Chapter2

Introduction to Statistical Investigations Test Bank

Note: TE = Text entry TE-N = Text entry - Numeric

Ma = Matching MS = Multiple select

MC = Multiple choice TF = True-False

DD = Drop-down

CHAPTER 2 LEARNING OBJECTIVES

2-1: Describe how to select a random sample.

2-2: Understand and predict sampling distributions for a sample proportion and a sample mean.

2-3: Perform both a simulation-based and theory-based test for a single mean.

2-4: Describe how to create a bootstrap sampling distribution for a single mean.

Section 2.1: Sampling from a Finite Population: Proportions

2.1-1: Identify parameters (long-run proportion) and statistics (sample proportion) in a statistical study.

2.1-2: Identify which statistics (proportions) and graphs (bar graph) are appropriate for categorical variables, construct graphs and calculate statistics with use of technology, and interpret appropriately.

2.1-3: Identify the (finite) population and the sample in a statistical study.

2.1-4: Identify whether a sampling method is likely to be biased and explain the potential impact.

2.1-5: Describe how to select a random sample and recognize that one advantage of a random sample is that it is likely to be representative of the population regardless of sample size.

2.1-6: Fill in a data-table where rows are the observational units and columns are the variables.

2.1-7: Predict the mean, standard deviation, and shape of the sampling distribution of a sample proportion from a random sample of size n, where the population proportion π is known.

2.1-8: Apply simulation- and theory-based inference methods for a population proportion to research studies involving random samples from finite populations.

2.1-9: Identify whether a study may be impacted by non-sampling concerns and explain the potential impact.

Questions 1 through 5: In June of 2012, New York City Mayor Michael Bloomberg proposed banning the sale of any regular soda or other sugary drinks in containers larger than 16 oz. A fast food chain polled a random sample of 2500 of its regular customers and found that 63% opposed the ban, 28% supported the ban, and 9% had no opinion.

1. What type of plot would be most appropriate to display these data?
   1. Histogram
   2. Bar graph
   3. Dot plot
   4. Box plot
2. Is this sample representative of the population of ALL New York City residents?
   1. Yes, because it is a random sample.
   2. Yes, because the sample size was large.
   3. No, because the entire population was not surveyed.
   4. No, because the people in this sample would be more likely to drink regular soda or other sugary drinks in containers larger than 16 oz than the general population.
3. To which population should this sample be generalized?
   1. All New York City residents
   2. All New York City residents who drink regular soda or other sugary drinks
   3. All regular customers of the fast food chain
   4. Only the 2500 people surveyed
4. The value 63% is a___.
   1. Population
   2. Sample
   3. Parameter
   4. Statistic
5. The 2500 regular fast food chain customers surveyed are the ___.
   1. Population
   2. Sample
   3. Parameter
   4. Statistic

Questions 6 through 8: For a survey on affirmative action, researchers selected a random sample of 20 faculty members from a large research university. Faculty in the sample were asked, “Do you favor or oppose using race, religion, sex, color, ethnicity, or national origin as a criterion for college admission?”

1. Which statistic is appropriate for summarizing these data?
   1. The proportion of the 20 faculty members surveyed that answered “Yes”
   2. The proportion of all faculty at this large research university that would have answered “Yes” to the survey question if asked
   3. The number of faculty members surveyed
   4. The mean number of the 20 faculty members surveyed that answered “Yes”
2. Is this sample representative of the entire population of faculty at this large research university?
   1. Yes, because it is a random sample.
   2. No, because the sample size is too small.
   3. No, because it is a convenience sample.
   4. No, because they did not survey all faculty.
3. The proportion of all faculty at this large research university that would have answered “Yes” to the survey question if asked is the___.
   1. Population
   2. Sample
   3. Parameter
   4. Statistic

Questions 8 through 16: Suppose that the proportion of all American adults that fit the medical definition of being obese is 0.23. A large medical clinic would like to determine if the proportion of their patients that are obese is higher than that of all American adults. The clinic takes a simple random sample of 30 of their patients and finds that 9 patients in the sample are obese.

