AP Statistics Unit 1 Complete Study Guide

Fundamental Concepts of Statistics & Data Collection

Definition of Statistics and Data:
- Statistics is the science of collecting, analyzing, presenting, and interpreting data.
- Data: Observations that have been measured, recorded, collected, analyzed, and reported for use. Data contains information about a specific group of individuals.
- Individuals: The objects or entities described by a set of data (e.g., people, animals, objects, transactions).
- Observational Unit: The specific individual or item from which a single piece of measurement or data value is collected.
- Datum vs. Data:
- Datum: A single piece of information or one specific value collected from an observational unit (e.g., a single test score of $92$ ).
- Data: A collection of data values; the plural form of datum (e.g., a set of test scores $92, 85, 78, 90$ ).
Populations and Samples:
- Population:
- The entire group of individuals that an investigator is interested in learning about.
- The total size of the population is denoted by the uppercase variable $N$ .
- Parameter: A fixed numerical value describing a characteristic of an entire population. Parameters are typically unknown and are estimated using sample statistics.
- Sample:
- A subset or smaller group selected from the population from which data is actually collected.
- The size of the sample is denoted by the lowercase variable $n$ .
- Statistic: A numerical value calculated from sample data that describes a characteristic of the sample.

Population and Sample Relationship Diagram

Rationale for Sampling:
- Studying an entire population is often impossible or impractical due to:
- The population being excessively large.
- Measuring every individual being too time-consuming.
- High costs associated with reaching or evaluating every member.
Population vs. Sample Contextual Examples:
- Example 1: Female Student Heights
- Scenario: The average height for a $15\text{-year-old}$ female at a high school is $5'4"$ . A sample of $10$ females at the school has an average height of $5'7"$ .
- Population: All $15\text{-year-old}$ females at the high school ( $N$ ).
- Sample: The $10$ females selected and measured ( $n = 10$ ).
- Parameter: The true average height of all $15\text{-year-old}$ females at the school ( $5'4"$ ).
- Statistic: The average height calculated from the sample of $10$ females ( $5'7"$ ).
- Example 2: High School Student Work Hours
- Scenario: High school students in a county work an average of $18\,\text{hours}$ per week. A sample of $50$ high school students in the county revealed an average of $23\,\text{hours}$ per week.
- Population: All high school students in the county ( $N$ ).
- Sample: The $50$ high school students surveyed ( $n = 50$ ).
- Parameter: The average work hours per week for all county high school students ( $18\,\text{hours/week}$ ).
- Statistic: The average work hours per week calculated from the sample ( $23\,\text{hours/week}$ ).

Variables and Data Classification

Definition of a Variable:
- A characteristic or attribute that can take on different values for different observational units.
Categorical Variables (Qualitative):
- Variables that place individuals into specific groups, categories, or non-numerical classifications.
- Arithmetic operations (such as addition or averaging) do not make sense for categorical data.
- Examples: Eye color, gender, blood type, fast food preference, zip code, jersey numbers, state of residence.
Quantitative Variables (Numerical):
- Variables that take on numerical values where standard arithmetic operations (such as calculating a mean) are meaningful.
- Quantitative variables are subdivided into discrete and continuous types:
- Discrete Quantitative Variables:
  - Numerical data where values are isolated points along a scale, typically obtained by counting.
  - Whole numbers make logical sense; decimal parts or fractional values between increments do not make sense in context.
  - Examples: Number of apps on a smartphone, count of alarm rings in the morning, number of siblings, number of text messages sent.
- Continuous Quantitative Variables:
  - Numerical data where values can take on any real value within a given interval, typically obtained by measuring.
  - Decimals and fractions make logical sense to describe measurements to higher precision.
  - Examples: Time spent on homework, height, weight, running time, temperature.
Special Classification Nuances:
- Numbers as Categorical Variables: Numerical identifiers that serve strictly as labels rather than measurable quantities are categorical (e.g., Area Codes, Zip Codes, Social Security Numbers).
- Classification Based on Usage: Certain variables like age can theoretically be measured continuously, but in practice are rounded to integer years. In such contexts, age is treated as a discrete variable unless measured continuously in precise time units.
Comprehensive Variable Analysis Scenario:
- Scenario: A school administrator wants to know how much time students spend on homework each night. A random sample of $n = 75$ students is surveyed, revealing an average of $96\,\text{minutes}$ of homework per night.
- Population: All students enrolled at the school ( $N$ ).
- Sample: The $75$ students surveyed ( $n = 75$ ).
- Variable Measured: Minutes spent on homework each night.
- Variable Type: Quantitative Continuous.
- Investigative Question: What is the average number of minutes spent on homework per night by students at the school?
- Result in Context: The sample of $75$ students spent an average of $96\,\text{minutes}$ on homework per night.

