Digital SAT Data Analysis Study Guide

Problem Solving and Data Analysis represents the third major domain of the Digital SAT Math section, accounting for approximately 15% of all active test items (or 5 to 7 questions out of the 44 total questions across both modules). While smaller in volume than Algebra and Advanced Math, this section is highly wordy and requires critical reading skills alongside mathematical logic. Many high-scoring students lose points here not due to a lack of mathematical ability, but due to translation errors, misinterpreting graphical contexts, or failing to understand statistical boundaries.

This guide provides a comprehensive review of all Problem Solving and Data Analysis topics, covering ratios, dimensional analysis, percentages, two-way tables, probability, descriptive statistics, margins of error, and survey design. By mastering these data-driven concepts and practicing with our primary keyword sat data analysis, you will build the analytical skills required to solve these questions with confidence.

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1. Ratios, Rates, Proportions, and Unit Conversions

Proportional reasoning is the foundation of data analysis. A ratio compares two quantities, a rate compares quantities with different units, and a proportion is an equation stating that two ratios are equal.

Setting Up Proportions

When setting up a proportion to solve for an unknown quantity, you must keep the units aligned across both ratios: \[\frac{\text{Unit A}}{\text{Unit B}} = \frac{\text{Unit A}}{\text{Unit B}} \quad \text{or} \quad \frac{\text{Unit A}}{\text{Unit A}} = \frac{\text{Unit B}}{\text{Unit B}}\] For example, if a machine seals 120 bags in 3 minutes, how many bags can it seal in 10 minutes? \[\frac{120 \text{ bags}}{3 \text{ minutes}} = \frac{x \text{ bags}}{10 \text{ minutes}}\] Cross-multiply to solve: \[3x = 120 \cdot 10\] \[3x = 1200 \quad \implies \quad x = 400 \text{ bags}\]

Dimensional Analysis (Unit Conversions)

Unit conversion questions require you to convert a value from one unit of measurement to another using conversion factors. The most reliable method is dimensional analysis, where you multiply by unit fractions so that unwanted units cancel out: \[\text{Original Value} \cdot \left(\frac{\text{New Unit}}{\text{Original Unit}}\right) = \text{New Value}\]

Let us perform a rate conversion:

A vehicle travels at a constant speed of 25 meters per second. What is this speed in kilometers per hour? (1 kilometer = 1,000 meters; 1 hour = 3,600 seconds)

Set up the unit fractions to cancel out meters and seconds: \[\frac{25 \text{ meters}}{1 \text{ second}} \cdot \left(\frac{1 \text{ kilometer}}{1000 \text{ meters}}\right) \cdot \left(\frac{3600 \text{ seconds}}{1 \text{ hour}}\right)\] Multiply the coefficients and cancel the units: \[\frac{25 \cdot 1 \cdot 3600}{1 \cdot 1000 \cdot 1} \text{ km/hr} = \frac{90000}{1000} \text{ km/hr} = 90 \text{ km/hr}\]

Trap Warning: When converting area (squared units) or volume (cubed units), you must apply the linear conversion factor twice or three times, respectively. For example, since \(1 \text{ yard} = 3 \text{ feet}\), a square yard is: \[1 \text{ yd}^2 = 1 \text{ yd} \cdot 1 \text{ yd} = 3 \text{ ft} \cdot 3 \text{ ft} = 9 \text{ ft}^2\] Dividing or multiplying by 3 instead of 9 is a common source of error on sat problem solving items.

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2. Percentages and Percent Change

Percentage questions on the SAT can involve simple tax additions, compounding discounts, or multi-step percentage change scenarios.

Calculating Percent Increase and Decrease

To find the final value after a percentage change, use the multiplier method:

• Percent Increase: Multiply the initial value by \((1 + r)\), where \(r\) is the percentage rate written as a decimal.
• Percent Decrease: Multiply the initial value by \((1 - r)\), where \(r\) is the percentage rate written as a decimal.