1. What is the population of interest?
   1. All American adults
   2. All patients at the large medical clinic
   3. The 30 patients surveyed
   4. The 9 patients that are obese
2. Describe the parameter of interest in the context of this problem. What is the appropriate statistical symbol for this parameter?
   1. proportion of the random sample of 30 patients that are obese
   2. proportion of all patients at the large medical clinic that are obese
   3. proportion of the random sample of 30 patients that are obese
   4. proportion of all patients at the large medical clinic that are obese
3. Write the null and alternative hypotheses in proper notation.
   1. versus
   2. versus
   3. versus
   4. versus

The researchers created the following simulated null distribution with 1000 trials using the One Proportion applet:

1. What is the appropriate label for the x-axis of the randomization distribution plot?
   1. Proportion of obese patients in a sample of 30 patients assuming the true proportion of obese patients is greater than 0.23
   2. Proportion of obese patients in a sample of 30 patients assuming the true proportion of obese patients is equal to 0.23
   3. Number of obese patients in a sample of 30 patients assuming the true proportion of obese patients is greater than 0.23
   4. Number of obese patients in a sample of 30 patients assuming the true proportion of obese patients is equal to 0.23
2. Where is the distribution centered? Explain why this makes sense.
   1. The distribution is centered around 7, since 23% of 30 is around 7.
   2. The distribution is centered around 7, since 30% of 30 is around 7.
   3. The distribution is centered around 0.23, since we are assuming the null hypothesis.
   4. The distribution is centered around 0.30, since we are assuming the null hypothesis.
3. Calculate the p-value using the table of counts provided.

Solution: (109 + 73 + 29 + 19 + 4 + 1 + 1)/1000 = 236/1000 = 0.236

LO: 2.1-8; Difficulty: Hard; Type: TE-N

1. Based on this p-value, what is your conclusion for the hypothesis test in the context of this problem?
   1. There is little to no evidence that more than 23%of all patients at this clinic are obese.
   2. There is strong evidence that 23% of all patients at this clinic are obese.
   3. There is little to no evidence that more than 30%of all patients at this clinic are obese.
   4. There is strong evidence that 30% of all patients at this clinic are obese.
2. If the clinic had collected a sample of 100 patients instead of 30, would the standard deviation of the simulated null distribution increase, decrease, or remain the same?
   1. Increase
   2. Decrease
   3. Remain the same

Section 2.2: Quantitative Data

2.2-1: Describe the distribution of a quantitative variable in terms of shape, center, variability, and unusual observations.

2.2-2: Use the skewness of the distribution to predict the relationship between the mean and median.

2.2-3: Identify parameters (such as long-run means or long-run medians) and statistics (such as sample means or sample medians) in a statistical study.

2.2-4: Predict the mean, standard deviation, and shape of the sampling distribution of a sample mean from a random sample of size n, where the population mean, μ, and population standard deviation, σ, are known.

Questions 17 through 21: A hot dog manufacturer asserts that one of its brands of hot dogs has an average fat content of 18grams per hot dog. Fat contents in hot dogs from this manufacturer follow an approximately normal distribution with a standard deviation of 1.1grams. Define the statistic = mean fat content in a random sample of 10 hot dogs.

1. Is the value 18 grams a statistic or a parameter?
   1. Statistic
   2. Parameter
2. If the hot dog manufacturer’s claim is true, what would be the mean of the sampling distribution of ? ________ grams

Solution: The mean is the same as the mean of the population, 18 grams.

LO: 2.2-4

Difficulty: Easy

Type: TE-N

1. If the hot dog manufacturer’s claim is true, what would be the standard deviation of the sampling distribution of ?

Solution: = 0.348

LO: 2.2-4

Difficulty: Medium

Type: TE-N

1. Is the sampling distribution of approximately normal?
   1. No, since the sample size is too small for the Central Limit Theorem to hold.
   2. No, since we do not know if the population of hot dogs is more than 20 times the size of the sample.
   3. Yes, since the population of fat contents in hot dogs is approximately normal.
   4. We do not have enough information to determine if the sampling distribution of is approximately normal.
2. If we were to select a random sample of 50 hot dogs rather than 10, which of the following features of the sampling distribution of would change?

