AP Stat FRQ Practice

Section II: Free-Response

1. Khaled’s Reading Assignment

• Context: Khaled is tracking reading time and pages left among classmates.
• Data Collection: Each student records hours reading and pages remaining.
• Scatterplot Description:
    - Assessment of Relationship: Examine correlation between hours read and pages left.
• Regression Analysis:
    - Regression Equation: $ext{number of pages left} = 592.504 - 21.512 imes ( ext{hours spent reading})$
    - Slope Interpretation:
      - The slope $-21.512$ indicates that for each additional hour spent reading, the number of pages left decreases by approximately 21.5 pages.
• Khaled’s Reading Performance:
    - Khaled read for 4 hours and has 540 pages left.
    - (i) Residual Calculation:
      - Predicted pages left: $592.504 - (21.512 imes 4)$
      - Residual: $540 - ext{predicted pages left}$, calculated value to be shown.
    - (ii) Residual Interpretation:
      - The residual indicates how much Khaled’s reading performance deviated from the prediction made by the model. A negative residual suggests he has fewer pages left than predicted for his reading time, indicating efficient reading.

2. Psychologist Experiment on Lighting Conditions

• Research Objective: Investigate lighting effects on work productivity.
• Methodology:
    - Sample Size: 40 employees randomly selected.
    - Assignments: 20 in bright light, 20 in dim light.
    - Response Measure: Quality rated from 0 (low) to 5 (high).
• Study Components:
    - Treatments: Bright light vs. dim light.
    - Experimental Units: 40 employees in total.
    - Response Variable: Quality of work, rated from 0 to 5.
• Generalizability of Results:
    - Threatened by the small and specific sample size; may not represent all employees in the corporation due to unknown variance in productivity across different roles or departments.
• Proposed Change in Design:
    - Design Type: Matched pairs design.
    - Statistical Advantage: Reduces variability by pairing similar employees, thus isolating the effect of lighting conditions more clearly.

3. Airline Flight Delay Probability Analysis

• Claimed Distribution: Minutes late (mean = 2.4 minutes, SD = 2.1 minutes).
• Delay Definition: More than 5 minutes late indicates a delay.
• (i) Probability Calculation:
    - Find P(X > 5) using the normal distribution with the stated parameters:
      - Standardizing: $Z = \frac{X - ext{mean}}{ ext{SD}}$
      - Specifically: $Z = \frac{5 - 2.4}{2.1}$
• Kelly’s Flights Monitoring:
    - (ii) Random Variable Definition: Let $Y$ denote the number of delayed flights, where $Y ext{ follows } B(20, p)$, with p = P(X > 5).
    - Expectation of Delayed Flights: Expect $20p$ delayed flights based on the value calculated from the probability.
• Kelly’s Experience of Delays:
    - (i) Probability of 7 or More Delayed Flights:
      - Use binomial probability formula or normal approximation:
      - Calculate $P(Y ext{ ≥ } 7)$ using binomial (or normal approx. if applicable).
    - (ii) Evidence Review:
      - Compare Kelly's observations against the expected probability from the airline claims to assess if her experience provides statistical evidence of more frequent delays.

4. Educational Research on Teaching Methods

• Research Design: 100 students paired by previous test scores, assigned to two different teaching methods.
• Method Details:
    - Pairing Basis: Two lowest scores paired, the two highest scores paired, etc.
    - Method Comparison: New vs. traditional.
• Findings Summary:
    - Sample Size: Both methods have n = 50.
    - Mean Scores: New (mean = 81.4), Traditional (mean = 78.1), Difference (3.3).
    - Standard Deviations: New (8.6), Traditional (6.5), Difference (7.7).
• Statistical Evidence Evaluation:
    - Analyze means and variances, apply significance level of $ext{α} = 0.05$ to conclude if the new method outperforms the traditional method statistically validly.

5. Attitudes towards AI Technology by Age Group

• Survey Overview: Two age groups (18 to 44 years, 45 years and older) surveyed on AI attitudes.
• Mosaic Plot Construction: Visual representation of responses across age groups (positive, neutral, negative).
• Association Description:
    - Examine if attitudes significantly differ between the age groups based on summarized survey data.
• Chi-Square Test for Homogeneity:
    - Hypotheses: Null: no difference; Alternative: difference exists.
    - Test statistic calculation: Show formula and calculate for the respective counts.
    - Interpretation of Test Statistic: Contextualize what the test statistic indicates regarding age differences in AI attitudes.
    - Observed Frequencies:
      - Positive responses: 24 (18-44) and 6 (45+).
      - Neutral: 18 and 12.
      - Negative: 6 and 14.

6. Mateo's AP Course Enrollment Study

• Student Sampling: Mateo sampled 200 students, finding 82 enrolled in AP courses.
• Confidence Interval Construction: Seek to estimate population proportion of students enrolled.
• Inference Conditions Check: Confirm random sampling and success/failure condition status for validity of inferences.
• Normal Approximation Justification:
    - Based on sample size and success/failure counts for standard normal approximation of the sampling distribution.
• Type I Error Definition: Incorrectly rejecting the null hypothesis that the true proportion of AP enrollment equals 35%.
• Confidence Level Impact: Higher confidence level (95% vs. 90%) results in wider intervals and lower Type I error rate, thus balanced decision making.
• Confidence Interval Outcomes:
    - 90% confidence interval: (0.353, 0.467) - decision based on the hypothesis.
    - 95% confidence interval: (0.342, 0.478) - different conclusion may emerge based on levels.
• P-value Range Estimate:
    - Discuss based on confidence intervals without numeric calculations, influencing statistical significance assessments.