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Try These AP Statistics Practice Test Questions
Exploring One-Variable Data and Collecting Data Practice Test
Question
As part of his school's needs assessment, the principal of an elementary school wanted to estimate the percentage of students who eat breakfast at the school on a regular basis. From the 790 students who attend the school, a random sample of 80 boys and 95 girls was selected. Of the 175 students selected, 25 indicated that they eat breakfast at school on a regular basis. What are the population and the sample of the needs assessment?
| A. The population is the 175 students who were selected, and the sample is the 25 students who indicated that they eat breakfast at school on a regular basis. | |
| B. The population is the 175 students who were selected, and the sample is whether each student in the assessment eats breakfast at school on a regular basis. | |
| C. The population is the 790 students who attend the school, and the sample is the 25 students who indicated that they eat breakfast at school on a regular basis. | |
| D. The population is the 790 students who attend the school, and the sample is the 175 students who were selected. |
Hint:
A population includes all individuals of a target group, whereas a sample is a subset of individuals randomly selected from the population.
Explanation
A population includes all individuals of a target group.
A sample is a subset of individuals selected from the population.
The school principal selected 175 students (80 boys and 95 girls) from the 790 students who attend the elementary school to estimate the percentage of students who eat breakfast at school on a regular basis.
Therefore, the population is the 790 students who attend the school, and the sample is the 175 students who were selected.
(Choices A and B) The 175 students who were selected represent the sample (not the population) of students taken from the school.
(Choice C) The 25 students who indicated that they eat breakfast at school on a regular basis are part of the sample taken from the population, but the sample consists of all 175 selected students.
Things to remember:
A population includes all individuals of a target group, whereas a sample is a subset of individuals randomly selected from the population.
Question
The response times, in minutes, for 30 ambulances are shown below.
Which of the following best describes the shape of the distribution of response times for ambulances to arrive at the location of an emergency?
| A. Approximately normal | |
| B. Bimodal | |
| C. Skewed to the left (negatively skewed) | |
| D.Skewed to the right (positively skewed) |
Hint:
Use the frequency table to draw a histogram of the distribution of data and examine its general pattern.
Explanation
The symmetry (or lack of symmetry) and the number of peaks (modes) in a distribution of data define the distribution's general pattern.
The given table shows the distribution of ambulance response times.
Draw a histogram to visualize the shape of the distribution. Let the horizontal axis be the first row of the table (time) and the vertical axis be the second row of the table (frequency or count of ambulances).
Notice that the distribution has a single central peak (40–44 minutes) and a longer tail on the left than on the right of the distribution.
Therefore, the shape of the distribution of response times is negatively skewed (skewed to the left).
Note: It is also possible to represent the data with a dotplot to identify the shape of the distribution.
(Choice A) An approximately normal distribution is roughly symmetrical around a central peak (mode), but a histogram or dotplot of the given data shows that the distribution of ambulance response times is skewed to the left. Therefore, the distribution is asymmetric, not symmetric.
(Choice B) A bimodal distribution has two distinct peaks (modes), but a histogram or dotplot of the given data shows that the distribution of time it takes to arrive to an emergency has a single peak.
(Choice D) A positively skewed distribution has a longer tail on the right than on the left, but a histogram or dotplot of the given data shows a distribution with a longer tail on the left than on the right. Therefore, the distribution is negatively skewed (not positively skewed).
Things to remember:
- The frequency table of a negatively skewed (skewed to the left) distribution has lower frequencies at the left side of the distribution and higher frequencies at the right side of the distribution.
- A histogram of a negatively skewed (skewed to the left) data has a single peak (mode) and a longer tail on the left than on the right.
Question
A statistics teacher sent an electronic survey to a random sample of her students through their personal school email addresses, and she is concerned about possible sources of bias. Which of the following statements is the best example of nonresponse bias?
| A. Students who responded to the survey may not be similar to those selected to participate but who chose not to respond. | |
| B. The number of questions in the survey led students to submit incomplete surveys. | |
| C. The sampling procedure failed to include students who do not use their personal school email addresses. | |
| D. The wording of the survey questions led students to respond in socially acceptable ways. |
Hint:
Nonresponse bias in sampling occurs when responders differ in meaningful ways from nonresponders.