Formulating Investigative Questions

Requirements of an Investigative Question:
- Must have a clearly defined population of interest.
- Must be decided and finalized prior to collecting data.
- Must not change based on the collected data or intermediate results.
Key Characteristics of Strong Investigative Questions:
- Identifies the target population explicitly.
- Identifies the specific variable being measured.
- Is clear, precise, and objective.
- Can be answered directly using gathered data.
Evaluation of Investigative Questions:
- Good Question: "What is the average number of hours of sleep per night for $12\text{th}$ grade students at North High School?"
- Strengths: Explicitly specifies the population ( $12\text{th}$ grade students at North High School) and a clear, measurable quantitative variable (hours of sleep per night).
- Bad Question: "Do teens sleep enough?"
- Weaknesses: Highly vague, does not specify a precise population, lacks a clearly defined measurable variable, and relies on subjective criteria ("enough").
Meaning of "In Context":
- Every numerical value, statistic, summary parameter, or conclusion must be explicitly tied back to the real-world scenario, including relevant units of measurement and population descriptions.
- Non-Contextual Statement: "The mean is $6.2$ ."
- Contextual Statement: "The mean number of hours of sleep for the sample of high school seniors is $6.2\,\text{hours}$ per night."

Graphical Representations of Data

Displays for Categorical Data:
- Bar Graphs: Displays the count or proportion/relative frequency of categorical data using separate bars for each category. Bar order can be rearranged arbitrarily without altering the data.
- Pie Charts: Displays categorical distributions as proportional slices of a circle. The central angle for each slice is calculated as: $\text{Central Angle} = \text{Relative Frequency} \times 360^\circ$
Displays for Quantitative Data:
- Dotplots: Shows each individual data value as a dot placed above a standardized number line. Preserves individual data values explicitly.
- Stemplots (Stem-and-Leaf Plots):
- Organizes quantitative data by separating values into a "stem" (leading digits) and a "leaf" (trailing single digit).
- Stems do not strictly represent the tens place; they can represent hundreds, ones, tenths, or any scale defined by a mandatory plot KEY.
- Preserves exact individual data points while revealing distribution shape.
- Histograms:
- Groups quantitative data into adjacent bins/class intervals along a continuous numerical axis.
- The vertical axis displays frequency or relative frequency.
- Bins must touch (no spaces between bars unless an interval has a count of zero).
- Changing the width of class intervals alters the visual appearance and apparent shape of the distribution.
Selecting Appropriate Visual Graphs:
- Minutes to get ready for school: Quantitative Continuous $\rightarrow$ Histogram.
- Number of smartphone apps: Quantitative Discrete $\rightarrow$ Stemplot.
- Favorite fast food venue: Categorical $\rightarrow$ Bar Graph or Pie Chart.
- Count of morning alarm rings: Quantitative Discrete $\rightarrow$ Dotplot.
Analysis of Misleading Visual Displays:
- Graphs can distort perceptions if axes are manipulated, inappropriate baseline metrics are chosen, or categorical comparisons are conflated.

