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Unit 1: Exploring One-Variable Data
Essential Content Notes
1. Types of Data: - Categorical: Places individuals into groups or categories (e.g., zip code, eye color). - Quantitative: Takes numerical values for which it makes sense to find an average (e.g., height). 2. Describing Distributions (SOCS): - Shape: Symmetric, Skewed Right (mean > median), Skewed Left (mean < median), Bimodal. - Outliers: Values falling outside the overall pattern. Rule: x < Q 1 − 1_._ 5( IQR ) or x > Q 3 + 1_._ 5( IQR ). - Center: Mean (use for roughly symmetric), Median (resistant to outliers, use for skewed). - Spread (Variability): Standard Deviation (use with mean), IQR = Q 3 − Q 1 (use with median), Range. 3. Graphs for Quantitative Data: - Histograms, Dotplots, Stem-and-leaf plots, Boxplots (shows 5-number summary: Min, Q1, Median, Q3, Max). 4. Normal Distributions: - Symmetric, single-peaked, bell-shaped density curves described by mean μ and stan- dard deviation σ. - Empirical Rule (68-95-99.7): Approx. 68% of observations fall within 1 σ , 95% within 2 σ , and 99.7% within 3 σ of the mean. - Z-Scores: Standardized score indicating how many standard deviations a value is from the mean. z = x − σ μ.
Test-Taking Tip: Always label your axes and provide a key for stem-and-leaf plots! When comparing two distributions, use comparative language (e.g., "greater than", "less variable than") rather than listing their SOCS side-by-side.
Unit 1: Multiple Choice Questions
- The heights of American adult males are approximately normally distributed with a mean of 70 inches and a standard deviation of 3 inches. Which of the following represents the height of a male who is at the 16th percentile?
A. 64 inches B. 67 inches C. 70 inches D. 73 inches E. 76 inches
- A teacher calculates the grades on a recent exam and finds that the median is 85 and the mean is 78. Which of the following is the most likely shape of the distribution of exam scores?
A. Skewed left B. Skewed right C. Symmetric D. Bimodal E. Uniform
- Which of the following summary statistics are strictly resistant to outliers?
A. Mean and Standard Deviation B. Median and Interquartile Range (IQR) C. Mean and Interquartile Range (IQR) D. Median and Range E. Range and Standard Deviation
- The 5-number summary for a dataset of housing prices in a city is: $120,000, $180,000, $250,000, $350,000, $800,000. According to the 1.5 × IQR rule, which of the following is true?
A. There are no outliers in this dataset. B. $800,000 is definitely an outlier, and there may be others. C. $120,000 and $800,000 are both outliers. D. The dataset is perfectly symmetric. E. The mean is exactly $250,000.
- A student’s score on a national chemistry exam has a z-score of -1.2. What does this value mean in context?
Unit 1: Free Response Questions
- A consumer advocacy group is testing the battery life of two different brands of smartphones, Brand A and Brand B. They randomly select 20 phones of each brand and record the number of hours the battery lasts under heavy use. Brand A 5-Number Summary: Min = 4.5, Q1 = 6.2, Median = 7.5, Q3 = 8.8, Max =
Brand B 5-Number Summary: Min = 3.2, Q1 = 5.0, Median = 6.8, Q3 = 7.5, Max =
(a) Determine if there are any outliers for Brand A. Justify your answer statistically.
(b) Compare the distributions of battery life for Brand A and Brand B based on the provided statistics.
- The weights of a certain breed of adult golden retrievers are approximately normally dis- tributed with a mean of 65 pounds and a standard deviation of 4.5 pounds.
(a) What proportion of these adult golden retrievers weigh between 60 and 70 pounds?
(b) A veterinarian categorizes a golden retriever as "underweight" if its weight is in the lowest 10% of the distribution. What is the maximum weight a dog can have and still be categorized as underweight?
Unit 2: Multiple Choice Questions
- Which of the following statements about the correlation coefficient r is TRUE?
A. A correlation of r = 0_._ 8 means that 80% of the data points fall perfectly on the regression line. B. The correlation coefficient will change if units are changed from inches to centimeters. C. A correlation of r = 0 indicates there is no relationship of any kind between the variables. D. The correlation coefficient is highly resistant to extreme outliers. E. The correlation coefficient measures the strength and direction of the linear relationship between two quantitative variables.
