PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022

Surveys and data samples
Tables, statistics, and probability
Data analysis

LEARNING OBJECTIVES

After this lesson, you will be able to:

· Determine whether a survey is valid or biased

· Draw inferences about surveys and data samples

To answer a question like this:

A county commerce department conducted a survey of 200 customers of one of the county’s pizzerias. The purpose of the survey was to estimate the average number of take-out pizzas purchased each year by residents of the county. The average number of pizzas purchased per year by these 200 customers was 43. Which of the following statements must be true based on this information?

A. The survey is representative, so 43 is a reasonable estimate of the average number of pizzas purchased annually by the county’s residents.

B. The survey is not representative of the target population due to a bias inherent in the sampling method.

C. The average number of pizzas purchased by the county’s residents is fewer than one per week.

D. The average number of pizzas purchased per county resident per year cannot be determined from such a small sample.

You need to know this:

You will see occasional questions on the PSAT Math sections that do not require any calculations or even test your ability to interpret numerical data. Instead, these questions test your ability to draw logical conclusions about surveys and data sampling methods.

Answering these questions correctly hinges on your ability to tell whether a data sample is representative of the larger population. A representative sample is a small group that shares key characteristics with a larger group you are trying to draw conclusions about.

A sample that is selected truly at random is generally representative of the larger group. For example, a scientist who wants to learn the average height of the penguins in a colony of 200 might measure the heights of a random sample of only 20 penguins. As long as the 20 penguins are selected at random, their average height will approximate the average height of the birds in the entire colony.

On the other hand, a sample that is not selected at random may not be representative and may lead to a biased conclusion. For instance, imagine that a small town uses volunteer firefighters and that a stipulation for becoming a volunteer firefighter is living within a mile of the fire station. If you wanted to know what percent of households in the town include at least one volunteer firefighter, you would need to survey a random sample of households from the entire town, not just a sample of households within a mile of the fire station. A sample of households within a mile of the fire station would be a biased sample and would lead to an erroneous conclusion (namely, that the percent of households in the town that include at least one volunteer firefighter is higher than it actually is).

You need to do this:

· Check whether the data sample represents the larger population. If it doesn’t, the survey is biased.

· In questions that ask you to draw a conclusion from a random (unbiased) sample, look for the answer choice for which the representative sample accurately reflects the larger population. For example, in a question asking for a conclusion based on a sample of librarians, the correct answer will match the sample to a larger population of librarians, not to a population of, say, accountants.

Explanation:

The sample in this question consists of the 200 customers of a pizzeria. This is not a randomly selected sample. It is likely that frequent purchasers of pizza will be overrepresented at a pizzeria. Thus, the survey is biased, so (B) is correct.

Try on Your Own

Directions: Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

13. A polling company wanted to determine whether American voters would support a constitutional amendment that requires a person running for the U.S. Senate to have a minimum of a bachelor’s degree. To do this, the company conducted a survey by sending 20,000 text messages across the entire United States to randomly selected phones with text-messaging capabilities. For every text that the company sent, it received a response to the survey. Which of the following best explains why this random sample is unlikely to be a representative sample of the American population’s opinion in an actual election?

A. The survey was biased because most Americans who own a cell phone have a bachelor’s degree.

B.Most Americans don’t care about this issue, which is likely to skew the results.

C.Surveys conducted via text messaging are illegal and as such are not considered reliable.

D. There is no way to verify whether the respondents to the survey were U.S. citizens who were registered to vote.

14. A medical testing company conducted an experimental study to determine which of three antihistamines is most effective for alleviating allergy symptoms. If the only allergies treated in the course of the study were pollen allergies, which of the following is true?

A. The antihistamine that is found to be the most effective will work for all allergies.

B.The antihistamine that is found to be the most effective will work only for pollen allergies.

C.The study will be able to produce results concerning the effects of the antihistamines only on pollen allergies.

D. The study is clearly biased and, therefore, not relevant to determining which antihistamine is most effective.

HINT: For Q15, first find the percent of the 200 students polled who want to see movies.

15. The PTA is planning to sponsor a cultural arts day at school. It asked 200 randomly selected students what the focus of the day should be. Of those students asked, 42 recommended having a speaker, 48 asked for an art display, 60 wanted to taste foods from around the world, and the rest said they would like to watch cultural movies. If there are 1,260 students in the school, about how many would you expect to want to watch cultural movies?

HINT: For Q16, make sure you’re answering the question you’re actually being asked.

16. SoFast Internet is hoping to expand its services to three new counties in rural Virginia. According to its research, a total of approximately 86,400 homes in the three counties currently have Internet service. SoFast surveys a sample of 500 randomly selected households with Internet service and finds that 225 are not satisfied with their current provider. SoFast would be the only other Internet service provider in the area, and it is confident that it will be able to acquire 80% of the dissatisfied households. Based on this information and the results of the sample survey, about how many new customers should SoFast be able to acquire?

A. 31,104

B.38,880

C.41,608

D. 69,120

17. A psychology professor at a large university is conducting a research project on pre-law students’ study habits for courses required for their majors versus their study habits for strictly elective courses. His original plan was to randomly select 250 third-year pre-law students and 250 fourth-year pre-law students and ask them to estimate the amount of time they spend studying for the two course types. Due to a printing error, only 200 survey copies were made; 50 of these went to third-years and 150 went to fourth-years. Assuming 100% of the surveys are returned, what effect(s) will the printing error have on the data collected?

A. The mean study times will be skewed toward third-year students and the margins of error will increase.

B.The mean study times will be skewed toward fourth-year students and the margins of error will increase.

C.The mean study times will be skewed toward third-year students and the margins of error will decrease.

D. The mean study times will be skewed toward fourth-year students and the margins of error will decrease.