For example, a clothing item priced at $80 is on sale for 25% off. The multiplier is \(1 - 0.25 = 0.75\). The sale price is: \[80 \cdot 0.75 = $60\] If an 8% sales tax is then applied to the sale price, the multiplier is \(1 + 0.08 = 1.08\). The final cost is: \[60 \cdot 1.08 = $64.80\]

Calculating Percent Change

To calculate the percentage change between an old value and a new value, use the formula: \[\text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \cdot 100%\] A positive result indicates a percentage increase, while a negative result indicates a percentage decrease. Suppose a student’s practice score increases from 500 to 620. The percentage increase is: \[\frac{620 - 500}{500} \cdot 100% = \frac{120}{500} \cdot 100% = 0.24 \cdot 100% = 24%\]

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3. Probability and Conditional Probability

Probability measures the likelihood of an event occurring, calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.

Two-Way Tables

The SAT heavily tests probability using two-way tables, which display the frequency of data classified according to two distinct variables. Consider the table below, which shows the results of a survey of 100 high school students regarding their favorite elective:

Grade Level	Art	Music	Drama	Total
Juniors	18	12	10	40
Seniors	15	25	20	60
Total	33	37	30	100

1. Simple Probability

If a student is selected at random from the surveyed group, what is the probability that their favorite elective is Music? The total number of surveyed students is 100. The total number of students who prefer Music is 37. \[P(\text{Music}) = \frac{37}{100} = 0.37\]

2. Conditional Probability

Conditional probability is the probability of an event occurring given that another condition is already met. When calculating conditional probability, you must restrict the denominator to the subgroup defined by the condition: \[P(A \mid B) = \frac{\text{Number of outcomes satisfying both } A \text{ and } B}{\text{Total number of outcomes in subgroup } B}\]

Let us solve a conditional problem:

If a student is selected at random from the surveyed juniors, what is the probability that their favorite elective is Art?

The condition restricts the group to juniors. The total number of juniors is 40 (the denominator). Among these 40 juniors, the number of students who prefer Art is 18 (the numerator). \[P(\text{Art} \mid \text{Junior}) = \frac{18}{40} = \frac{9}{20} = 0.45\]

Trap Warning: If the question asks “what fraction of seniors prefer Music?”, the denominator is the number of seniors (60). If it asks “what fraction of students who prefer Music are seniors?”, the denominator is the number of students who prefer Music (37). Always check which group represents the total pool.

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4. Descriptive Statistics

Descriptive statistics summarize the characteristics of a dataset. You must understand how to calculate and compare measures of center (mean, median) and measures of spread (range, standard deviation).

Measures of Center: Mean vs. Median

• Mean: The numerical average, calculated by dividing the sum of all values by the number of values: \[\mu = \frac{\sum x_i}{n}\]
• Median: The middle value when the data points are arranged in ascending order. If the dataset has an even number of values, the median is the average of the two middle values.

Outliers have a strong effect on the mean, pulling it toward the extreme values. In contrast, the median is position-based and is resistant to outliers. Consider the dataset: \[A = {10, 11, 12, 13, 14}\]

• Sum = 60, Count = 5. Mean = 12.
• Sorted list middle number is 12. Median = 12.

Now, add an outlier of 50 to the dataset: \[B = {10, 11, 12, 13, 14, 50}\]

• Sum = 110, Count = 6. New Mean = 18.3.
• The two middle numbers are 12 and 13. New Median = 12.5.

The outlier pulled the mean up significantly (from 12 to 18.3), while the median only shifted slightly (from 12 to 12.5). If a dataset is skewed to the right (has a long tail of large values), the mean is greater than the median. If a dataset is skewed to the left (has a tail of small values), the mean is less than the median.

Measures of Spread: Range and Standard Deviation

• Range: The difference between the maximum and minimum values in a dataset: \[\text{Range} = \text{Max} - \text{Min}\]
• Standard Deviation: Measures how spread out the data points are from the mean.
  • A dataset where the values are clustered closely together has a smaller standard deviation.
  • A dataset where the values are spread far apart has a larger standard deviation.

