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

How much do you know?
Tables, statistics, and probability
Data analysis

LEARNING OBJECTIVES

After completing this chapter, you will be able to:

· Draw inferences about data presented in a variety of graphical formats

· Find an unknown value given the average

· Calculate mean, median, mode, and range

· Define standard deviation and margin of error

· Determine whether a survey is valid or biased

· Draw inferences about surveys and data samples

· Calculate probabilities based on data sets

60/600 SmartPoints^® (High Yield)

How much do you know?

Directions: Try the questions that follow. Show your work so that you can compare your solutions to the ones found in the Check Your Work section immediately after this question set. The “Category” heading in the explanation for each question gives the title of the lesson that covers how to solve it. If you answered the question(s) for a given lesson correctly, and if your scratchwork looks like ours, you may be able to move quickly through that lesson. If you answered incorrectly or used a different approach, you may want to take your time on that lesson.

Questions 1 and 2 refer to the following information.

1. Assuming that all students are required to take exactly one language, which student class contains the greatest percentage of students enrolled in a Romance (French or Italian) language?

A. Freshmen

B.Sophomores

C.Juniors

D. Seniors

2. A group of 150 freshmen were randomly selected for a survey that asked whether they were excited to continue their foreign language studies in the next school year. Of this group, 93 said they were excited to continue, 37 said they were not excited to continue, and 20 said they did not know yet whether they were excited to continue. Using the data from this survey and the table, which of the following is most likely a valid conclusion?

A. About 130 freshman students do not know yet whether they are excited to continue foreign language studies next year.

B.About 240 freshman students are not excited to continue foreign language studies next year.

C.About 440 freshman students are not excited to continue foreign language studies next year.

D. About 740 freshman students are excited to continue foreign language studies next year.

3. Liam is conducting a market research study to determine the effectiveness of advertisements on social networking sites. He surveyed a randomly selected group of 2,000 young adults on how likely they are to purchase from a company based on its advertisements on social networking sites. Respondents rated their likelihood to purchase on a scale from 1 to 7, with 1 being “Not at all likely” and 7 being “Definitely.” The results are summarized in the bar graph above. Which of the following statements is NOT true based on the survey results?

A. mode = 3

B.median = 3

C.mean < median

D. 2 < mean < 5

4. The average (arithmetic mean) of the data list {2, 4, 5, 8, 9, x, 3} is 5. What is the value of x ?

Questions 5 and 6 refer to the following information.

The table below summarizes the results of a survey about the favorite sports of a group of high school students. Assume that every student has a favorite sport and that students could select only one favorite.

5. The research group that conducted the survey wants to select one participant at random for a follow-up survey. Given that the selected participant is not a freshman, what is the probability of the research group randomly selecting someone who chose baseball as his or her favorite sport?

A. 0.1622

B.0.2164

C.0.2399

D. 0.2948

6. Based on the table provided, which of the following is NOT a true statement?

A. About 27% of the entrants in the freshman prize drawing voted for basketball as their favorite sport.

B.About 23% of the entrants in the sophomore prize drawing voted for basketball as their favorite sport.

C.About 27% of the entrants in the junior prize drawing voted for basketball as their favorite sport.

D. About 30% of the entrants in the senior prize drawing voted for basketball as their favorite sport.

Check Your Work

1. C

Difficulty: Medium

Category: Tables and Graphs

Getting to the Answer: Focus on the columns of the table that you need and block out the rest. Start by identifying how many students in each class are enrolled in either French or Italian (freshmen: 342 French, 489 Italian; sophomores: 276 French, 324 Italian; juniors: 190 French, 252 Italian; seniors: 158 French, 219 Italian). Next, extract the total foreign language enrollment of each class from the table (freshmen: 1,195; sophomores: 853; juniors: 579; seniors: 498). To find the percentages, divide the Romance language part by the language total, then multiply by 100% to convert to a percent:

The junior class has the highest percent enrollment in Romance languages, so (C) is correct.

2. D

Difficulty: Medium

Category: Surveys and Data Samples

Strategic Advice: Find what you need about the group from the follow-up survey. Then, extrapolate to see which answer choice matches your calculations.

Getting to the Answer: First, determine the number of freshmen who fall into each of the three groups of the follow-up survey:

Percent excited to continue:

Percent not excited to continue:

Percent who do not know yet whether they are excited to continue:

There are 1,195 freshmen in the original survey, making the number excited to continue 0.62 × 1,195 = 740.9 ≈ 741; the number not excited to continue 0.2467 × 1,195 = 294.8 ≈ 295; and the number who do not know yet whether they are excited to continue 0.1333 × 1,195 = 159.3 ≈ 159. Choice (D) states that about 740 freshmen are excited to continue their foreign language studies, which matches these findings. Choice (D) is therefore correct.

3. C

Difficulty: Medium

Category: Statistics

Getting to the Answer: Examine the bar chart to evaluate the choices.

(A): The bar for “Unlikely” (which is 3 on the scale given in the question) is clearly the tallest, so this is the mode. Eliminate (A).

(B): There are 2,000 responses, so the median is the average of responses 1,000 and 1,001. The total of the first two bars is about 600. Since the third bar is about 700, that means that the median response is indeed within this bar. Eliminate (B).

(C): The data is skewed to the right, meaning that the extreme values on the right will weight the mean calculation to make the mean greater than the median. Thus, (C) is not true and is the correct choice.

(D): If you were not certain that (C) was the correct choice, you could see that the substantial majority of values are between 2 and 5 inclusive, so the mean has to be somewhere within that wide range. Eliminating (D) confirms (C) as correct.

4. 4

Difficulty: Medium

Category: Statistics

Strategic Advice: When the goal is to find a missing value in a set of data and the average is given, consider using the balance approach. We’ll demonstrate both approaches starting with the average formula.

Getting to the Answer: The question requires finding the value of x in the given data set using the average of the data set. The values given in the data, besides x, are: 2, 4, 5, 8, 9, and 3. The average is given as 5. To find the missing value x, plug the known values into the average formula and solve for x:

Grid in 4 as the correct answer.

Alternatively, to use the balance approach, write down how much each value is above or below the average of 5. For example, the first value of 2 is 3 below the average: 2 − 5 = −3.

Now, observe that, excluding the variable, the values are −3 + (−1) + 0 + 3 + 4 + (− 2) = 1. Without the variable, the total is 1 more than what you’d expect based on the average. So for the values to balance out to the average, the variable value must be one less than the average of 5, or 5 − 1 = 4. Grid in 4.

5. B

Difficulty: Medium

Category: Probability

Getting to the Answer: The question says that the selected candidate is not a freshman, so focus on the sophomore, junior, and senior columns. This makes the total possible outcomes for this drawing 1,355 + 1,349 + 1,340 = 4,044. The total number of upperclassmen who voted for baseball as their favorite sport (number of desired outcomes) is 343 + 249 + 283 = 875. Divide this by the total to find the probability: , which makes (B) correct.

6. D

Difficulty: Easy

Category: Tables and Graphs

Getting to the Answer: The question asks for the answer choice that is NOT a true statement, so test each one. Use the table to determine the correct percents for each choice. Notice that all choices are about the estimated percent of students in each class who voted for basketball as their favorite sport, so use the percent formula, . In each case, the part is the number of students in each class who voted for basketball, while the whole is the total number of students in that class:

Hence, (D) must be correct since the statement was about 30% and NOT about 22%.