Explanation
Nonresponse bias in sampling occurs when responders differ from nonresponders in meaningful ways.
This type of bias may occur when individuals selected for a sample do not provide data because they cannot be contacted, are unwilling or unable to participate in the study, or are otherwise unavailable.
Consider that nonresponse bias would occur if students who respond to the survey (responders) differ from those who do not (nonresponders).
Therefore, the best example of nonresponse bias is the following:
| Students who responded to the survey may not be similar to those selected to participate but who chose not to respond. |
(Choices B and D) These are examples of response bias rather than nonresponse bias. Response bias occurs when responders provide random, incomplete, or inaccurate responses in a systematic manner.
(Choice C) This is an example of undercoverage bias rather than nonresponse bias. Undercoverage bias occurs when certain groups in the population are less likely to be selected into the sample.
Things to remember:
- Nonresponse bias occurs when responders in a study differ from nonresponders and from the target population in important ways.
- Nonresponse bias is common in observational studies such as surveys.
Probability, Random Variables and Probability Distributions Practice Test
Question
The table shows the responses from 156 adults in the U.S. when asked if they plan to take time to celebrate Thanksgiving this year.
One adult from those who responded will be selected at random. Which of the following is closest to the probability that the adult selected will be someone who responded no, given that the adult selected is age 65 or older?
| A. 0.013 | |
| B. 0.049 | |
| C. 0.103 | |
| D. 0.125 |
Hint:
If all outcomes in a sample space are equally likely, then the probability of an event B is the number of desired outcomes divided by the total number of possible outcomes.
Explanation
If all outcomes in a sample space are equally likely, then the probability of an event B is the number of desired outcomes divided by the total number of possible outcomes.
It is given that one adult from those who responded will be selected at random. The question asks for the probability that the adult responded no, given that they are age 65 or older.
To find the total number of possible outcomes, identify the column total for the "Age 65 or older" column.
To find the desired outcomes, identify the intersection of the "No" row and the "Age 65 or older" column.
The total number of adults age 65 or older is 41, and the number of adults age 65 or older who responded no is 2. Plug in these values into the probability formula.
| Probability formula | |
| Plug in values | |
| Simplify |
Therefore, the probability that the adult selected will be someone who responded no, given that the adult selected is age 65 or older, is 0.049.
(Choice A) 0.013 is the probability that the selected adult is age 65 or older and responded no .
(Choice C) 0.103 is the probability that the selected adult responded no .
(Choice D) 0.125 is the probability that the selected adult is age 65 or older, given that they responded no .
Things to remember:
If all outcomes in a sample space are equally likely, then the probability of an event B is equal to the number of outcomes in event B divided by the total number of outcomes in the sample space.
The question asks for the probability that the randomly selected adult responded no, given that they are age 65 or older.
This is the conditional probability of event B (adult responded no), given event A (age 65 or older), denoted P(B|A). Use the following formula to determine the conditional probability.
First calculate P(A), the probability that the selected adult is age 65 or older.
Divide the number of adults age 65 or older (41) by the total number of adults (156) to see that P(A) is equal to .
Now calculate P(A ∩ B), the probability that the selected adult is age 65 or older and responded no.
Divide the number of adults who are age 65 or older and responded no (2) by the total number of adults (156) to see that P(A ∩ B) is equal to .
Plug and into the conditional probability formula.
| Conditional probability formula | |
| Plug in and | |
| Cancel the denominators | |
| Simplify |
Therefore, the probability that the adult selected will be someone who responded no, given that the adult selected is age 65 or older, is 0.049.
Things to remember:
- The conditional probability P(B|A) is the probability that an event B occurs given event A.
- P(B|A) can be calculated using the formula .
Question
The probability that a treadmill bought online will arrive later than promised to a customer is 0.06. The probability that a treadmill bought online arrives later than promised and with damage is 0.03. The probability that a treadmill bought online arrives with damage is 0.15. Given that a treadmill bought online arrives later than promised to a customer, what is the probability that it arrives with damage?
| A. 0.03 | |
| B. 0.15 | |
| C. 0.20 | |
| D.0.50 |
Hint:
The conditional probability is equal to the probability that both events A and B occur, divided by the probability of B:
Explanation
Let event L be that the treadmill arrives late, and let event D be that the treadmill arrives with damage.