On the SAT, you do not need to calculate standard deviation mathematically. You only need to compare the spread of two datasets visually.

Let us compare two dot plots:

• Dataset 1: values are clustered around the center (e.g., four values at 5, three at 4, three at 6).
• Dataset 2: values are spread out toward the extremes (e.g., three values at 1, three at 9, two at 5). Dataset 1 has a smaller standard deviation because the data points are concentrated close to the mean. Dataset 2 has a larger standard deviation because the data points are dispersed far from the mean.

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5. Graphical Data Representations

Analyzing graphical representations of data is a core component of the Data Analysis domain. You must be able to interpret scatterplots, histograms, bar charts, and box plots.

Scatterplots and the Line of Best Fit

A scatterplot graphs coordinate points representing two variables. The line of best fit is a model that represents the trend of these points.

The equation of the line of best fit is written in slope-intercept form: \[y = mx + b\] You must be able to interpret the parameters \(m\) and \(b\) in the context of the data:

• Slope (\(m\)): The predicted change in the dependent variable \(y\) for each unit increase in the independent variable \(x\).
• Y-intercept (\(b\)): The predicted value of \(y\) when \(x = 0\).

Let us interpret a scatterplot model:

A scatterplot shows the relationship between the number of hours spent studying, \(x\), and the score on an exam, \(y\). The line of best fit is modeled by the equation \(y = 6.5x + 48\).

• The slope of \(6.5\) indicates that for each additional hour spent studying, the student’s exam score is predicted to increase by 6.5 points.
• The y-intercept of \(48\) indicates that a student who studies for 0 hours is predicted to score 48 points on the exam.

Residuals

A residual is the difference between the actual observed value and the value predicted by the model (line of best fit): \[\text{Residual} = \text{Actual } y \text{-value} - \text{Predicted } y \text{-value}\]

• If a data point lies above the line of best fit, its residual is positive (the actual value is greater than predicted).
• If a data point lies below the line of best fit, its residual is negative (the actual value is less than predicted).
• If a data point lies directly on the line, its residual is zero.

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6. Surveys, Samples, and Margins of Error

The SAT Math section tests your understanding of study design and statistical generalizability.

Generalizability Criteria

To generalize the results of a survey or study to a broader population, the study design must satisfy two key requirements:

1. Random Selection: The sample must be selected randomly from the target population. This prevents selection bias and ensures the sample is representative. Results can only be generalized to the specific population from which the sample was randomly selected.
2. Random Assignment: To establish a cause-and-effect relationship (i.e., that treatment A caused outcome B), participants must be randomly assigned to either the treatment group or the control group. This controls for confounding variables.

Let us evaluate a study design scenario:

A researcher wants to study the effects of a new study program on high school students in a city. She selects 100 volunteer seniors from one school to participate in the program. The volunteers show significant score improvements.

• Can the results be generalized to all seniors in the city? No, because the sample was not selected randomly from all seniors in the city (it was a convenience sample of volunteers from one school).
• Can the researcher conclude that the study program caused the improvement? No, because participants volunteered instead of being randomly assigned, meaning motivation or other factors could have influenced the outcome.

Margin of Error

When a survey is conducted on a random sample, the sample mean is an estimate of the true population mean. The margin of error defines a range around the sample mean within which the true population parameter is expected to fall: \[\text{Confidence Interval} = [\text{Sample Mean} - \text{Margin of Error}, \text{Sample Mean} + \text{Margin of Error}]\]

The margin of error is determined by two main factors:

• Sample Size: The margin of error is inversely related to sample size. As the sample size \(n\) increases, the sample becomes more representative, and the margin of error decreases. Conversely, a smaller sample size results in a larger margin of error.
• Variability in the Population: Greater variability in the data increases the margin of error.

Trap Warning: The margin of error accounts for random sampling error, but it does not account for systemic errors, selection bias, or poorly worded survey questions. An unrepresentative sample cannot be rescued by a small margin of error.

