Lecture Notes: Practice Test Review and Statistical Concepts

Overview

• Review of practice test questions for statistics
• Discussion on various statistical concepts and problem-solving techniques

Question 1: Proportion of Days Selling Less Milk

• Problem: Mean milk sales: 194.35 gallons, SD: 22.74 gallons
• Goal: Calculate proportion of days selling <175.32 gallons
  • Use z distribution for normally distributed data
  • Formula: z = (x - μ) / σ
  • Calculation: z = (175.32 - 194.35) / 22.74 = -0.84
  • Proportion: 0.2005 from z-table (0.20134 from z-calculator)

Question 2: Population Proportion and Standard Error

• Survey: 148 graduates, 122 say parking is limited
• Proportion Estimate: 122/148 = 0.824
• Standard Error: sqrt(p(1-p)/n) = 0.313

Question 3: GM Vehicles Driven by Males

• Given: Probability of male driving = 0.5421, n = 4213
• Standard Error Calculation: sqrt(0.5421(1-0.5421)/4213) = 0.0077
• Mean Cars Driven by Males: 0.5421 * 4213 = 2283.8673
• Standard Error of Mean: 0.0077 * 4213 = 32.3386

Question 4: New Mean for Fuel Economy

• Current Mean: 23.86 mpg, SD: 1.134 mpg
• Increase: Each observation +5.49 mpg
• New Mean: 23.86 + 5.49 = 29.35 mpg
• Standard deviation remains unchanged

Question 5: Probability of Rolling a 3 on a Die

• Scenario: 195 rolls, probability of rolling a 3 >50 times
• Use: Binomial approximation with z-score
• Mean (μ): n * p = 32.5
• Standard Deviation (σ): sqrt(n * p * (1-p)) = 5.204
• Z-Score: (50 - 32.5) / 5.204 = 3.36
• Probability: 1 - 0.9996 = 0.0004

Question 6: Interval for Fines

• Mean Fine: 125.25, SD: 13.59
• Empirical Rule: 68% within ±1 SD
• Interval: 125.25 ± 13.59 = [111.66, 138.84]

Question 7: Interquartile Range

• Calculation: IQR = Q3 - Q1 = 12.85 - 8.96 = 3.89

Question 8: Skewness of Distribution

• Distribution: Left skewed (median > mean)

Question 9: Probability of Correct Answers

• Multiple Choice: 50 questions, p(correct)=0.25
• Goal: Probability 9-11 correct answers (inclusive)
• Continuity Correction: z = (x - np) / sqrt(np(1-p))
• Result: Calculated probability = 0.276

Question 10: Mean Height of Sample

• Sample Size: 28 males
• Mean Height: 64.58 inches, SD: 4.357 inches
• Calculation: z-score for mean height < 65.28
• Result: Probability = 0.564

Question 11: Decreasing Standard Error

• Sample Size: 324
• Goal: Reduce standard error by factor of 5
• Required Sample Size: 324 * 25 = 8100

Question 12: Central Limit Theorem

• Concept: As n increases, distribution of sample means -> normal

Question 13: Probability of Selling Drinks

• Average Sales: 6,783,456 drinks, SD: 6,445.769
• Goal: Probability sales < 6,765,958
• Z-Score: Calculated probability = 0.0034

Question 14: Variable Types

• Employee Category: Categorical
• Income Tax Bracket: Categorical

Question 15: Histogram Shape

• Observation: Left skewed histogram

Question 16: Exam Score Percentile

• Exam Stats: Mean = 87.54, SD = 4.23
• Problem: Find score < 93.07 percentile
• Calculation: Use z-score to solve for x

Question 17: Business Audits

• Scenario: 437 businesses, audit rate = 43.54%
• Goal: Probability < 37.45% audited
• Calculation: Use z-score = -2.57

Question 18: PGA Golfer Scores

• Average Score: 69.25, SD: 2.347
• Sample Size: 54
• Goal: Probability mean between 69.25 and 69.55
• Result: Probability = 0.3264

Conclusion

• Series aimed to prepare students for statistics exams with detailed problem-solving strategies.