The probability that a treadmill arrives with damage given that it arrives late is the conditional probability of L given D, denoted by P(D|L). Write the conditional probability formula in terms of events L and D.
It is given that the probability that a treadmill arrives late is 0.06 and that the probability a treadmill arrives late and with damage is 0.03, so P(L) = 0.06 and P(L and D) = 0.03.
Plug these values into the conditional probability formula.
| Formula for conditional probability | |
| Plug in P(L and D) = 0.03 and P(L) = 0.06 | |
| Divide |
Given that a treadmill bought online arrives later than promised to a customer, the probability that it arrives with damage is 0.50.
(Choice A) 0.03 is the probability that a treadmill arrives late and with damage, P(L and D). However, the question asks for the conditional probability that a treadmill arrives with damages given that it arrives late, P(D|L).
(Choice B) 0.15 is the probability that a treadmill will arrive with damage, P(D). This value may result from mistakenly assuming that the events are independent and that P(D|L) = P(D). However, P(L and D) ≠ P(L) × P(D) , so the events are dependent.
(Choice C) 0.20 is the probability that a treadmill arrives late given that it arrives with damage, P(L|D). However, the question asks for the conditional probability that a treadmill arrives with damages given that it arrives late, P(D|L).
Things to remember:
-
The conditional probability of an event is the probability that the event will occur given that another event has already occurred.
- The conditional probability is equal to the probability that both events A and B occur divided by the probability of B: .
Question
The number of marbles in each of 100 bags of marbles is recorded. The histogram below shows the distribution of the number of marbles for the 100 bags.
Which of the following values is the closest to the standard deviation of the number of marbles in the 100 bags of marbles?
| A. 3 marbles | |
| B. 4 marbles | |
| C. 5 marbles | |
| D. 8 marbles |
Hint:
Evaluate the overall pattern of the distribution and consider its properties to estimate the standard deviation of the distribution.
Explanation
To determine the standard deviation (SD) of the number of marbles in 100 bags, first notice that the given histogram is symmetric, single-peaked, and bell-shaped, so the distribution is approximately normal.
A normal distributions follow the "68-95-99.7" rule. This rule states that about 99.7% of a normal distribution falls within 3 SDs of the mean.
The mean of the given sampling distribution is about 20 (the highest point), and the entire distribution falls between 12 (the minimum) and 28 (the maximum).
The minimum (12) is about 3 SDs below the mean, and the maximum (28) is about 3 SDs above the mean, so there are about 6 SDs (3 below + 3 above) between 12 and 28.
To find an approximate value of the SD, calculate 28 minus 12 to determine the range of the distribution and divide the result by 6 (the total number of SDs between 12 and 28).
Therefore, among the answer choices, the closest value to the SD of the number of marbles in 100 bags of marbles is 3.
(Choices B, C, and D) 4, 5, and 8 may result from dividing the range by an incorrect number of SDs. However, it is necessary to divide the range by 6 to approximate the SDs of a normal distribution.
Things to remember:
-
According to the "68-95-99.7" rule for normal distributions, about 99.7% of all observations fall within 3 standard deviations (SDs) from the distribution mean.
-
To estimate the SDs of a normal distribution, divide the range of the distribution by 6.
Inference for Categorical Data: Proportions Practice Test
Question
In a simulation for a population proportion of 0.5, three different graphs were created by selecting 1,000 samples from the population. The three graphs below show the approximate sampling distribution of the sample proportion based on a different sample size n.
Which graph was based on samples with the largest sample size?
| A. Graph I | |
| B. Graph II | |
| C. Graph III | |
| D. The sample size is the same for each of the sampling distributions. |
Hint:
The spread of the sampling distribution of a mean depends on the sample size taken from the population.
Explanation
Consider that the graphs have a similar center but different spreads.
The spread of a distribution is measured by its standard deviation (SD). The greater the spread of a distribution, the greater its SD.
The SD of the sampling distribution of a sample proportion depends on the population proportion (p) and the sample size (n).