Google Search vs GenAI Visits Misleading Bar Graph

Case Study: Search Destination Comparison (Dec 2024):
- Graph Representation: Displays Google web visits ( $16.5\text{ Billion}$ ) side-by-side with Generative AI tools like ChatGPT ( $517.0\text{ Million}$ ), Bing ( $376.6\text{ Million}$ ), Gemini ( $42.9\text{ Million}$ ), and Perplexity ( $21.9\text{ Million}$ ), claiming Google drew $31.8\times$ more visitors than the next biggest GenAI destination.
- Why Misleading: Conflates general web search navigation with specific Generative AI queries. Most Google visits are traditional navigational or transactional searches rather than GenAI prompts.
- Intended Spin: Misleads viewers into believing Google dominates the standalone Generative AI search space to an extreme degree.
- Omitted Factor: Search intent and query classification (traditional search vs. generative prompt interaction).

Describing Distributions of Quantitative Variables

The SOCS Framework: When describing a single quantitative distribution, four critical components must be fully detailed in context:
- Shape: The visual configuration of the graph.
- Center: The numerical representation of a typical value.
- Spread: The extent of variability within the dataset.
- Unusual Features: Outliers, gaps, or structural clusters.
Categorization of Shapes:
- Modality: Unimodal (1 prominent peak), Bimodal (2 distinct peaks), Multimodal (3 or more peaks), or Uniform (bars are approximately equal height across all intervals).
- Symmetry vs. Skewness:
- Symmetric: The left and right halves of the graph are approximate mirror images.
- Left (Negatively) Skewed: The tail of the distribution extends farther toward smaller values on the left.
- Right (Positively) Skewed: The tail of the distribution extends farther toward larger values on the right.

Common Distribution Shapes

Rules for Choosing Measures of Center and Spread:
- Symmetric Distributions:
- Center: Use the Mean ( $\bar{x}$ ).
- Spread: Use the Standard Deviation ( $s$ ).
- Skewed Distributions or Distributions with Extreme Outliers:
- Center: Use the Median ( $med$ ).
- Spread: Use the Interquartile Range ( $IQR = Q_3 - Q_1$ ).
- No Summary Statistics: Use the full Range ( $max - min$ ).

Describing Distributions Framework Table

Unusual Features Definition and Identification:
- Outliers: Observations that fall unusually far above or below the general pattern of the data.
- Gaps: Regions of the distribution where no data observations occur.
- Clusters: Groupings of data values separated from other groupings by gaps.
- Outlier Example (Quiz Scores): Data set $9, 8, 9, 10, 8, 9, 9, 2$ exhibits an outlier at $2\,\text{points}$ , separated by a gap from the cluster at $8\text{--}10\,\text{points}$ .
- Cluster/Gap Example (Text Messages): Data set $2, 3, 4, 5, 6, 20, 21, 22$ exhibits two distinct clusters ( $2\text{--}6$ texts and $20\text{--}22$ texts) separated by a large gap spanning $7\text{--}19$ texts.
Detailed Worked Example: Word Length Distribution:
- Data: Word length in letters for Taylor Swift's song "I Can Do It with a Broken Heart".
- Summary Statistics: $\bar{x} = 3.5\,\text{letters}$ , $s = 1.94\,\text{letters}$ , $min = 1\,\text{letter}$ , $Q_1 = 2\,\text{letters}$ , $med = 3\,\text{letters}$ , $Q_3 = 4\,\text{letters}$ , $max = 10\,\text{letters}$ .

Word Length Histogram for Taylor Swift's Song Summary Statistics Table for Word Length

Complete Description: The distribution of word lengths in the song is unimodal and right-skewed. The typical word length is centered at a median of $3\,\text{letters}$ . The middle $50\%$ of word lengths have an interquartile range of $IQR = Q_3 - Q_1 = 4 - 2 = 2\,\text{letters}$ . There is a potential high outlier with a maximum word length of $10\,\text{letters}$ .