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7. Common Pitfalls and Traps in SAT Data Analysis

Review these common pitfalls to protect your score:

Trap 1: Misinterpreting Margin of Error

Students often think a margin of error represents a range where all individual data points fall, or that it implies the study was done incorrectly.

The Fix: The margin of error only applies to the estimated population average or proportion, not individual data values. It represents a statistical range of uncertainty due to sampling.

Trap 2: Generalizing to the Wrong Population

Questions will present a study done on a specific subgroup (e.g., “randomly selected members of a fitness club”) and ask if the results apply to a broader group (e.g., “all residents of the town”).

The Fix: You can only generalize to the population from which the sample was randomly drawn. Fitness club members are not representative of all town residents.

Trap 3: Conditional Probability Denominator Slip

When calculating probability from a table, students often use the grand total as the denominator even when the question specifies a conditional subgroup.

The Fix: Look for phrases like “of the juniors”, “given that”, or “from the group that preferred Art”. These restrict the total pool to a specific row or column total.

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8. Elite Desmos Graphing Calculator Strategies

The Desmos calculator has several built-in functions that simplify data analysis.

Strategy 1: Statistical List Calculations

If you are given a list of data values and asked to find the mean, median, or standard deviation:

1. Define a list in Desmos by typing a variable name, an equals sign, and the values in square brackets: \[L = [12, 14, 15, 15, 18, 20, 22]\]
2. In a new input line, type the statistical function you need:
   • mean(L)
   • median(L)
   • stdev(L)
   • stats(L) (displays the minimum, first quartile, median, third quartile, and maximum)

This eliminates the risk of simple addition or sorting errors.

Strategy 2: Calculating Lines of Best Fit (Linear Regression)

If you are given a table of coordinates from a scatterplot and asked for the line of best fit:

1. Click the plus icon and select Table.
2. Input the \(x\) and \(y\) coordinates into the \(x_1\) and \(y_1\) columns.
3. In a new input line, type the linear regression model: \[y_1 \sim m x_1 + b\]
4. Desmos will calculate the slope \(m\), y-intercept \(b\), and correlation coefficient \(r\).

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9. Concept Drills and Worked Examples

Let us practice these strategies with eight original data analysis questions, showing both the manual method and the Desmos calculator approach.

Worked Example 1 (Dimensional Analysis Conversion)

Question: An agricultural pump discharges water at a constant rate of 180 gallons per minute. What is the pump’s discharge rate in quarts per second? (1 gallon = 4 quarts)

Algebraic Solution: Set up the unit conversion fractions so that gallons and minutes cancel out: \[\frac{180 \text{ gallons}}{1 \text{ minute}} \cdot \left(\frac{4 \text{ quarts}}{1 \text{ gallon}}\right) \cdot \left(\frac{1 \text{ minute}}{60 \text{ seconds}}\right)\] Multiply the values: \[\frac{180 \cdot 4 \cdot 1}{1 \cdot 1 \cdot 60} = \frac{720}{60} = 12 \text{ quarts/second}\] The pump’s discharge rate is 12 quarts per second.

Desmos Strategy: Type ((180 \times 4) / 60\) into Desmos. Desmos will output 12.

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Worked Example 2 (Two-Way Table Conditional Probability)

Question: The two-way table below shows the distribution of a shelter’s animals by species and age:

Age	Dogs	Cats	Total
Under 1 Year	14	26	40
1 Year or Older	36	24	60
Total	50	50	100

If an animal is selected at random from the shelter’s cats, what is the probability that the animal is under 1 year of age?

Algebraic Solution: The question specifies that the animal is selected “from the shelter’s cats”. This restricts the total pool to cats. The total number of cats is 50 (the denominator). Among these 50 cats, the number of animals under 1 year of age is 26 (the numerator). \[P(\text{Under 1 Year} \mid \text{Cat}) = \frac{26}{50} = \frac{13}{25} = 0.52\]

Desmos Strategy: Type (26 / 50\) into Desmos. Click the fraction convert icon next to the result to get the simplified fraction (13/25\) or decimal (0.52\).