Notice that the population proportion (p) is equal to 0.5 for each graph.
When the population proportion p is fixed, the sample size n and are inversely related. The value of decreases as n increases in value (and vice versa).
The graph with the least spread (smallest SD) results from the largest sample size n. Therefore, the graph that was based on samples with the largest sample size is Graph III.
(Choice A) The SD (spread) of the sample proportion is inversely related to the sample size n, so the graph with the greatest spread has the smallest sample size (not the largest sample size).
(Choices B and D) These choices may result from a misconception about the relationship between sample size and the spread of the sampling distribution of a proportion.
Things to remember:
Larger sample sizes result in sampling distributions of with less spread than smaller sample sizes.
Question
Alana wants to estimate the proportion of employees who are in favor of a new lighting system at work. She interviews the 50 most readily available employees of the 178 in her work area and finds that 43 are in favor of a new lighting system. She plans to use a z-interval to estimate the proportion of all employees in her work area who are in favor of a new lighting system. Which of the following statements is true about this situation?
| A. A z-interval is valid because of the size of the sample relative to the population. | |
| B. A z-interval is valid because the number who are in favor of a new lighting system is greater than 10. | |
| C. A z-interval is valid because the size of the sample is greater than 30. | |
| D.A z-interval is not valid for these data. |
Hint:
Consider the conditions for which a z-interval for a proportion is valid.
Explanation
Notice that all answer choices describe reasons why a z-interval is valid or invalid for the given data. The z-interval uses a sample proportion to estimate a population proportion p.
To estimate a population proportion p from a random sample of size n from population N, it is necessary to verify that the sample is random, the observations are independent, and the sampling distribution is approximately normal.
To determine which statement is true, evaluate whether the conditions for the z-interval are met for the given data. The z-interval is not valid if any condition is not met.
The sample is random, and observations are independent
This condition requires that the sample be randomly selected from the population and that observations in the sample be independent. First determine whether the sample is random.
It is given that Alana interviewed readily available employees in her work area, so she used a nonrandom convenient sample. Therefore, the sample may not represent the entire population from which it was drawn.
Now determine whether the observations in the sample are independent. Verify that the size is less than or equal to 10% of the population from which it was selected (n ≤ 0.10N).
It is given that 50 of the 178 employees were interviewed, so n = 50 and N = 178. Use the given values to verify that n ≤ 0.10N is true.
| n ≤ 0.10N | 10% sample size condition |
| 50 ≤ 0.10(178) | Plug in the given values |
| 50 ≤ 17.8 | Simplify and verify the inequality |
| False |
The sample size of employees represents more than 10% of its population, so the observations might not be entirely independent.
The sample is NOT random, and the observations might not be entirely independent, so this condition is NOT met.
Note: It is possible to divide the sample size (50) by the population (178) of employees and convert the result to a percentage to see that the sample size represents 28.1% of the population.
The sampling distribution of is approximately normal
This condition requires that the number of successes and failures both be equal to or greater than 10 in the sample. Let a "success" be an employee in favor, and a "failure" be an employee not in favor.
It is given that of the 50 employees interviewed, 43 were in favor. Therefore, 50 − 43 = 7 were not in favor.
The number of failures (7) is less than 10, so the sampling distribution of is not approximately normal. Therefore, this condition is NOT met.
The conditions that a z-interval for a proportion requires to be valid are NOT met. Therefore, a z-interval is not valid for these data.
Note: It is also possible to answer this question more efficiently by recognizing that the z-interval will be valid only if all conditions are met (not just one).
(Choice A) The relative size of the sample of employees is greater than 10% of its population, so this condition is not met, and the z-interval is invalid.
(Choice B) The z-interval for a proportion would require that both the number of employees who favor the lighting system (43) and the number of employees who do not favor the lighting system (7) be at least 10.
(Choice C) The z-interval for a proportion does not require that the size of the sample be greater than 30; rather, it requires that the size of the sample be large enough to guarantee that the counts of successes and failures are both at least 10.
Things to remember:
- The z-interval uses a sample proportion to estimate a population proportion p.
