Advanced Placement Statistics Exam Quiz

1) In a survey of 1000 people, 600 said they prefer apples over oranges. What is the margin of error for the proportion of people who prefer apples, with a confidence level of 95%?

±3.1%

±4.5%

±5.6%

±6.2%

The margin of error for a proportion can be calculated using the formula margin of error = z * sqrt(p(1-p)/n), where z is the z-score corresponding to the desired confidence level, p is the proportion, and n is the sample size. For a 95% confidence level, the z-score is approximately 1.96. Using the given values (p = 600/1000 = 0.6, n = 1000), the margin of error is approximately 1.96 * sqrt(0.6*0.4/1000) = ±3.1%.

Explanation

The margin of error for a proportion can be calculated using the formula margin of error = z * sqrt(p(1-p)/n), where z is the z-score corresponding to the desired confidence level, p is the proportion, and n is the sample size. For a 95% confidence level, the z-score is approximately 1.96. Using the given values (p = 600/1000 = 0.6, n = 1000), the margin of error is approximately 1.96 * sqrt(0.6*0.4/1000) = ±3.1%.

2) In a clinic, 50 patients with sleep disorders are randomly assigned to one of two different groups. Patients in one group are given medication before bedtime. Patients in another group are given blindfolds and played soft music at bedtime. Each patient is attached to a machine that records breathing patterns. From the patterns, it is possible to determine if the patient is awake or asleep. The data will be used to decide which method is more effective in helping patients with sleep disorders. Which of the following statements is correct in the context of this experiment?

This is a double blind experiment because the patients were divided into the two treatment groups randomly.

It is not possible for this experiment to be single-blind or double-blind, because human subjects are involved.

In a single-blind experiment, the participants are aware of the treatment they are receiving, but the experimenters are not. In this case, the patients will know whether they are receiving medication or blindfolds and soft music, but the examining doctor may not be aware of the treatment. Therefore, this experiment can be classified as a single-blind experiment.

Explanation

In a single-blind experiment, the participants are aware of the treatment they are receiving, but the experimenters are not. In this case, the patients will know whether they are receiving medication or blindfolds and soft music, but the examining doctor may not be aware of the treatment. Therefore, this experiment can be classified as a single-blind experiment.

3) Data were collected on two variables X and Y and a least-squares regression line was fitted to the data. The estimated equation for this data is y = -2.29 + 1.70x. One point has x = 5 and y = 6. What is the residual for this point?

7.91

6.21

0.21

-0.21

-2.91

The residual for a data point is the difference between the actual value of the dependent variable (y) and the predicted value of the dependent variable based on the regression line. In this case, the estimated equation is y = -2.29 + 1.70x. Plugging in x = 5, we can calculate the predicted value of y as follows: y = -2.29 + 1.70(5) = -2.29 + 8.50 = 6.21. Since the actual value of y is given as 6, the residual is calculated as 6 - 6.21 = -0.21. Therefore, the correct answer is -0.21.

Explanation

The residual for a data point is the difference between the actual value of the dependent variable (y) and the predicted value of the dependent variable based on the regression line. In this case, the estimated equation is y = -2.29 + 1.70x. Plugging in x = 5, we can calculate the predicted value of y as follows: y = -2.29 + 1.70(5) = -2.29 + 8.50 = 6.21. Since the actual value of y is given as 6, the residual is calculated as 6 - 6.21 = -0.21. Therefore, the correct answer is -0.21.

4) Two random variables, X and Y, are independent. X has an expected value 2.5 and a standard deviation 0.3, while Y has an expected value 4.7 and standard deviation 0.4. Which of the following is true?

The mean of X + Y is 6.2.

The standard deviation of X + Y is 0.7.

The variance of X + Y is 0.7.

The standard deviation of X - Y = 0.1

The standard deviation of X - Y = 0.5

Since X and Y are independent random variables, the standard deviation of their difference (X - Y) is equal to the square root of the sum of their variances. Given that X has a standard deviation of 0.3 and Y has a standard deviation of 0.4, the standard deviation of X - Y can be calculated as follows:

Standard deviation of X - Y = sqrt(0.3^2 + 0.4^2) = sqrt(0.09 + 0.16) = sqrt(0.25) = 0.5

Therefore, the standard deviation of X - Y is indeed 0.5.

Explanation

Since X and Y are independent random variables, the standard deviation of their difference (X - Y) is equal to the square root of the sum of their variances. Given that X has a standard deviation of 0.3 and Y has a standard deviation of 0.4, the standard deviation of X - Y can be calculated as follows:

Standard deviation of X - Y = sqrt(0.3^2 + 0.4^2) = sqrt(0.09 + 0.16) = sqrt(0.25) = 0.5

Therefore, the standard deviation of X - Y is indeed 0.5.