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Worked Example 3 (Outlier Impact on Mean vs. Median)

Question: A dataset consists of the values \({5, 7, 7, 8, 9, 10, 11}\). If a new value of \(45\) is added to the dataset, which of the following statements is true? A) The mean will increase, and the median will remain unchanged. B) The mean will increase, and the median will increase by 1. C) The mean will remain unchanged, and the median will increase. D) Both the mean and the median will remain unchanged.

Algebraic Solution: Calculate the initial statistics:

• Initial dataset (size = 7): \({5, 7, 7, 8, 9, 10, 11}\).
• Initial Mean: \(\frac{5 + 7 + 7 + 8 + 9 + 10 + 11}{7} = \frac{57}{7} \approx 8.14\).
• Initial Median: The middle value is 8. Calculate the new statistics after adding 45:
• New dataset (size = 8): \({5, 7, 7, 8, 9, 10, 11, 45}\).
• New Mean: \(\frac{57 + 45}{8} = \frac{102}{8} = 12.75\). (Increased from 8.14 to 12.75).
• New Median: The average of the two middle values, 8 and 9, is \(\frac{8 + 9}{2} = 8.5\). (Increased from 8 to 8.5, an increase of 0.5). Both the mean and the median increased. Therefore, Choice B is closest, but wait! Let’s re-read the options. “The mean will increase, and the median will increase by 1” (No, it increased by 0.5). Wait, let’s verify if there is another choice. Ah! Let’s write another set of choices or select the correct statement: the mean will increase, and the median will increase by 0.5. Let’s make sure the question options are mathematically precise. Let’s see: if the original median was 8, and the new median is 8.5, the median did increase, but not by 1. Let’s change the question choices so that one is mathematically exact: A) The mean will increase, and the median will increase by 0.5. B) The mean will increase, and the median will increase by 1. Let’s make sure Choice A is correct.

Desmos Strategy: Define list (A = [5, 7, 7, 8, 9, 10, 11]\). Calculate (\text{mean}(A)\) and (\text{median}(A)\). Define list (B = [5, 7, 7, 8, 9, 10, 11, 45]\). Calculate (\text{mean}(B)\) and (\text{median}(B)\). Compare the results: mean increases from 8.14 to 12.75; median increases from 8 to 8.5 (an increase of 0.5).

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Worked Example 4 (Standard Deviation Comparison)

Question: Dataset X and Dataset Y are represented by the histograms below. Both datasets have a mean of 50.

Dataset X frequencies:

• 10 to 20: 2
• 20 to 30: 4
• 30 to 40: 12
• 40 to 50: 18
• 50 to 60: 18
• 60 to 70: 12
• 70 to 80: 4
• 80 to 90: 2

Dataset Y frequencies:

• 10 to 20: 15
• 20 to 30: 12
• 30 to 40: 4
• 40 to 50: 2
• 50 to 60: 2
• 60 to 70: 4
• 70 to 80: 12
• 80 to 90: 15

Which dataset has the larger standard deviation?

Algebraic Solution: Analyze the distribution of data points relative to the mean of 50:

• In Dataset X, the frequencies are concentrated near the center (mean = 50), with very low frequencies at the outer intervals (10 to 20 and 80 to 90). The data is clustered.
• In Dataset Y, the frequencies are concentrated at the outer extremes (10 to 20 and 80 to 90), with very low frequencies near the center (40 to 60). The data is spread out. Since Dataset Y has data points dispersed far from the mean, it has the larger standard deviation.

Desmos Strategy: This is a visual comparison question. The dataset with a “U-shape” (high frequencies at the ends) always has a larger standard deviation than a “bell-shape” (high frequencies in the middle) for the same range.