- A researcher is studying the relationship between hours studied per week ( x ) and GPA ( y ). The LSRL is y ˆ = 1_._ 2 + 0_._ 08 x. What is the correct interpretation of the slope?
A. For every additional hour studied per week, a student’s GPA will increase by 1.2. B. For every additional hour studied per week, the predicted GPA increases by 0.08. C. If a student studies 0 hours, their predicted GPA is 0.08. D. The correlation between hours studied and GPA is 0.08. E. 8% of the variation in GPA is explained by the number of hours studied.
- A scatterplot of a dataset reveals a distinct curved pattern. The correlation coefficient is calculated to be r = 0_._ 02. Which of the following is the most appropriate conclusion?
A. There is a very weak linear relationship, but there may be a strong non-linear relation- ship. B. There is absolutely no relationship between the variables. C. A linear model is appropriate because r is close to 0. D. The standard deviation of the residuals will be very small. E. A calculation error must have occurred, as curved patterns always have negative corre- lation.
- A student has a predicted test score of 82 based on the number of hours they slept. Their actual score on the test was a 76. What is the value of their residual?
A. 6 B. - C. 158 D. 0. E. Cannot be determined without the standard deviation.
- Which of the following is the primary purpose of a residual plot?
A. To determine if the correlation is positive or negative. B. To calculate the exact value of the y-intercept. C. To assess whether a linear model is appropriate for the data. D. To find the mean of the explanatory variable. E. To identify the units of measurement for the variables.
- The coefficient of determination between a car’s weight and its fuel efficiency is r^2 = 0_._ 81. The correlation between these two variables is negative. What is the value of the correlation coefficient, r?
A. 0. B. 0. C. -0. D. -0. E. 0.
- A marketing analyst is investigating the relationship between advertising spend (in thousands of dollars) and sales revenue (in tens of thousands of dollars) for a retail chain. The scatterplot shows a positive, somewhat linear relationship. A residual plot is created and shows a clear U-shaped pattern.
(a) Based on the residual plot, is a linear model appropriate for describing the relationship between advertising spending and sales revenue? Explain your reasoning.
(b) If the analyst uses the linear model to predict sales revenue for a store that spent significantly more on advertising than any store in the original dataset, what statistical concern arises? Explain why this is a problem.
Unit 3: Collecting Data
Essential Content Notes
1. Sampling Methods: - Simple Random Sample (SRS): Every group of size n has an equal chance of being selected. - Stratified Random Sample: Divide population into homogeneous groups (strata), then take an SRS from each. Reduces variability. - Cluster Sample: Divide population into heterogeneous groups (clusters), randomly select whole clusters, and survey everyone in them. 2. Types of Bias: - Undercoverage: Some members of the population cannot be chosen. - Nonresponse: Individuals chosen for the sample cannot be contacted or refuse to participate. - Response Bias: Systematic pattern of inaccurate answers (e.g., due to wording, interviewer effect). 3. Observational Studies vs. Experiments: - Observational Study: Observes individuals and measures variables but does not attempt to influence the responses. Cannot establish causation due to confounding. - Experiment: Deliberately imposes some treatment on individuals to measure their responses. - Principles of Experimental Design: Comparison, Random Assignment (creates roughly equivalent groups), Control (keeps other variables constant), Replication (using enough subjects).
Test-Taking Tip: Whenever asked to describe how to randomly assign or select, be specific enough that someone else could follow your exact instructions (e.g., "Put names on identically sized slips of paper, mix thoroughly in a hat...").
- A researcher is comparing two different diets for dogs. Because different dog breeds might react differently to the diets, the researcher separates the dogs by breed first, and then randomly assigns dogs within each breed to the two diets. This is an example of a:
A. Completely randomized design B. Randomized block design C. Matched pairs design D. Stratified random sample E. Cluster sample
- Why is a double-blind procedure often used in medical experiments?
A. It prevents both subjects and researchers who interact with them from knowing which treatment is being given, reducing bias. B. It ensures that the placebo effect does not occur. C. It doubles the sample size of the experiment. D. It guarantees that the results will be statistically significant. E. It controls for all lurking variables.