- To estimate a population proportion p from a random sample of size n from population N, it is necessary to verify that the sample is random, the observations are independent, and the sampling distribution is approximately normal.
Question
A local company offered a workshop on effective communication during conflict to its 50 employees. All employees who participated were asked to complete a quick satisfaction survey after the workshop. The possible responses to the survey were: satisfied, neither satisfied nor dissatisfied, and dissatisfied. A chi-square test for independence is used to determine whether employee satisfaction with the workshop depends on gender. Assuming that all conditions for inference have been met, which of the following values represents the degrees of freedom for the chi-square test?
| A. 2 | |
| B. 5 | |
| C. 6 | |
| D. 49 |
Hint:
To determine the degrees of freedom for a chi-square test for independence, consider the number of rows and the number of columns in the two-way table used to display the two categorical variables.
Explanation
The chi-square test for independence determines whether there is an association between two categorical variables that may be displayed in a two-way table.
The formula to determine the degrees of freedom (df) for the chi-square test for independence is:
Use a two-way table to display the two categorical variables: gender (female, male) and satisfaction (satisfied, neither satisfied nor dissatisfied, dissatisfied).
The gender categories are displayed in 2 rows, and the satisfaction categories are displayed in 3 columns.
Plug these values into the formula for the df for a chi-square test for independence and solve.
| df = (number of rows − 1)(number of columns − 1) | Formula for df for a chi-square test for independence |
| df = (2 − 1)(3 − 1) | Input number of rows = 2 and number of columns = 3 |
| df = 2 | Simplify |
Therefore, the chi-square test for independence should be based on 2 degrees of freedom.
Note: If the two-way table had been formatted such that gender was displayed in columns and satisfaction in rows, the df for the chi-square test for independence would have been the same: (3 − 1)(2 − 1) = 2.
(Choices B and C) 5 and 6 may result from adding (2 + 3 = 5) and from multiplying (2 × 3 = 6) the number of categories of both categorical variables.
(Choice D) 49 may result from subtracting 1 from the number of employees surveyed (50 − 1 = 49).
Things to remember:
The formula to determine the degrees of freedom (df) for the chi-square test for independence is: df = (number of rows − 1)(number of columns − 1).
Inference for Quantitative Data: Means Practice Test
Question
There were 12,246 previously owned vehicles sold by the government last year. The distribution of the sales prices of these vehicles was strongly left-skewed, with a mean of $10,750 and a standard deviation of $3,250. If 1,000 simple random samples of size 100 are drawn from this population and the mean is computed for each of these samples, which of the following describes the sampling distribution of the sample mean?
| A. Strongly left-skewed with mean $10,750 and standard deviation $32.50 | |
| B. Strongly left-skewed with mean $10,750 and standard deviation $325 | |
| C. Approximately normal with mean $10,750 and standard deviation $32.50 | |
| D. Approximately normal with mean $10,750 and standard deviation $325 |
Hint:
To describe the sampling distribution of when the distribution of the population is not normal, consider the central limit theorem.
Explanation
The sampling distribution of the sample mean is the distribution of means from all possible random samples of size n taken from a large population with mean μ and standard deviation (SD) σ.
To describe the sampling distribution of , determine its shape (symmetric or skewed), center (mean), and spread (SD).
Shape
The shape of the sampling distribution of is normal if the population distribution is normal.
If the population distribution is not normal, the central limit theorem (CLT) states that the sampling distribution of will be approximately normal when n ≥ 30.
It is given that random samples of size 100 were taken from a population, so n = 100. Therefore, the sampling distribution of is approximately normal because the sample size is large enough (n ≥ 30).
Eliminate Choices A and B because they describe a sampling distribution that is strongly left-skewed.
Center
Notice that all answer choices have the same center (mean of $10,750).
Spread
When the population size N is at least ten times larger than the sample size n, the SD of the sampling distribution of depends on the population SD σ and the sample size n as follows:
The population size (12,246) is at least 10 times larger than the given sample size (100), so use the formula for the SD of the sampling distribution of .
Plug σ = 3,250 and n = 100 into the formula for the SD of the sampling distribution of , and simplify
| Formula for the SD of the sampling distribution of | |
| Plug in σ = 3,250 and n = 100 | |
| Simplify |
The sampling distribution of has a SD of $325, eliminate Choices A and C.
