Statistics Exam Notes Review

Regression Line and Residuals

• Regression Line

  • The equation of a regression line is used to predict values.
  • Example question: Find the regression line for given data points.
  • Given data points:
    • (-5, -10), (-3, -8), (4, 9), (1, 1), (-1, -2), (-2, -6), (0, -1), (2, 3), (3, 3), (6, -4), (-4, -8)
  • Possible equations:
    • A) y^{\wedge} = 2.097x + 0.552
    • B) y^{\wedge} = -0.552x + 2.097
    • C) y^{\wedge} = 0.522x - 2.097
    • D) y^{\wedge} = 2.097x - 0.552

• Residual Calculation

  • The residual is calculated as:
  • \text{Residual} = \text{Observed value} - \text{Predicted value}
  • Example: Given y^{\wedge} = -1.885x + 0.758, find the residual for x=2, y=-4.
  • Choices:
    • A) -7.012
    • B) -6.298
    • C) -3.012
    • D) -0.988

Probability Calculations

• Probability from Inventory Data

  • Problem: Determine probability of selecting a shaft from warehouses other than Warehouse 1.
  • Provided percent distribution of shaft types across warehouses:
  • Regular: 19% (W1), 14% (W2), 9% (W3)
  • Stiff: 8% (W1), 12% (W2), 18% (W3)
  • Extra Stiff: 4% (W1), 16% (W2), 0% (W3)
  • Total for Warehouse 1 is 31%.
  • Probability: 1 - 0.31 = 0.69.
  • Choices:
    • A) 0.42
    • B) 0.80
    • C) 0.31
    • D) 0.69

• Conditional Probability

  • Given a manager is rated as "fair", what is the probability of having a college degree?
  • Table of manager ratings:
  • Good: 39 total
  • Fair: 87 total
  • Poor: 34 total
  • P(\text{College | Fair}) = \frac{\text{College & Fair}}{\text{Total Fair}} = \frac{44}{87}
  • Choices:
    • A) \frac{11}{40}
    • B) \frac{44}{87}
    • C) \frac{9}{20}
    • D) \frac{24}{29}

Combinatorics and Probability Distributions

• Distinct Arrangements of Coins
  • A man has 12 coins (3 pennies, 4 nickels, 5 quarters). Calculate distinct arrangements.
  • The formula for arrangements of multiset:
  • \frac{12!}{3! \cdot 4! \cdot 5!}
• Committee Selection Probability
  • Select a committee of 6 from 8 parents and 4 teachers.
  • Example: Calculate probability of selecting 3 parents and 3 teachers.
• Discrete Probability Distribution
  • A baseball player swings 100 sets of 4 pitches. Given P(x) for number of hits, calculate mean:
  • E(X) = \sum (xi \cdot P(xi))
• Binomial Distribution
  • Problem: Probability that only 2 out of 10 users order something online, with 50% probability of ordering.

Normal Distribution and Confidence Intervals

• Normal Distribution of Bag Weights
  • Mean bag weight = 24.5 oz, SD = 0.2 oz. Find the 67th percentile.
• Parking Spot Search Time
  • Normal distribution (mean = 4.5 min, SD = 1 min). Find the cutoff time exceeded by 75.8% of students.
• Kitten Birth Probability
  • Approximate binomial distribution to find probability of 55 or more females in 100 births (0.5 female).

Samples and Confidence Intervals

• Golfers' Scores
  • Average score of 49 golfers = 76, mean = 75, SD = 3.5. Calculate probability that the average exceeds 76.
• Birth Weight Estimation Sample Size
  • Desired 99% confidence within ±2 ounces (SD = 7 oz). Calculate necessary sample size.
• Understanding P-values
  • Hypothesis testing for pro-shop tension claims (P-value = 0.07), decision-making using significance level (α = 0.05). The conclusion must consider both P-value and α.