determine the proportion p of apples in the shipment that are rotten. She will only accept the shipment if there is convincing evidence that this proportion is less than 0.06. She selects a random sample of 50 apples from the over 20,000 apples on the truck to test the hypotheses H₀: p = 0.06, H_a: p < 0.06. Which of the following conditions for inference have not been met in this situation?

• The data are a random sample from the population of interest.

• The sample size is less than 10% of the population size.

• 1. np ≥ 10 and n(1−p) ≥ 10

• The population distribution is approximately Normal.

• More than one condition is violated.

Answer explanation

n=50

p=0.06

np=50(0.06)=3<10

A 95% confidence interval for the proportion of viewers of a certain reality television show who are over age 30 is (0.26, 0.35). The show’s producers want to test the hypothesis H₀: p = 0.25 against H_a: p ≠ 0.25. Which of the following is an appropriate conclusion for them to draw?

• Reject H₀ at both α = 0.01 and α = 0.05.

• Reject H₀ at α = 0.05.

• Fail to reject H₀ at α = 0.05 but reject at α = 0.01.

• Reject H₀ at α = 0.10 , fail to reject for any smaller α level.

• The producers do not have enough information to perform a test of significance.

Answer explanation

We reject the null because the interval did not capture the null proportion at a 95% confidence.

95% confidence level →α=0.05

1. Mr. Mastrogiacomo is testing the hypothesis that the proportion of households in a large town that have high-speed internet service is equal to 0.7 against the alternative that the proportion is different from 0.7. What is the chief advantage of using a confidence interval to test this hypothesis rather than a significance test?

• 1. A confidence interval can be one-sided or two-sided but the significance test is always two-sided.

• The conditions for using a confidence interval are less restrictive than for a significance test.

• 1. A confidence interval has more power than the significance test.

• 1. A confidence interval gives a set of plausible values for the true proportion.

• There is no advantage. A significance test should be used, not a confidence interval.