Probability and Expected Values Quiz

Question 1: Probability and Expected Values

• Exam consists of 30 multiple-choice questions with 3 possible answers each.
• If a student guesses, the expected number of correct answers, E(X) or μ, is calculated as (μ = \frac{1}{3} \times 30 = 10).
• Variance, Var(X), can be computed using the formula for a binomial distribution: (Var(X) = n \cdot p \cdot (1 - p) = 30 \cdot \frac{1}{3} \cdot \frac{2}{3} = 20).
• Standard deviation is the square root of variance: (SD(X) = \sqrt{Var(X)} = \sqrt{20} \approx 4.47).

Question 2: Probability in Games of Chance

• Winning probability in a game is 0.35.
• Aim is to find the minimum number of games to play for winning at least twice with a probability greater than 0.9.

Question 3: Sample Proportion and Margin of Error

• A survey indicates 70% of primary school children use social media.
• Sample proportion (p̂ = 0.70).
• Margin of error (E) at 95% confidence level is expressed as (E = z \cdot \sqrt{\frac{p̂(1 - p̂)}{n}}), where (z) corresponds to the z-value for 95% confidence.
• Tripling the sample size (n) reduces the margin of error (E) by about 29%.

Question 4: Normal Distribution of Heights

• Heights in a class are normally distributed with mean 175 cm and standard deviation 6 cm.
• To find the expected number of students shorter than 168 cm, utilize the z-score formula and the normal distribution table.

Question 5: Probability with Public Transport

• 60% of students use public transport; random selection of 4 students.
• Probability that at least one student caught public transport can be found using (P(X ≥ 1) = 1 - P(X = 0)).

Question 6: Probability of Kicking Goals in AFL

• Probability of a goal for a full forward is 0.15 with 10 kicks.
• Compute probabilities for:
  • Kicking a goal every time: (P(X = 10) = (0.15)^{10}).
  • Kicking at least one goal: Use complementary probability.
  • Kicking more than one goal given at least one goal: Conditional probability (P(X > 1 | X ≥ 1)).

Question 7: Continuous Random Variable

• (f(x) = kx(4 - x)) for (0 ≤ x ≤ 4), find value of constant (k).
• Calculate expected value (E(X)) using the probability function.

Question 8: Driver's License Probability

• 60% of surveyed individuals have a driver's license.
• Approximate probability for more than 65% of a sample of 200 people using normal approximation to the binomial distribution.

Question 9: Normal Distribution of Coke Cans

• 15% of cans < 372 mL, and 10% > 379 mL.
• Find mean and standard deviation of coke amount in a can.
• To determine the proportion of cans with less than 377 mL, use the derived normal distribution parameters.

Question 10: Confidence Intervals for Postgraduate Intent

• Survey of 500 individuals shows 237 want to pursue postgraduate studies.
• Calculate confidence intervals for proportions at 90%, 95%, and 99% levels, comparing the intervals statistically.