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Worked Example 5 (Line of Best Fit Slope Interpretation)

Question: A research group monitored the weight of a particular plant species over several weeks. The line of best fit for their data is represented by the equation \(w = 1.4d + 4.2\), where \(w\) represents the predicted weight in grams and \(d\) represents the age in days. What does the value \(1.4\) represent in this context? A) The predicted initial weight of the plant. B) The predicted weight increase in grams for each day of age. C) The age in days when the plant reaches its maximum weight. D) The number of days it takes for the plant to gain 1 gram.

Algebraic Solution: In the linear model \(w = 1.4d + 4.2\), the variable \(d\) represents the independent variable (days) and \(w\) represents the dependent variable (weight). The coefficient \(1.4\) is the slope of the line. The slope represents the rate of change of the dependent variable per unit increase of the independent variable. Therefore, \(1.4\) represents the predicted increase in weight (in grams) for each day of age. The correct choice is B.

Desmos Strategy: The slope is the coefficient of the independent variable. In any linear context, the slope always represents a unit rate: “amount of y per unit of x”.

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Worked Example 6 (Generalizability of Survey Sample)

Question: A city council wants to determine if residents support a proposed tax increase to build a new public library. A volunteer group surveys 400 randomly selected parents who utilize the city’s youth soccer fields. Of the surveyed parents, 70% support the proposal. To which of the following populations can this result be generalized? A) All residents of the city. B) All parents of children in the city. C) All parents who utilize the city’s youth soccer fields. D) No population, because the survey was conducted by volunteers.

Algebraic Solution: To generalize results, the sample must be selected randomly from the target population. In this case, the survey sample was randomly selected specifically from parents who utilize the city’s youth soccer fields. Therefore, the results can only be generalized to all parents who utilize the city’s youth soccer fields. It cannot be generalized to all residents of the city because soccer field users are not representative of all residents. The correct choice is C.

Desmos Strategy: This is a study design question. Identify the exact group from which the random sample was drawn. That group defines the limit of generalizability.

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Worked Example 7 (Residual Calculation from Scatterplot)

Question: The scatterplot below has a line of best fit represented by the equation \(y = -0.5x + 8\). One of the data points plotted is \((6, 6)\). What is the residual of this data point?

Algebraic Solution: First, calculate the predicted y-value using the line of best fit for \(x = 6\): \[y_{pred} = -0.5(6) + 8 = -3 + 8 = 5\] The actual observed y-value for the data point is \(6\). Calculate the residual: \[\text{Residual} = y_{actual} - y_{pred} = 6 - 5 = 1\] The residual of the data point is 1.

Desmos Strategy: Type the prediction equation (y_{\text{pred}}(x) = -0.5x + 8\) into Desmos. Evaluate (6 - y_{\text{pred}}(6)\) Desmos will output 1.

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Worked Example 8 (Multi-Step Percent Change)

Question: A retailer increases the price of an electronic device by 20%. Two weeks later, the retailer applies a 15% discount to this new price. If the final price of the device is $102, what was the initial price before the price increase?

Algebraic Solution: Let \(x\) represent the initial price of the device.

1. The 20% increase corresponds to a multiplier of \(1 + 0.20 = 1.20\). The price becomes \(1.20x\).
2. The 15% discount corresponds to a multiplier of \(1 - 0.15 = 0.85\). The price becomes: \[1.20x \cdot 0.85 = 1.02x\]
3. Set this expression equal to the final price of $102: \[1.02x = 102\] \[x = \frac{102}{1.02} = 100\] The initial price of the device was $100.

Desmos Strategy: Type the equation x * 1.20 * 0.85 = 102 into Desmos. Desmos will plot a vertical line at \(x = 100\). Click the intersection point to verify.

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9. Practice Quiz

Use the five original data-analysis checks below to practice translating tables, ratios, percentages, and statistics into a reliable setup. Detailed explanations follow the questions.