The mean would ______. (1)

The shape would ______. (2)

The standard deviation would ______. (3)

Drop-down options for (1), (2), and (3):

• increase
• decrease
• remain the same

LO: 2.2-4

Difficulty: Medium

Type: DD

Questions 22 through 24: A rental car company has noticed that the distribution of the number of miles customers put on rental cars per day is skewed to the right, with some occasional high outliers. The distribution has a mean of 80 miles and a standard deviation of 50 miles.

1. Which of the following distributions most likely match the shape of the distribution of the number of miles customers put on rental cars per day for this rental car company?

1. Which of the following distributions would have the smallest standard deviation?
   1. Number of miles put on a rental car in one day
   2. Average number of miles put on a rental car per day across 5 customers
   3. Average number of miles put on a rental car per day across 25 customers
   4. Distributions of all of the above would have the same standard deviation.
2. Which of the following distributions would have the smallest mean?
   1. Number of miles put on a rental car in one day
   2. Average number of miles put on a rental car per day across 5 customers
   3. Average number of miles put on a rental car per day across 25 customers
   4. Distributions of all of the above would have the same mean.

Questions 25 through 29: The following histogram shows the distribution of the difference between the actual and “ideal” weights for 119 female students. Ideal weights are responses to the question “What is your ideal weight”? The difference = actual – ideal.

1. What is the approximate shape of the distribution?
   1. Nearly symmetric
   2. Skewed to the left
   3. Skewed to the right
   4. Bimodal
2. The median of the distribution is approximately___
   1. –10 pounds.
   2. 10 pounds.
   3. 30 pounds.
   4. 50 pounds.
3. Most of the women in this sample felt that their actual weight was
   1. about the same as their ideal weight.
   2. less than their ideal weight.
   3. greater than their ideal weight.
   4. no more than 2 pounds different from their ideal weight.
4. How does the mean compare to the median in this data set?
   1. The mean is larger than the median.
   2. The median is larger than the mean.
   3. The median and mean are approximately equal.
   4. We are unable to compare the mean and the median without calculations.
5. If the observation at -20 pounds was removed from the data set, the mean would___
   1. increase.
   2. decrease.
   3. remain the same.
6. Which of the following is most likely to match the shape of the distribution of heights among teachers and students at a large elementary school?

1. True or False: If a distribution is skewed left, then the mean will be smaller than the median.

Section 2.3: Theory-based Inference for a Population Mean

2.3-1: Conduct a simulation-based analysis to conduct a single test involving the mean of a single quantitative variable.

2.3-2: Anticipate the relative magnitude of the standard error of a sampling distribution of sample means based on the formula .

2.3-3: Carry out a theory-based analysis (one-sample t-test) involving the mean of a single quantitative variable, including checking relevant validity conditions.

2.3-4: Recognize that even with appropriately gathered and analyzed data there is a chance of arriving at an incorrect decision, and depending on the conclusion drawn these decision errors are very different.

Questions 32 through 41: Most experts recommend college students get at least 8 hours of sleep per night. You collect a random sample of 112 students at your college and ask how many hours of sleep they got last night. The sample mean amount of sleep last night was 7.6 hours, with a sample standard deviation of 2.1 hours. The sample distribution of hours of sleep was slightly skewed right. Does this provide evidence that students at your college are getting less than the recommended amount of sleep per night?

1. What are the observational units?
   1. Hours
   2. Students
   3. Whether a student got 8 hours of sleep last night
   4. Experts
2. What are the measurement units?
   1. Hours
   2. Students
   3. Whether a student got 8 hours of sleep last night
   4. Experts
3. Is the value 7.6 hours a parameter or a statistic? What is its appropriate symbol?
   1. Statistic,
   2. Statistic,
   3. Parameter,
   4. Parameter,
4. What is the null hypothesis?
   1. The mean amount of sleep per night among all college students in the U.S. is equal to 8 hours.
   2. The mean amount of sleep per night among all students at your college is equal to 8 hours.
   3. The mean amount of sleep per night among the 112 students surveyed is equal to 8 hours.
   4. The mean amount of sleep per night among all students at your college is 7.6 hours.
5. Use the sample results to estimate the standard error of the sampling distribution of sample means.