Unit 3: Free Response Questions
- A local gym is considering adding a new type of high-intensity interval training (HIIT) class. The gym manager wants to conduct an experiment to see if the HIIT class improves cardio- vascular fitness more than the current standard aerobics class. They have 60 volunteers who are members of the gym.
(a) Describe a completely randomized design for this experiment. Include the method you would use for random assignment.
(b) The manager realizes that age might be a confounding variable, as younger members might recover faster. Explain how you would modify your design to incorporate blocking to account for age.
Unit 4: Probability, Random Variables, & Distributions
Essential Content Notes
1. Probability Rules: - General Addition Rule: P ( A or B ) = P ( A ) + P ( B ) − P ( A and B ). - Mutually Exclusive (Disjoint): P ( A and B ) = 0. They cannot happen at the same time. - Conditional Probability: P ( A | B ) = P^ ( A P^ and( B )^ B ). - Independence: Knowing one event occurred does not change the probability of the other. P ( A | B ) = P ( A ). If independent, P ( A and B ) = P ( A ) × P ( B ). 2. Discrete Random Variables: - Expected Value (Mean): E ( X ) = μx =
∑ xi · P ( xi ).
- Transformations: If Y = a + bX , then μY = a + bμX and σY = | b | σX. (Adding a constant does not change variance/standard deviation).
- Combining: μX + Y = μX + μY. If independent, σ X^2 + Y = σ^2 X + σ^2 Y (always add variances, never standard deviations!). 3. Binomial vs. Geometric Distributions:
- Binomial (BINS): Binary outcomes, Independent trials, fixed Number of trials ( n ), Same probability of success ( p ). Mean: μ = np , Std Dev: σ =
√ np (1 − p ).
- Geometric: Looking for the number of trials until the first success. Mean: μ = 1 /p.
Test-Taking Tip: When adding or subtracting random variables, remember the "Pythagorean Theorem of Statistics"—you must convert standard deviations to variances (square them), add the variances, and then take the square root!
Unit 4: Multiple Choice Questions
- Events A and B are mutually exclusive. If P ( A ) = 0_._ 3 and P ( B ) = 0_._ 4 , what is P ( A or B )?
A. 0. B. 0. C. 0. D. 0. E. 1.
- A carnival game costs $2 to play. You have a 10% chance of winning $10, a 20% chance of winning $5, and a 70% chance of winning nothing. What is your expected net gain/loss for playing this game?
A. -$2. B. -$0. C. $0. D. $1. E. $2.
- In a factory, 8% of the widgets produced are defective. A quality control inspector selects widgets at random and tests them. What is the probability that the first defective widget she finds is the 5th one tested?
A. (0_._ 08)^5 B. (0_._ 92)^5 C. (0_._ 92)^4 (0_._ 08) D.
( 5 1
) (0_._ 08)^1 (0_._ 92)^4 E.
( 5 4
) (0_._ 08)^4 (0_._ 92)^1
- Let X be a random variable with a mean of 15 and a standard deviation of 4. Let Y = 3 X − 5. What are the mean and standard deviation of Y?
A. Mean = 40, Std Dev = 7 B. Mean = 40, Std Dev = 12 C. Mean = 45, Std Dev = 12 D. Mean = 40, Std Dev = - E. Mean = 15, Std Dev = 12
- An airplane has 100 seats. The probability that a ticketed passenger actually shows up for the flight is 0.90, and passengers arrive independently of one another. The airline sells 105 tickets. What distribution can be used to model the number of passengers who show up?
Unit 4: Free Response Questions
- A high school offers two advanced math classes: AP Calculus and AP Statistics. Let C be the event that a randomly selected senior takes AP Calculus, and S be the event that a randomly selected senior takes AP Statistics. The following probabilities are known: - P ( C ) = 0_._ 40 - P ( S ) = 0_._ 25 - P ( C and S ) = 0_._ 15
(a) Are the events "taking AP Calculus" and "taking AP Statistics" independent? Justify your answer mathematically.
(b) Find the probability that a randomly selected senior takes AP Calculus GIVEN that they are taking AP Statistics.
- A professional basketball player makes 82% of his free-throw attempts. Assume that each free-throw attempt is independent of the others. During a particular game, he shoots 10 free throws.
(a) What is the probability that he makes exactly 8 of the 10 free throws?
(b) What is the mean and standard deviation of the number of free throws he makes out of 10? Interpret the mean in context.