Comparing Quantitative Distributions

Principles of Comparative Writing:
- Comparing two or more distributions requires explicit comparative phrasing (e.g., greater than, less than, similar to, larger variability than) rather than listing separate isolated descriptions.
- Comparisons must cover Shape, Center, Spread, and Unusual Features in context.
Comparative Case Study: AP vs. Non-AP Homework Time:
- AP Group ( $n = 21$ ): $\bar{x} = 88.905\,\text{min}$ , $s = 25.894\,\text{min}$ , $min = 15\,\text{min}$ , $Q_1 = 82.5\,\text{min}$ , $med = 95\,\text{min}$ , $Q_3 = 106.5\,\text{min}$ , $max = 120\,\text{min}$ , $IQR = 24\,\text{min}$ .
- Non-AP Group ( $n = 20$ ): $\bar{x} = 47.3\,\text{min}$ , $s = 17.357\,\text{min}$ , $min = 5\,\text{min}$ , $Q_1 = 39\,\text{min}$ , $med = 51\,\text{min}$ , $Q_3 = 60\,\text{min}$ , $max = 70\,\text{min}$ , $IQR = 21\,\text{min}$ .

Time Spent on Homework Histograms for AP vs Non-AP Students AP Students Homework Summary Statistics Non-AP Students Homework Summary Statistics

Synthesized Comparison Paragraph: Both distributions of homework time appear left-skewed. The AP student distribution features a low visual outlier near $15\,\text{minutes}$ , whereas the non-AP student distribution displays no apparent outliers. The center of the AP distribution is higher than the non-AP distribution, as shown by the median of $95\,\text{minutes}$ for AP students compared to $51\,\text{minutes}$ for non-AP students. Additionally, the AP distribution exhibits greater spread/variability than the non-AP distribution, with an $IQR$ of $24\,\text{minutes}$ compared to an $IQR$ of $21\,\text{minutes}$ for non-AP students.

Standardizing Data & Z-Scores

Concept of Standardizing:
- Standardizing is the process of converting observations from different scales to a common standardized scale by calculating how many standard deviations an observation falls above or below the distribution mean.
- Standardizing enables direct performance comparisons across distributions with different means and standard deviations.
Formula for Z-Scores: For an observation $x$ from a distribution with mean $\mu$ (or $\bar{x}$ ) and standard deviation $\sigma$ (or $s$ ): $z = \frac{x - \mu}{\sigma} \quad \text{or} \quad z = \frac{x - \bar{x}}{s}$

Z-Score Definition and Direction Rules

Properties of Z-Scores:
- Indicates the exact number of standard deviations an observation lies from the mean and in what direction.
- Observations larger than the mean produce positive z-scores.
- Observations smaller than the mean produce negative z-scores.
- Observations equal to the mean have a z-score of $0$ .
Standardizing Application Examples:
- Example 1: Mallory's Physics Tests
- Test 1: Score $x = 86\%$ , Mean $\mu = 83\%$ , Standard Deviation $\sigma = 4.5\%$ . $z_1 = \frac{86 - 83}{4.5} = \frac{3}{4.5} \approx 0.67$ Interpretation: Mallory scored $0.67$ standard deviations above the class mean.
- Test 2: Score $x = 82\%$ , Mean $\mu = 78\%$ , Standard Deviation $\sigma = 5.2\%$ . $z_2 = \frac{82 - 78}{5.2} = \frac{4}{5.2} \approx 0.77$ Interpretation: Mallory scored $0.77$ standard deviations above the class mean.
- Conclusion: Test 2 was the more impressive score relative to her peers because her z-score was higher ( $0.77 > 0.67$ ).
- Example 2: Female Shoe Size Interpretation
- Scenario: Female shoe sizes have $\mu = 8$ and $\sigma = 0.75$ . Calculate and interpret the z-score for a size $6$ shoe.
- Calculation: $z = \frac{6 - 8}{0.75} = \frac{-2}{0.75} \approx -2.67$
- Interpretation: A shoe size of $6$ is $2.67$ standard deviations below the mean female shoe size.
- Example 3: Corbin's SAT vs. ACT Performance
- SAT Score: $x = 1100$ , $\mu = 1083$ , $\sigma = 194$ . $z_{\text{SAT}} = \frac{1100 - 1083}{194} = \frac{17}{194} \approx 0.0876$
- ACT Score: $x = 23$ , $\mu = 21$ , $\sigma = 1.62$ . $z_{\text{ACT}} = \frac{23 - 21}{1.62} = \frac{2}{1.62} \approx 1.2346$
- Conclusion: Corbin performed significantly better on the ACT ( $z = 1.235$ ) than on the SAT ( $z = 0.088$ ) relative to test-takers nationwide.

