One-Variable Data and Graphs
A study records the highest education level completed (High School, Bachelor's, Master's, Doctorate) for 100 job applicants. What type of variable is education level?
Categorical
A quantitative dataset has a first quartile Q1 = 60 and a third quartile Q3 = 80. According to the 1.5 x IQR rule, what is the lower cutoff threshold below which a value is considered an outlier?
30
IQR= 20
Lower boundary = Q1-1.5(IQR)
Lower boundary = 60-1.5(20) = 30
A school administrator selects a sample by taking an alphabetical student directory and picking every 10th student on the list. What sampling method was used?
Systematic random sampling
If Events A and B are mutually exclusive (disjoint), what is the value of P(A and B)?
0
(Mutually exclusive events cannot occur simultaneously).
A researcher calculates the sample mean score xbar = 82 for 50 randomly chosen students to estimate the true population mean score mu of all students nationwide. What is the statistical term for x-bar, and what is the term for mu?
x-bar is a statistic
mu is a parameter
When describing a single quantitative distribution from a graph, what four key features must always be addressed in AP Statistics to earn full credit?
Center, Shape, Spread, and Outliers/Unusual Features -SOCS or CUSS (all described in context)
A student scores 82% on an exam where the class mean is 70% and the standard deviation is 6%. What is the student's z-score, and what does it mean in context?
z = +2.0
The student's score is 2 standard deviations above the class mean score.
A researcher wants to sample 60 students from a high school. She divides the school into four grade levels (9th, 10th, 11th, 12th) and randomly selects 15 students from each grade. What sampling method was used?
Stratified random sampling.
Given P(A) = 0.40, P(B) = 0.50, and P(A and B) = 0.10, calculate the union probability, P(A or B).
0.80
P(A or B) = P(A) + P(B) - P(A and B)
= 0.40+0.50-0.10
In a two-way table categorizing 100 students by grade level and sports participation, 40 students play a sport (Event S). What is P(S^C), the probability of selecting a student from the complement of Event S?
1 - P(S) = 1 - 0.40 = 0.60
A distribution of home values in a city is strongly skewed to the right. Which numerical measure of center will be larger: the mean or the median?
The mean (extreme high values in the right tail pull the mean up).
Daily temperatures in Celsius are transformed to Fahrenheit using F = 1.8C + 32. If the mean temperature in Celsius is 10 with a standard deviation of 5, what are the mean and standard deviation in Fahrenheit?
Mean = (1.8 x 10 +32) = 50 F
StDev = (1.8 x 5) = 9F
(adding a constant of 32 doesn't affect spread)
What essential element distinguishes an experiment from an observational study, and what specific type of conclusion does it allow researchers to draw?
Random assignment of treatments to experimental units; it allows for a cause-and-effect conclusion.
In a class of 30 students, 18 take Spanish, 12 take French, and 6 take both. If a randomly selected student takes French, what is the probability they also take Spanish?
0.50
P(Spanish given French) = 6/12
An experiment randomly assigns 60 adult volunteers from a single city running club to two hydration strategies and finds a significant difference in recovery time. Why cannot these results be generalized to all adults in the city?
The participants were non-randomly selected volunteers from a specific running club, not a random sample selected from the general city population.
A dotplot of test scores shows two distinct peaks around 65% and 85%. How would you describe the modality of this distribution, and what might this structure suggest about the sample?
Bimodal; it suggests the sample may be composed of two distinct sub-populations (e.g., students who studied vs. students who did not).
A student's height is at the 85th percentile of all high school seniors. What percentage of high school seniors are taller than or equal to this student?
15% (85% of values lie at or below this height).
Explain the difference in purpose between stratification in sampling and blocking in experimental design.
Stratification divides a population into groups before random sampling to ensure representation; blocking divides experimental units into groups before random assignment of treatments to reduce variability caused by known extraneous variables.
Events A and B are independent with P(A) = 0.30 and P(B) = 0.40.
Find P(A and B) and P(A given B).
P(A and B) = 0.30 x 0.40 = 0.12
P(A given B) = P(A) (independent)
= 0.30
To address an investigative question about whether a coin is biased toward heads, a student uses a computer simulation to model 100 fair coin tosses 1,000 times. What primary purpose does this simulation serve?
It establishes the expected outcome distribution due to chance variation alone, providing a baseline to evaluate if real sample data is unusual.
You compare two classes' exam scores using parallel boxplots. Class A's boxplot is shifted further right than Class B's, but Class B's boxplot has a much wider IQR. Compare the center and variability of the two classes.
Class A has a higher median score than Class B, but Class B has greater variability in its middle 50% of scores (a larger IQR).
A dataset of 10 employee salaries has a mean of $50,000 and a median of $45,000. The CEO's salary of $500,000 is added to the dataset. Which measure of center will change more, and why?
The mean; the mean is non-resistant to extreme outliers, whereas the median is resistant.
A study finds that people who drink more coffee have lower rates of heart disease. However, coffee drinkers in the study also reported exercising more regularly. What is the statistical term for exercise frequency in this study, and why?
A confounding variable; exercise is associated with coffee drinking (explanatory variable) and affects heart disease (response variable), making it impossible to isolate the effect of coffee alone.
Can two events with non-zero probabilities be both mutually exclusive and independent? Explain why or why not.
No.
If events are mutually exclusive, P(A and B)=0.
But if they are independent, P(A and B)=P(A) x P(B) > 0.
Knowing one event occurred gives full information that the other did not occur.
Consider the investigative question: "Is the proportion of high school freshmen who participate in school sports greater than the proportion of seniors who do?" Identify the population(s), the variable of interest, and the variable type.
Populations: High school freshmen and high school seniors.
Variable of interest: Sports participation status (participates or does not participate).
Variable type: Categorical.