Solution:

LO: 2.3-2; Difficulty: Medium; Type: TE-N

1. The sample mean of 7.6 hours lies _______ (1) standard errors ______ (2) the recommended value of 8 hours.

(1) = blank for calculation

(2) = drop-down menu with options:

• above
• below

LO: 2.3-3; Difficulty: Hard; Type: (1) TE-N, (2) DD

1. Is a theory-based approach reasonable for these data?
   1. Yes, since the sample size is larger than 20 and the sample distribution is not strongly skewed.
   2. Yes, since there were at least 10 students that slept more than 8 hours and at least 10 that slept less than 8 hours.
   3. No, since the sample distribution was slightly skewed right.
   4. Unable to determine.
2. Using the Theory-Based Inference applet to determine the probability of observing a sample mean of 7.6 hours or less, assuming the true mean hours of sleep per night is 8 hours.

Solution: Use Theory-Based Inference applet for one mean with n = 112, = 7.6, and s = 2.1.

LO: 2.3-3; Difficulty: Hard; Type: TE-N

1. Do these results provide evidence that students at your college are getting less than the recommended amount of sleep per night?
   1. No, since 7.6 hours is not very far below 8 hours.
   2. Yes, since 7.6 hours is less than 8 hours.
   3. Yes, since the p-value is small.
   4. No, since the p-value is large.
2. Suppose that instead of measuring the amount of sleep each student got last night, you measured whether each student got at least 8 hours of sleep last night in your sample. Could you use these data to carry out a one-sample t-test to determine if students at your college are getting less than the recommended amount of sleep per night?
   1. No, since whether each student got at least 8 hours of sleep last night is a categorical variable.
   2. Yes, since we know whether each student got less than the recommended amount of sleep.
   3. No, since the sample size is too small.
   4. Yes, since we can calculate the proportion of students who got less than the recommended amount of sleep.
3. True or False: If your p-value is very small, you have proven the alternative hypothesis.
4. True or False: If your p-value is very large, your results prove the null hypothesis.

Questions 44 through 50: Adult male polar bears are expected to weigh, on average, 370 kg. A polar bear’s primary source of food is seals and other marine animals, which they hunt from a platform of sea ice. Scientists are concerned that global warming is melting these platforms earlier in the year, reducing the time polar bears are able to hunt and forcing them inland without the necessary fat reserves built up to survive summer and fall. The US Geological Survey (USGS) conducted a study in the Southern Beaufort Sea to investigate whether climate change has appeared to negatively impact the weight of polar bears. Eighty-three adult male polar bears were captured between March and May of the years 1990 and 2006 and their weights were recorded. The sample mean weight was 324.6 kg, and the sample standard deviation was 88.3 kg.

1. A histogram of the 83 polar bear weights in the sample is shown below.

Are the validity conditions for theory-based inference met for these data?

• 1. Yes, since there were at least 10 polar bears weighing more than 370 kg and at least 10 weighing less than 370 kg.
  2. Yes, since the sample size is larger than 20 and the sample distribution is not strongly skewed.
  3. No, since the sample distribution is slightly skewed left.
  4. Unable to determine.

1. What are the appropriate null and alternative hypotheses to investigate whether climate change has appeared to negatively impact the weight of polar bears?
   1. vs
   2. vs
   3. vs
   4. vs
2. Use the sample results to estimate the standard error of the sampling distribution of sample mean weight.

Solution:

LO: 2.3-2; Difficulty: Medium; Type: TE-N

1. The sample mean of 324.6 kg lies _______(1) standard errors ______ (2) the expected average polar bear weight of 370 kg.

(1) = blank for calculation

(2) = drop-down menu with options:

• above
• below

Solutions (1):

LO: 2.3-3; Difficulty: Hard; Type: (1) TE-N, (2) DD

1. Use the Theory-Based Inference applet to determine the p-value to investigate whether climate change has appeared to negatively impact the weight of polar bears.
   1. p-value = 0.0468
   2. p-value = 1
   3. p-value = 0.05
   4. p-value < 0.00001
2. Do these results provide evidence that the true mean weight of polar bears is less than 370 kg?
   1. No, since the sample results could have happened by chance.
   2. Yes, since 324.6kg is less than 370 kg.
   3. Yes, since the p-value is small.
   4. No, since the p-value is large.
3. To which population can we generalize these results?
   1. All polar bears
   2. All adult male polar bears
   3. All adult male polar bears in the Southern Beaufort Sea
   4. Adult male polar bears that are similar to those captured for the sample

Section 2.4: Other Statistics

2.4-1: Anticipate the impact of an additional data value or change in a data value on the mean, median, and standard deviation.