Therefore, the sampling distribution of is approximately normal with mean $10,750 and SD $325.
Note: The values for the mean and the SD of hold true regardless of the shape of the population distribution.
(Choice A) This choice may result from a combination of the errors described in Choices B and C.
(Choice B) The sampling distribution of is approximately normal (not strongly left-skewed) because random samples of size 100 are large enough (n ≥ 30) to apply the central limit theorem.
(Choice C) A standard deviation (SD) of $32.50 may result from mistakenly using the number of samples taken (1,000) to calculate the SD of the sampling distribution of , rather than the size of the samples taken (100).
Things to remember:
The sampling distribution of a sample mean :
- is approximately normal (in most cases) when the population is not normal and n ≥ 30.
- has mean and standard deviation , where is valid if the population size N is at least ten times larger than the sample size n.
Question
A test of the hypotheses H0: μ = 94 versus Ha: μ ≠ 94 was conducted using a sample of size 9. The test statistic was t = −1.713. Assuming all conditions for inference are met, which of the following is closest to the p-value of the test?
| A. 0.0604 | |
| B. 0.1208 | |
| C. 0.1250 | |
| D.0.1304 |
Hint:
A p-value is the probability of obtaining a test statistic at least as extreme as the one observed (in the direction of the alternative hypothesis Ha) when the null hypothesis H0 is assumed to be true.
Explanation
A p-value is the probability of obtaining a test statistic at least as extreme as the one observed (in the direction of the alternative hypothesis Ha) when the null hypothesis H0 is assumed to be true.
The test that compares a population mean μ to a hypothesized value μ0 (94) and results in a t-statistic (t = −1.713) is the one-sample t-test for a mean. The given null H0 and alternative Ha hypotheses are:
When H0 is true, the t-statistic follows a t-distribution with n − 1 degrees of freedom (df). Notice that Ha is two-sided (Ha: μ ≠ 94), so the test and its p-value are two-sided.
The two-sided p-value is the sum of the areas to the left of t = −1.713 and to the right of t = 1.713 under the t-distribution curve. These areas are equal, so the p-value is twice the area to the left of t = −1.713.
Note: The areas to the left of t = −1.713 and to the right of t = 1.713 are equal because the t-distribution is symmetric about 0.
It is given that the test was conducted using a sample size of 9, so the p-value for the test is twice the area to the left of −1.713 under a t-distribution with 9 − 1 = 8 df.
| To find the area to the left of −1.713 on a t-distribution with 8 df, locate the t cumulative distribution function (tcdf) on a calculator. Input a lower limit (ex. −1,000 to represent negative infinity), the upper limit (the test statistic, −1.713), and the df (8). |
The area under the curve of a t-distribution with 8 df to the left of t = −1.713 is 0.0625. Therefore, the p-value for the two-sided test is approximately 2 × 0.0625 = 0.1250.
(Choice A) 0.0604 may result from mistakenly using a t-distribution with degrees of freedom df = 9 (instead of a t-distribution with df = 8) and then not multiplying the calculated probability by 2 to get a two-sided p-value.
(Choices B and D) 0.1208 and 0.1304 may result from mistakenly using a t-distribution with degrees of freedom equal to the sample size (n) or 2 less than the sample size (n − 2), respectively. However, the degrees of freedom for the t-statistic is equal to 1 less than the sample size (n − 1).
Things to remember:
-
The one-sample t-test for a mean compares a single mean μ to a hypothesized value μ0. When conditions for inference are met, the test statistic follows a t-distribution with n − 1 degrees of freedom.
- The p-value is the probability of obtaining a test statistic at least as extreme as the one observed (in the direction of the alternative hypothesis Ha) when the null hypothesis H0 is assumed to be true.