Practice Problems & Conceptual Assessments

True / False Conceptual Analysis:
- a) A histogram can display either categorical or quantitative data, depending on how intervals are chosen. $\rightarrow$ False. Histograms are strictly reserved for quantitative data.
- b) Changing the width of class intervals in a histogram can change the apparent shape of the distribution. $\rightarrow$ True. Binned representations alter peak visibility and perceived skewness.
- c) In a stemplot, the stems must always represent the tens place. $\rightarrow$ False. Stems can represent any place value defined by the plot key.
- d) In a bar graph, the order of categories can be rearranged without changing the data displayed. $\rightarrow$ True. Category placement on a nominal scale is arbitrary.
- e) A stemplot is not appropriate for data with a large range and many observations. $\rightarrow$ True. Stemplots become overcrowded and unreadable with massive datasets.
- f) A dotplot makes it easier than a histogram to identify individual data values. $\rightarrow$ True. Dotplots plot every exact individual point.
- g) A dotplot is inappropriate for quantitative data that includes decimal values. $\rightarrow$ False. Axis scales on dotplots accommodate decimals.
- h) A pie chart is useful for comparing the distribution of a quantitative variable across different intervals. $\rightarrow$ False. Pie charts represent categorical proportions of a whole.
- i) When constructing a histogram, increasing the number of class intervals will always make the distribution easier to interpret. $\rightarrow$ False. Excessive intervals create empty bins and obscure overall shape.
- j) A variable measured in whole numbers is always discrete. $\rightarrow$ False. Continuous variables can be reported rounded to the nearest whole integer.
- k) Two different graphical displays of the same dataset may emphasize different features of the distribution. $\rightarrow$ True.
Multiple Choice Questions & Detailed Walkthroughs:
- Question 1: Netflix Data Classification
- Variables: I. Movie Genre (Categorical), II. Streaming Time (Quantitative), III. Viewer Age (Quantitative), IV. Movie Title (Categorical), V. View Count (Quantitative).
- Correct Answer: (C) I and IV are categorical and the rest are quantitative.
- Question 2: Daycare Ages Histogram Analysis
- Data: Distribution is right-skewed. Mode is at age interval $1\text{--}2$ , while tail extends right to age $7$ .
- In right-skewed distributions, $\text{Mean} > \text{Median}$ , and $\text{Mode} < \text{Mean}$ .
- Correct Answer: (C) I and III only.
- Question 3: Classroom Mean Age Adjustment
- Initial state: $10$ students, $\text{Mean} = 16\,\text{years}$ . Sum of ages = $10 \times 16 = 160\,\text{years}$ .
- One $7\text{-year-old}$ leaves: New sum = $160 - 7 = 153\,\text{years}$ .
- Remaining $9$ students mean = $\frac{153}{9} = 17\,\text{years}$ .
- Correct Answer: (A) $17\,\text{years old}$ .
- Question 4: Standard Deviation Evaluation
- Option A ( $2, 2, 2, 2, 2$ ): $s = 0$ .
- Option B ( $1, 1, 1, 10, 10$ ): Extreme distance of values from $\bar{x} = 4.6$ , maximizing $s$ .
- Option C ( $5, 6, 7, 8, 9$ ): Small uniform spread.
- Option D ( $9, 9, 9, 10, 10$ ): Tightly clustered.
- Correct Answer: (B) $1, 1, 1, 10, 10$ .
- Question 5: Movie Length Back-to-Back Stemplot

Back-to-Back Stemplot of Action vs Comedy Movie Lengths

- Action data ( $n=10$ ):  $109, 113, 115, 132, 134, 143, 150, 155, 157, 163$ .