2.4-2: Explain how population shape does not affect shape of sampling distribution of sample mean when sample size is large.

2.4-3: Understand that shape of sampling distribution for sample median is not predictable regardless of population shape.

2.4-4: Understand that bootstrapping is sampling with replacement from the original sample and how a bootstrap distribution is used in a test of significance.

Questions 51through 53: The following 19 test scores are based on a random sample of final exam scores (sorted in increasing order) for the population of students who took an elementary statistics course at a large university over a 5 year period:

61 65 69 70 71 73 75 77 77 82 82 83 84 85 8689 92 92 94

Answer the following questions without performing any calculations.

1. Suppose the largest test score in the sample changed from a 94 to a 90. Describe how each of the following statistics would change.

The median would ______. (1)

The mean would ______. (2)

The standard deviation would ______. (3)

Drop-down options for (1), (2), and (3):

• increase
• decrease
• remain the same

LO: 2.4-1; Difficulty: Medium; Type: DD

1. Suppose we removed the smallest test score (61) from the sample. Describe how each of the following statistics would change.

The median would ______. (1)

The mean would ______. (2)

The standard deviation would ______. (3)

Drop-down options for (1), (2), and (3):

• increase
• decrease
• remain the same

LO: 2.4-1; Difficulty: Medium; Type: DD

1. Suppose the population of test scores is skewed right. What would be the shape of the sampling distribution for sample mean test scores from samples of 50 students?
   1. Skewed right
   2. Skewed left
   3. Approximately normal
   4. Unable to determine
2. True or False: If the sample size is larger than 20 and the sample distribution is not strongly skewed, then the sampling distribution of the sample median will be approximately normal.

Questions 55 through 60: Noxious weeds are invasive plants that quickly spread and deplete the soil of nutrients that are required for all aspects of farming and ranching. With much at stake, the state of Montana has hired a team of researchers to explore the issue. They want to know the average number of noxious weeds per acre in North Central Montana. To do so, researchers divided the area of interest into 1,000,000 acres and randomly selected 24 of these acres. For each selected acre, they counted and removed the noxious weeds. The sample mean was 103.47 weeds, and the sample standard deviation was 46.034 weeds. Below is the simulated bootstrap distribution of 1,000 sample means they created from resampling the observed data.

1. What is the population of interest?
   1. All acres in North Central Montana
   2. The 24 selected acres in the sample
   3. The true mean number of noxious weeds per acre in North Central Montana
   4. The sample mean number of noxious weeds per acre
2. How was one bootstrapped sample generated in the plot above?
   1. Randomly select 24 acres from the area of interest and calculate the mean number of noxious weeds.
   2. Randomly select 24 acres from the acres selected for the sample, without replacement, and calculate the mean number of noxious weeds.
   3. Randomly select 24 acres from the acres selected for the sample, with replacement, and calculate the mean number of noxious weeds.
   4. Randomly select 24 acres from a hypothetical population of acres and calculate the mean number of noxious weeds.
3. How could you use the bootstrapped distribution to calculate a p-value to test whether the mean number of noxious weeds in North Central Montana is larger than 90 weeds per acre?
   1. Calculate the proportion of bootstrapped sample means that are equal to or larger than 90 on the bootstrap distribution above.
   2. Calculate the proportion of bootstrapped sample means that are equal to or larger than 103.47 on the bootstrap distribution above.
   3. Shift the bootstrap distribution to the left so that it’s centered at 90, then calculate the proportion of shifted bootstrapped sample means that are equal to or larger than 90.
   4. Shift the bootstrap distribution to the left so that it’s centered at 90, then calculate the proportion of shifted bootstrapped sample means that are equal to or larger than 103.47.
4. Can we use this study to generalize our results to the population of interest?
   1. No, since we only observed 24 acres.
   2. Yes, since we randomly selected the 24 acres from all 1,000,000 acres.
   3. No, since we only generated 1,000 bootstrapped samples.
   4. Yes, since our results were statistically significant.
5. Suppose we had selected 240 acres rather than 24, but the values of our sample statistics were the same. Describe how each of the following would change.