Question
A biology student reads that the life span of female butterflies is different than the total life span of male butterflies of the same species. To test this, he collects information on the time spent in the caterpillar, pupa, and adult stages, measured in days, from a random sample of female butterflies and a random sample of male butterflies. Which of the following might be an appropriate null hypothesis for this study?
| A. The average life span for female butterflies is different than the average life span for male butterflies. | |
| B. The average life span for female butterflies is greater than the average life span for male butterflies. | |
| C. The average life span for female butterflies is less than the average life span for male butterflies. | |
| D. The average life span for female butterflies is the same as the average life span for male butterflies. |
Hint:
The null and alternative hypotheses are statements typically about a parameter (ex. mean, proportion) from one or more populations.
Explanation
In statistical inference, the statistical hypotheses are two mutually exclusive statements typically about parameters (ex. mean, proportion) from one or more populations.
The null hypothesis H0 is a statement of no difference; it states the opposite of what is expected.
The alternative Ha is a statement of difference; it states what is expected and may be one- or two-sided.
It is given that the life span of female butterflies is different than the life span of male butterflies, so H0 must be a statement of no difference in total life span between female and male butterflies.
It is possible to eliminate Choices A, B, and C because they define H0 as a statement of direction ("not equal to").
Therefore, the appropriate null hypothesis H0 for this study is that the average life span (in days) for female butterflies is the same as the average life span (in days) for male butterflies.
(Choices A, B, and C) These choices describe the alternative hypotheses Ha for a two-sided test (Choice A) and one-sided tests (Choices B and C). However, the question asks for the null hypothesis for the study.
Things to remember:
-
The null (H0) and alternative (Ha) hypotheses are two mutually exclusive statements typically about parameters from one or more populations.
-
H0 is a statement of no difference; it states the exact opposite of what is predicted or expected.
-
Ha is a statement of difference; it states exactly what is predicted or expected. Ha may be one- or two-sided, depending on how a population parameter is expected to be different from a null value.
-
Regression Analysis Practice Test
Question
A scatterplot is an appropriate way to represent data consisting of which of the following?
| A. Political party affiliation and gender of a random sample of eligible voters | |
| B. Distance people commute to work and primary mode of transportation | |
| C. Number of messages posted on three different social media platforms | |
| D. Age of a female chicken and average number of eggs laid per week |
Hint:
A scatterplot shows the relationship between two quantitative variables.
Explanation
A scatterplot shows the relationship between two quantitative variables x and y. Each point on a scatterplot shows two numeric values for an observation: one value from the x-variable and one value from the y-variable.
A quantitative variable takes on numerical values for a measured quantity (ex. weight in pounds, height in feet) or a counted quantity (ex. number of births, number of patients seen in a day).
Only one choice describes data consisting of two quantitative variables, and the other three choices describe data consisting of at least one categorical variable.
Therefore, a scatterplot is an appropriate way to represent data consisting of the age of a female chicken and average number of eggs laid per week.
(Choices A, B, and C) These choices describe data consisting of at least one categorical variable, not two quantitative variables. Therefore, a scatterplot is not an appropriate way to represent these data sets.
Things to remember:
A scatterplot shows the relationship between two quantitative variables x and y.
Question
Crickets chirp as an important means of recognizing members of their own species. Entomologists believe that the frequency of chirping varies according to the temperature of the environment. A sample of 25 crickets was taken, and the number of chirps in 14 seconds (sec) for each cricket and the ambient temperature, in degrees Fahrenheit, was recorded. The output shown in the table is from a least-squares regression to predict temperature given the number of chirps in 14 sec.
Suppose a cricket chirps 30 times in 14 sec at a temperature of 65 degrees Fahrenheit. Based on the residual, does the regression model underestimate or overestimate the temperature?
| A. The residual is negative, so the model underestimates the temperature. | |
| B. The residual is negative, so the model overestimates the temperature. | |
| C. The residual is positive, so the model underestimates the temperature. | |
| D.The residual is positive, so the model overestimates the temperature. |
Hint:
In linear regression, a residual is the difference between the observed value of a response variable and the value for the response variable predicted by the regression model :
Explanation
The residual of a data point is the difference between the observed value of the response variable and the value that is predicted by the regression line , so the residual is equal to .
A positive residual means that the regression model underestimates the predicted value of a response variable.
A negative residual means that the regression model overestimates the predicted value of a response variable.
To calculate the residual, first find the predicted value of the response variable . Determine the equation of the estimated regression line to find .