$Q_1 = 115$ , $Q_3 = 155$ , $IQR = 40$ , $\text{Range} = 163 - 109 = 54$ . - Comedy data ( $n=10$ ): $84, 87, 89, 95, 97, 100, 108, 115, 124, 134$ . $Q_1 = 89$ , $Q_3 = 115$ , $IQR = 26$ , $\text{Range} = 134 - 84 = 50$ . - Evaluated Statements: Comedy $Q_3 (115) = \text{Action } Q_1 (115)$ ; Action $IQR (40) > \text{Comedy } IQR (26)$ ; Action Range ( $54$ ) > Comedy Range ( $50$ ). - Correct Answer: (D) All the statements above are true.

Question 6: Retirement Investment Bar Graph to Pie Chart Conversion
- Counts ( $N = 200$ ): 401(k) = $60$ ( $30\%$ ), Mutual Funds = $50$ ( $25\%$ ), Individual Stocks = $40$ ( $20\%$ ), Real Estate = $30$ ( $15\%$ ), Crypto = $20$ ( $10\%$ ).
- Correct Answer: (C) Pie chart matching percentages $30\%$ , $25\%$ , $20\%$ , $15\%$ , $10\%$ .
Question 7: Golf Scores Summary Statistics Assessment
- Stats ( $N=20$ ): $\bar{x}=75.1$ , $med=75$ , $s=6.37$ , $Q_1=70$ , $Q_3=78.5$ , $min=65$ , $max=89$ .
- Outlier check: $IQR = 78.5 - 70 = 8.5$ . $1.5 \times IQR = 12.75$ . Lower Fence = $70 - 12.75 = 57.25$ ( $min = 65 > 57.25$ ). Upper Fence = $78.5 + 12.75 = 91.25$ ( $max = 89 < 91.25$ ). No outliers exist.
- Correct Answer: (D) All of the above statements are true.
Question 8: Home Prices Histogram Analysis

Home Prices Histogram

- Total sample =  $33\,\text{homes}$ . Bins in thousands of dollars.
- Bin  $0\text{--}500$  contains  $18\,\text{homes}$ . Since the median is the  $17\text{th}$  value, the median **must** fall inside the first bin ( $\le \$500,000$ ).
- *Correct Answer*: **(C)** The median price is not greater than  $\$500,000$ .

Question 9: Standard Deviation Definition in Context
- Correct Answer: (A) The approximate mean distance between the listed home price and the mean home price.
Question 10: Restaurant Bill Stemplot to Histogram Conversion

Restaurant Bill Stemplot

- Bin interval counts ( $20\,\text{dollar}$  bin widths):
  -  $[20, 40)$ :  $28, 32, 35, 38 \rightarrow \text{Count} = 4$ .
  -  $[40, 60)$ :  $41, 43, 45, 47, 49, 52, 54, 56, 58 \rightarrow \text{Count} = 9$ .
  -  $[60, 80)$ :  $60, 62, 65, 67, 69, 72, 74, 76, 79 \rightarrow \text{Count} = 9$ .
  -  $[80, 100)$ :  $82, 85, 88, 92, 95 \rightarrow \text{Count} = 5$ .
  -  $[100, 120)$ :  $105, 110, 118 \rightarrow \text{Count} = 3$ .
- *Correct Answer*: **(A)** Histogram with frequencies  $4, 9, 9, 5, 3$