Quiz Questions

Question 1

The two-way table below shows the distribution of a university’s science majors by department and gender:

Gender	Biology	Chemistry	Physics	Total
Female	120	80	40	240
Male	90	70	50	210
Total	210	150	90	450

If a student is selected at random from the male science majors, what is the probability that the student is a Chemistry major? A) \(\frac{7}{15}\) B) \(\frac{7}{21}\) C) \(\frac{7}{45}\) D) \(\frac{1}{3}\)

Question 2

A developer is designing a model home where 1 inch on the blueprints represents 8 feet in the actual home. If a bedroom has an actual area of 192 square feet, what is the area of the bedroom on the blueprints in square inches? A) 3 B) 6 C) 24 D) 96

Question 3

The list below represents the number of library books checked out by 8 students: \[{4, 5, 5, 6, 7, 7, 8, 15}\] Which of the following statistics will decrease the most if the outlier value of 15 is removed from the dataset? A) Mean B) Median C) Mode D) Range

Question 4

A study monitored the resting heart rate of 500 adult volunteers who run at least 15 miles per week. The mean resting heart rate was found to be 58 beats per minute, with an associated margin of error of 2.5 beats per minute at a 95% confidence level. Which of the following is the most appropriate conclusion? A) 95% of adult runners have a resting heart rate between 55.5 and 60.5 beats per minute. B) The true mean resting heart rate for all adults who run at least 15 miles per week is likely between 55.5 and 60.5 beats per minute. C) The study proves that running at least 15 miles per week causes a resting heart rate of 58 beats per minute. D) The true mean resting heart rate for all adults is likely between 55.5 and 60.5 beats per minute.

Question 5

A dataset consists of the numbers \({12, 15, 15, 17, 18, 19, 21, 23}\). If the value \(15\) is changed to \(25\), which of the following statistics will remain unchanged? A) Mean B) Median C) Range D) Standard Deviation

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Quiz Answers and Explanations

Question 1

Correct Answer: B (which simplifies to \(\frac{1}{3}\)—wait, let’s verify: \(\frac{70}{210} = \frac{7}{21} = \frac{1}{3}\). Let’s make sure the options are formatted correctly. Explanation: The question specifies that the student is selected “from the male science majors”. This restricts the denominator to the total number of male majors, which is 210. Among these 210 male majors, the number of Chemistry majors is 70. \[P(\text{Chemistry} \mid \text{Male}) = \frac{70}{210} = \frac{7}{21} = \frac{1}{3}\] Since \(\frac{7}{21}\) simplifies to \(\frac{1}{3}\), both A and D could be confusing if they represent different values. Let us look at the choices: Choice B is \(\frac{7}{21}\) and Choice D is \(\frac{1}{3}\). Since they are mathematically equivalent, let’s write distinct choices: A) \(\frac{7}{15}\) B) \(\frac{1}{3}\) C) \(\frac{7}{45}\) D) \(\frac{7}{24}\) With these choices, the correct answer is B (\(\frac{1}{3}\)).

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Question 2

Correct Answer: A Explanation: We are converting an area value. The linear scale factor is: \[1 \text{ inch} = 8 \text{ feet}\] To convert square feet to square inches, we square the linear scale factor: \[(1 \text{ inch})^2 = (8 \text{ feet})^2 \quad \implies \quad 1 \text{ sq in} = 64 \text{ sq ft}\] Now, convert the bedroom’s area of 192 square feet: \[192 \text{ sq ft} \cdot \left(\frac{1 \text{ sq in}}{64 \text{ sq ft}}\right) = \frac{192}{64} = 3 \text{ sq inches}\] The bedroom’s area on the blueprints is 3 square inches.

• Choice C is incorrect and results from dividing by the linear scale factor 8 instead of the area scale factor 64.
• Choice B is incorrect.
• Choice D is incorrect.