5) In which of the following situations is a binomial model not an appropriate model to describe the outcome?

The number of heads in three tosses of a coin.

The number of rainy days in a given week.

The number of girls in a randomly selected family of five children.

The number of students over five feet tall at a randomly selected middle school.

The number of defective computer parts randomly selected from an assembly line.

A binomial model is not appropriate to describe the outcome of the number of rainy days in a given week because the number of rainy days in a week is not a fixed number of trials with two possible outcomes (rainy or not rainy). It is a continuous variable that can take on any value from 0 to 7, depending on the weather conditions. Therefore, a binomial model, which is used for discrete variables with a fixed number of trials and two possible outcomes, is not suitable in this situation.

Explanation

A binomial model is not appropriate to describe the outcome of the number of rainy days in a given week because the number of rainy days in a week is not a fixed number of trials with two possible outcomes (rainy or not rainy). It is a continuous variable that can take on any value from 0 to 7, depending on the weather conditions. Therefore, a binomial model, which is used for discrete variables with a fixed number of trials and two possible outcomes, is not suitable in this situation.

6) Select the appropriate answer below.

A

B

C

D

E

7) A company sells laptops with a mean battery life of 5 hours and a standard deviation of 0.5 hours. If the battery life is normally distributed, what percentage of laptops can be expected to have a battery life of at least 4.5 hours?

84%

90%

94%

97.5%

The correct answer is D. To find the percentage of laptops with a battery life of at least 4.5 hours, we need to calculate the z-score corresponding to 4.5 hours and then find the percentage of values greater than that z-score. Using the given mean and standard deviation, we find the z-score to be (4.5 - 5) / 0.5 = -1. The percentage of laptops with a battery life of at least 4.5 hours is then 1 - P(Z < -1), which is approximately 97.5%.

Explanation

The correct answer is D. To find the percentage of laptops with a battery life of at least 4.5 hours, we need to calculate the z-score corresponding to 4.5 hours and then find the percentage of values greater than that z-score. Using the given mean and standard deviation, we find the z-score to be (4.5 - 5) / 0.5 = -1. The percentage of laptops with a battery life of at least 4.5 hours is then 1 - P(Z < -1), which is approximately 97.5%.

8) A study finds that the correlation coefficient between the number of hours students spend studying and their test scores is 0.75. What does this value indicate about the relationship between the two variables?

There is a strong positive relationship between the number of hours studied and test scores.

There is a weak negative relationship between the number of hours studied and test scores.

There is no relationship between the number of hours studied and test scores.

There is a moderate negative relationship between the number of hours studied and test scores.

A correlation coefficient of 0.75 indicates a strong positive linear relationship between the number of hours students spend studying and their test scores. This means that as the number of hours studied increases, the test scores also tend to increase, and vice versa.

Explanation

A correlation coefficient of 0.75 indicates a strong positive linear relationship between the number of hours students spend studying and their test scores. This means that as the number of hours studied increases, the test scores also tend to increase, and vice versa.

9) A coin is flipped 10 times, and it lands on heads 6 times. What is the probability of getting exactly 6 heads?

0.205

0.312

0.2051

0.3125

The probability of getting exactly 6 heads when a coin is flipped 10 times can be calculated using the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success on each trial. In this case, n = 10, k = 6, and the probability of getting heads on each flip (p) is 0.5. Plugging these values into the formula, we get P(X = 6) = (10 choose 6) * (0.5)^6 * (1-0.5)^(10-6) ≈ 0.2051. Therefore, the correct answer is C: 0.2051.

Explanation

The probability of getting exactly 6 heads when a coin is flipped 10 times can be calculated using the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success on each trial. In this case, n = 10, k = 6, and the probability of getting heads on each flip (p) is 0.5. Plugging these values into the formula, we get P(X = 6) = (10 choose 6) * (0.5)^6 * (1-0.5)^(10-6) ≈ 0.2051. Therefore, the correct answer is C: 0.2051.

10) In a hypothesis test, the null hypothesis is rejected if the p-value is:

Greater than the significance level

Equal to the significance level

Less than the significance level

Less than 0.05

In hypothesis testing, the null hypothesis is rejected if the p-value is less than the significance level (typically 0.05). This indicates that the observed data is unlikely to occur under the assumption that the null hypothesis is true.

Explanation

In hypothesis testing, the null hypothesis is rejected if the p-value is less than the significance level (typically 0.05). This indicates that the observed data is unlikely to occur under the assumption that the null hypothesis is true.