Advanced Placement: Quiz On Statistics! Trivia Questions

1. In a large high school, student leadership wants to randomly select a sample to survey the student body. They will first randomly select 25 freshman, then randomly select 25 sophomores, then randomly select 25 juniors, and then randomly select 25 seniors. This is an example of a

Cluster sample

Stratified sample

Simple random sample

Convenience sample

Systematic sample

The given scenario describes a stratified sample. In a stratified sample, the population is divided into distinct groups or strata, and a random sample is selected from each stratum. In this case, the student body is divided into four strata based on grade level (freshman, sophomore, junior, and senior), and a random sample of 25 students is selected from each stratum. This ensures that the sample represents the diversity of the student body across different grade levels.

Explanation

The given scenario describes a stratified sample. In a stratified sample, the population is divided into distinct groups or strata, and a random sample is selected from each stratum. In this case, the student body is divided into four strata based on grade level (freshman, sophomore, junior, and senior), and a random sample of 25 students is selected from each stratum. This ensures that the sample represents the diversity of the student body across different grade levels.

2. A least-squares regression line for predicting performance on a college entrance exam based on high school GPA is determined to be Predicted Score = 273.5 + 91.2(GPA). One student in the study had a high school GPA of 3.0, and an exam score of 510. What is the residual for this student?

26.2

43.9

-37.1

-26.2

37.1

The residual for a student is the difference between the actual value and the predicted value. In this case, the predicted score for a student with a high school GPA of 3.0 would be 273.5 + 91.2(3.0) = 547.1. The actual score for this student is 510. Therefore, the residual is 510 - 547.1 = -37.1.

Explanation

The residual for a student is the difference between the actual value and the predicted value. In this case, the predicted score for a student with a high school GPA of 3.0 would be 273.5 + 91.2(3.0) = 547.1. The actual score for this student is 510. Therefore, the residual is 510 - 547.1 = -37.1.

3. A recent poll reported that 43% of Californians approve of the job the Governor is doing, with a margin of sampling error of + or - 3.25% at a 95% level of confidence. Which of these correctly interprets that margin of error?

About 3.2% of all Californians approve of 95% of what the Governor does.

The correct answer correctly interprets the margin of error by stating that about 95% of polls conducted in this way will give a sample proportion within 3.2 percentage points of the actual proportion of all Californians who approve of the job the Governor is doing. This means that the reported approval rate of 43% could range from 39.8% to 46.2% in reality, with a high level of confidence.

Explanation

The correct answer correctly interprets the margin of error by stating that about 95% of polls conducted in this way will give a sample proportion within 3.2 percentage points of the actual proportion of all Californians who approve of the job the Governor is doing. This means that the reported approval rate of 43% could range from 39.8% to 46.2% in reality, with a high level of confidence.

4. Below is a cumulative relative frequency plot (or, an ogive) for the scores of a large university class on a 90-point calculus exam. Which of the following observations is correct?

The median score is at least 60 points.

The distribution of scores is skewed to the right.

The distribution of scores is skewed to the left.

The distribution of scores is roughly symmetric.

If a passing score is 60, most students passed the test.

Based on the cumulative relative frequency plot, the correct answer is that the distribution of scores is skewed to the right. This can be determined by observing that the curve of the plot rises more steeply on the left side and gradually levels off on the right side, indicating that there are more lower scores and fewer higher scores. This is characteristic of a right-skewed distribution.

Explanation

Based on the cumulative relative frequency plot, the correct answer is that the distribution of scores is skewed to the right. This can be determined by observing that the curve of the plot rises more steeply on the left side and gradually levels off on the right side, indicating that there are more lower scores and fewer higher scores. This is characteristic of a right-skewed distribution.

5. A test for heartworm in dogs shows a positive result in 96% of dogs that actually have heartworm, and shows negative results in 98% of dogs with no heartworm. If heartworm actually occurs in 10% of dogs, what is the approximate probability that a randomly selected dog that tested positive for heartworm actually has heartworm?

11%

18%

84%

88%

96%

The probability that a randomly selected dog that tested positive for heartworm actually has heartworm can be calculated using Bayes' theorem. The theorem states that the probability of an event A given event B can be calculated as the probability of event A and event B occurring divided by the probability of event B occurring. In this case, event A is the dog actually having heartworm and event B is the dog testing positive for heartworm. The probability of event A and event B occurring is 0.10 * 0.96 = 0.096, and the probability of event B occurring is 0.10 * 0.96 + 0.90 * 0.02 = 0.104. Therefore, the probability that a randomly selected dog that tested positive for heartworm actually has heartworm is 0.096 / 0.104 = 0.923, which is approximately 84%.

Explanation

The probability that a randomly selected dog that tested positive for heartworm actually has heartworm can be calculated using Bayes' theorem. The theorem states that the probability of an event A given event B can be calculated as the probability of event A and event B occurring divided by the probability of event B occurring. In this case, event A is the dog actually having heartworm and event B is the dog testing positive for heartworm. The probability of event A and event B occurring is 0.10 * 0.96 = 0.096, and the probability of event B occurring is 0.10 * 0.96 + 0.90 * 0.02 = 0.104. Therefore, the probability that a randomly selected dog that tested positive for heartworm actually has heartworm is 0.096 / 0.104 = 0.923, which is approximately 84%.