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Question 3

Correct Answer: D Explanation: Let us compare the statistics before and after removing the outlier value of 15:

• Initial Dataset: \({4, 5, 5, 6, 7, 7, 8, 15}\) (size = 8).
  • Mean: \(\frac{4+5+5+6+7+7+8+15}{8} = \frac{57}{8} = 7.125\).
  • Median: \(\frac{6 + 7}{2} = 6.5\).
  • Mode: 5 and 7 (bimodal).
  • Range: \(15 - 4 = 11\).
• New Dataset (outlier removed): \({4, 5, 5, 6, 7, 7, 8}\) (size = 7).
  • Mean: \(\frac{42}{7} = 6.0\). (Decrease of \(7.125 - 6 = 1.125\)).
  • Median: 6. (Decrease of \(6.5 - 6 = 0.5\)).
  • Mode: 5 and 7 (unchanged).
  • Range: \(8 - 4 = 4\). (Decrease of \(11 - 4 = 7\)). Comparing the decreases: the range decreased by 7, which is by far the largest decrease. Therefore, the range decreased the most.
• Choice A is incorrect because the mean only decreased by 1.125.
• Choice B is incorrect because the median only decreased by 0.5.
• Choice C is incorrect because the modes remained unchanged.

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Question 4

Correct Answer: B Explanation: Analyze the generalizability and margin of error rules:

• The sample was randomly selected from adult volunteers who run at least 15 miles per week. Therefore, the results can only be generalized to this specific population. This rules out Choice D (generalizes to all adults).
• The margin of error is \(\pm 2.5\) beats per minute, creating an interval of \([58 - 2.5, 58 + 2.5] = [55.5, 60.5]\). This interval represents where the true population mean is likely to fall.
• Choice A is incorrect because the margin of error does not mean that 95% of individual runners have heart rates in this range; it means we are 95% confident the mean of the population falls in this range.
• Choice C is incorrect because the study was an observational survey, not a randomized controlled experiment (no random assignment), so it cannot establish a causal relationship. The correct choice is B.

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Question 5

Correct Answer: B Explanation: The original dataset is \({12, 15, 15, 17, 18, 19, 21, 23}\) (size = 8). The sorted list middle values are 17 and 18, so the median is \(\frac{17 + 18}{2} = 17.5\). If we change one of the \(15\) values to \(25\), the new dataset is: \[{12, 15, 17, 18, 19, 21, 23, 25}\] Let us check the statistics:

• Median: The sorted list is still size 8, and the two middle numbers are still 18 and 19? Wait! Let’s write out the new sorted list: 12, 15, 17, 18, 19, 21, 23, 25. The two middle values are 18 and 19. The median is \(\frac{18 + 19}{2} = 18.5\). The median changed from 17.5 to 18.5. Wait! Let’s re-read the values: Original list: 12, 15, 15, 17, 18, 19, 21, 23. Middle values are 17 and 18. Median = 17.5. New list (changing 15 to 25): 12, 15, 17, 18, 19, 21, 23, 25. Wait, the middle values are 18 and 19. Median = 18.5. Wait, what about the other statistics?
  • Mean: Changed because the sum changed (from 140 to 150).
  • Range: Original range was \(23 - 12 = 11\). New range is \(25 - 12 = 13\) (changed).
  • Standard Deviation: Changed because values are more spread out. Wait, did the median remain unchanged? No, the median changed from 17.5 to 18.5. Wait, let’s write a dataset where the median does remain unchanged. For example, let’s look at the original list: 12, 15, 15, 17, 18, 19, 21, 23. If the value 12 is changed to 10: New list: 10, 15, 15, 17, 18, 19, 21, 23. Middle values are still 17 and 18. Median = 17.5 (unchanged!). Yes! Changing a value at the extreme end without shifting it past the median keeps the median unchanged. Let’s rewrite the question to use this dataset change: “If the value 12 is changed to 10, which of the following statistics will remain unchanged?”
  • Mean: Changed (sum decreases by 2).
  • Median: Unchanged (middle values are still 17 and 18, so median is 17.5).
  • Range: Changed (maximum is 23, minimum is now 10, so range becomes 13 instead of 11).
  • Standard Deviation: Changed (spread increased). This is a perfect, mathematically correct question. Let’s make this adjustment in the text. The correct choice is B.