The sample standard deviation would ______. (1)

The standard error of the sample mean would ______. (2)

The standard deviation of the bootstrap distribution would ______. (3)

The p-value would ______. (4)

Drop-down options for (1)-(4):

• decrease
• increase
• remain the same
• unable to determine

LO: 2.4-4; Difficulty: Hard; Type: DD Multipart

1. If we had generated a bootstrap distribution of 1,000 sample medians rather than 1,000 sample means, what would be the shape of the bootstrap distribution of sample medians?
   1. Approximately normal
   2. Skewed right
   3. Skewed left
   4. Unable to determine

Questions 61 through 63: Adult male polar bears are expected to weigh, on average, 370 kg. A polar bear’s primary source of food are seals and other marine animals, which they hunt from a platform of sea ice. Scientists are concerned that global warming is melting these platforms earlier in the year, reducing the time polar bears are able to hunt and forcing them inland without the necessary fat reserves built up to survive summer and fall. The US Geological Survey (USGS) conducted a study in the Southern Beaufort Sea to investigate whether climate change has appeared to negatively impact the weight of polar bears. Eighty-three adult male polar bears were captured between March and May of the years 1990 and 2006 and their weights were recorded. The sample mean weight was 324.6 kg, the sample median weight was 339.4 kg, and the sample standard deviation was 88.3 kg. A box plot of the data is shown below.

1. The mean weight in the data set is 324.6 kg, the median weight is 339.4 kg, and the sample standard deviation is 88.3 kg. Which of the following statements do we know to be true based on these values alone?

The average weight of all adult male polar bears in the Southern Beaufort Sea is 324.6 kg. ______ (1)

The distribution of weights in the sample is roughly symmetric, since the mean and median are close relative to the size of the standard deviation.______ (2)

There is evidence that the average weight of an adult male polar bear in the Southern Beaufort Sea is less than 370 kg.______ (3)

On average, each adult male polar bear captured weighed 88.3 kg more or less than 324.6 kg.______ (4)

Drop-down options for (1), (2), (3), (4):

• True
• False

1. Identify the appropriate parameter of interest for this study.
   1. The number of underweight adult male polar bears in the Southern Beaufort Sea
   2. The average weight of all adult male polar bears in the Southern Beaufort Sea
   3. The true proportion of adult male polar bears in the Southern Beaufort Sea that weigh less than 370 kg
   4. The weight of each adult male polar bear captured
2. Fill in the blanks below to explain how to use bootstrapping to create one simulated sample which assumes the null hypothesis is true.

Create ______ (1) cards.

On each card, you should write the ________ (2) after ________ (3) the value ________ (4).

Then draw ______ (5) cards ______ (6) replacement.

Plot the ______ (7).

Repeat 1000 times to create the simulated null distribution.

Drop-down options for (1), (4), and (5):

• 83
• 45.4
• 370
• 324.6

Drop-down options for (2) and (7):

• weight of each sampled polar bear
• mean weight of the resampled polar bears

Drop-down options for (3):

• adding
• subtracting

Drop-down options for (6):

• with
• without

(1) 83; (2) weight of each sampled polar bear; (3) adding; (4) 45.4; (5) 83; (6) with; (7) mean weight of the resampled polar bears

LO: 2.4-4; Difficulty: Hard; Type: DD

1. A histogram of a bootstrapped null distribution is shown below.

Based on these results, what is the strength of evidence against the null hypothesis?

• 1. Very strong evidence against the null
  2. Moderate evidence against the null
  3. Little to no evidence against the null
  4. We are not given enough information to determine the strength of evidence against the null hypothesis.

1. A histogram of a bootstrapped null distribution is shown below.

True or False: Since the bootstrapped null distribution is bell-shaped, the population of polar bear weights must also be bell-shaped.