The equation of a regression line that predicts the value of a response variable for a given value of an explanatory variable has the general form , where a is the y-intercept and b is the slope.
Notice that the number of chirps in 14 seconds (sec) predicts the ambient temperature, so the number of chirps is the explanatory variable and temperature is the response variable .
Use the given table (computer output) to identify the slope and y-intercept of the regression model. In a computer output, the constant coefficient is the y-intercept and the coefficient of the explanatory variable is the slope.
The y-intercept is 40.037 and the slope is 1.000. To find the value of (the predicted temperature for 30 chirps in 14 sec), plug these values and into the regression equation and simplify.
The predicted temperature for 30 chirps in 14 sec is about 70° F (Fahrenheit). It is given that the observed temperature for the same number of chirps is 65° F.
Plug the observed temperature and the predicted temperature into the residual formula and simplify.
| Definition of a regression residual | |
| Plug in and | |
| Simplify |
The regression residual is negative, so the temperature predicted by the model is higher than the observed temperature. Therefore, the regression model overestimates the ambient temperature.
(Choice A) A negative residual indicates that a regression model overestimates (not underestimates) the response variable. The model overestimates because the predicted value of the response variable is higher than the corresponding observed value .
(Choices C and D) The residual of a cricket that chirps 30 times in 14 sec at 65° F is negative, not positive. A positive residual may result from mistakenly swapping the order of the terms for the observed and predicted values of the response variable in the formula of a regression residual.
Things to remember:
-
The equation of a least-squares regression line is , where is the predicted response variable, is the y-intercept, is the slope, and is the explanatory variable.
- A regression residual is the difference between the observed response variable and the response variable predicted by the regression model : .
-
A negative residual indicates that a regression model overestimates the response variable.
-
A positive residual indicates that a regression model underestimates the response variable.
-
Question
A large company recorded the amount of sales (in thousands of dollars) and the amount spent on advertising (in thousands of dollars) in a random sample of months. The value of the correlation coefficient between amount of sales and amount spent on advertising for the chosen months was 0.9. Based on the least-squares regression line created from the data to predict amount of sales in a month based on the amount spent on advertising, which of the following is a correct statement?
| A. The average amount of sales in a month is 90% of the amount spent on advertising in a month. | |
| B. The least-squares regression line of amount of sales versus amount spent on advertising will have a slope of 0.9. | |
| C. The least-squares regression line will predict amount of sales based on amount spent on advertising 90% of the time. | |
| D. The proportion of the variation in amount of sales that is explained by a regression on amount spent on advertising is 0.81. |
Hint:
The coefficient of determination is the proportion (or percentage) of variation in the response variable that is explained by the explanatory variable in a simple regression model.
Explanation
The correlation coefficient r describes the direction and strength of the linear association between two variables x and y, but notice that none of the choices describe the direction and strength of the association.
In a simple linear regression of response variable y on an explanatory variable x, the coefficient of determination r2 is the square of r and determines the proportion of variation in y explained by x.
This means that r2 describes how close the observed values of the response variable y are to the regression line. The closer the observed values to the regression line, the greater the fit of the model.
It is given that a regression line that predicts the amount of sales in a month (y) based on the amount spent on advertising (x) had a correlation r = 0.9. This means that the coefficient of determination r2 is (0.9)2 = 0.81.
Therefore, the correct statement is that:
| The proportion of the variation in amount of sales that is explained by a regression on amount spent on advertising is 0.81. |
(Choice A) The correlation coefficient r is a measure of the strength of the linear association between the amount of sales and the amount spent on advertising. It does not describe the relationship between the variables in an individual month.
(Choice B) The correlation coefficient r is not equal to the slope b of the regression line. The correlation coefficient is a measure of the strength of the linear association between the amount of sales and the amount spent on advertising.
(Choice C) The correlation coefficient r (0.9) does not indicate the probability of predicting sales amounts. The probability that a regression model predicts the exact value of a continuous variable is 0%.
Things to remember:
- The coefficient of determination (r2) represents the proportion of variation in a response variable y that is explained by an explanatory variable x in a linear regression model.
- r2 is also the square of the correlation coefficient r between x and y in a simple regression model.
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