Free Response Exercises & Complete Solutions:
- Free Response Question 11: Fuel Efficiency Analysis
- Data:
  - Small Cars ( $n=14$ ): $33, 33, 33, 34, 34, 34, 34, 35, 36, 37, 37, 37, 45, 50$
  - Sporty Cars ( $n=27$ ): $29, 29, 29, 29, 29, 29, 29, 30, 30, 30, 30, 30, 30, 30, 31, 31, 31, 31, 31, 31, 32, 32, 32, 32, 33, 33, 33$
- Part a: Visual Graph Recommendation: Parallel boxplots or back-to-back stemplots are ideal for comparing these two distributions.
- Part b: Distribution Comparison:
  - Shape: Small cars distribution is strongly right-skewed; Sporty cars distribution is roughly symmetric.
  - Center: Small cars have a higher median fuel efficiency ( $34.5\,\text{mpg}$ ) than sporty cars ( $30\,\text{mpg}$ ).
  - Spread: Small cars display larger variability ( $IQR = 37 - 34 = 3\,\text{mpg}$ , $\text{Range} = 17\,\text{mpg}$ ) than sporty cars ( $IQR = 31.5 - 29 = 2.5\,\text{mpg}$ , $\text{Range} = 4\,\text{mpg}$ ).
  - Unusual Features: Small cars have two high potential outliers at $45\,\text{mpg}$ and $50\,\text{mpg}$ ; Sporty cars have no outliers.
- Part c: 1.5 x IQR Outlier Test for Small Cars:
  - Five-Number Summary: $min = 33$ , $Q_1 = 34$ , $med = 34.5$ , $Q_3 = 37$ , $max = 50$ .
  - $IQR = 37 - 34 = 3\,\text{mpg}$ .
  - $1.5 \times IQR = 1.5 \times 3 = 4.5\,\text{mpg}$ .
  - $\text{Upper Boundary Fence} = Q_3 + 4.5 = 37 + 4.5 = 41.5\,\text{mpg}$ .
  - $\text{Lower Boundary Fence} = Q_1 - 4.5 = 34 - 4.5 = 29.5\,\text{mpg}$ .
  - Conclusion: Values $45\,\text{mpg}$ and $50\,\text{mpg}$ exceed the upper fence of $41.5\,\text{mpg}$ . Thus, $45\,\text{mpg}$ and $50\,\text{mpg}$ are confirmed mathematical outliers.
- Free Response Question 12: Algebra 1 Final Exam Standardizing Comparison
- Data Table:
  - Morning Class: $\mu = 82.6$ , $\sigma = 4.65$
  - Afternoon Class: $\mu = 75.8$ , $\sigma = 7.48$
- Part a: Nicole ( $90\%$ Morning) vs. Steven ( $87\%$ Afternoon):
  - Nicole Z-Score: $z_{\text{Nicole}} = \frac{90 - 82.6}{4.65} = \frac{7.4}{4.65} \approx 1.591$
  - Steven Z-Score: $z_{\text{Steven}} = \frac{87 - 75.8}{7.48} = \frac{11.2}{7.48} \approx 1.497$
  - Explanation: Nicole's score was more impressive because her standardized z-score ( $1.591$ ) is higher than Steven's z-score ( $1.497$ ), meaning she scored farther above her class average in terms of standard deviations.
- Part b: Grade Conversion for Morning Score of 75:
  - Calculate Z-score of Morning Student: $z = \frac{75 - 82.6}{4.65} = \frac{-7.6}{4.65} \approx -1.6344$
  - Set equivalent Afternoon score equation: $-1.6344 = \frac{x - 75.8}{7.48}$
  - Solve for $x$ : $x = 75.8 + (-1.6344 \times 7.48) = 75.8 - 12.225 = 63.575\%$
  - Conclusion: A student would need a score of approximately $63.58\%$ in the afternoon class to have the same comparative grade.