STA2023 Practice Midterm 2: Step-by-Step Statistics Guidance

Q1. Sample Space and Classical Probability

Background

Topic: Probability, Sample Space, Counting Outcomes

This question tests your understanding of how to list all possible outcomes (sample space) for a sequence of coin flips, and how to calculate probabilities using classical probability rules.

Key Terms and Formulas:

• Sample Space: The set of all possible outcomes.

• Classical Probability: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

• "At least" means greater than or equal to the specified number.

Step-by-Step Guidance

1. List all possible outcomes for flipping a fair coin 3 times. Each flip can be Heads (H) or Tails (T).

2. Count the total number of outcomes in the sample space.

3. For part (b), identify all outcomes with exactly 2 heads. Count these outcomes.

4. For part (c), identify all outcomes with at least 1 tail. Count these outcomes.

5. Set up the probability calculations for each part using the classical probability formula.

Try solving on your own before revealing the answer!

Q2. Complement Rule and Multiplication Rule

Background

Topic: Probability with and without Replacement, Complement Rule

This question tests your ability to calculate probabilities when selecting items without replacement, and to use the complement rule for "at least one" type probabilities.

Key Terms and Formulas:

• Multiplication Rule (without replacement): $P(A \text{ and } B) = P(A) \times P(B|A)$

• Complement Rule: $P(\text{at least one}) = 1 - P(\text{none})$

Step-by-Step Guidance

1. For part (a), calculate the probability that the first monitor selected is not defective, then the probability that the second monitor selected is not defective (given the first was not defective).

2. Multiply these probabilities to find the probability that both are not defective.

3. For part (b), use the complement rule: $P(\text{at least one defective}) = 1 - P(\text{none defective})$.

4. Set up the calculation for $P(\text{none defective})$ using your answer from part (a).

Try solving on your own before revealing the answer!

Q3. Contingency Table, Conditional Probability, and Independence

Background

Topic: Contingency Tables, Marginal and Conditional Probability, Independence

This question tests your ability to read and interpret a contingency table, calculate marginal and conditional probabilities, and determine independence of events.

Key Terms and Formulas:

• Marginal Probability: $P(A) = \frac{\text{Total for A}}{\text{Grand Total}}$

• Joint Probability: $P(A \text{ and } B) = \frac{\text{Number in both A and B}}{\text{Grand Total}}$

• Conditional Probability: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$

• Independence: Events A and B are independent if $P(A|B) = P(A)$

Step-by-Step Guidance

1. For each probability, identify the relevant counts from the table (e.g., total females, total widowed, widowed females).

2. Calculate the marginal probabilities by dividing the relevant count by the total number of people.

3. For conditional probability, use the formula $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$.

4. For independence, compare $P(A|B)$ and $P(A)$ to see if they are equal.

Try solving on your own before revealing the answer!

Q4. Counting Techniques: Permutations and Combinations

Background

Topic: Counting, Permutations, Combinations

This question tests your understanding of when to use permutations (order matters) versus combinations (order does not matter) in counting problems.

Key Terms and Formulas:

• Permutation: $P(n, r) = \frac{n!}{(n - r)!}$

• Combination: $C(n, r) = \frac{n!}{r!(n - r)!}$

• Order matters for officer positions, but not for committee selection.

Step-by-Step Guidance

1. For part (a), recognize that selecting president, vice president, and secretary is a permutation problem (order matters).

2. Set up the permutation formula for selecting 3 officers from 12 members.

3. For part (b), recognize that selecting a committee is a combination problem (order does not matter).

4. Set up the combination formula for selecting 3 members from 12.

Try solving on your own before revealing the answer!

Q5. Discrete vs. Continuous Random Variables

Background

Topic: Types of Random Variables

This question tests your ability to distinguish between discrete and continuous random variables based on their definitions.

Key Terms:

• Discrete Random Variable: Takes countable values (e.g., 0, 1, 2, ...).

• Continuous Random Variable: Takes any value in an interval (can be measured, not counted).

Step-by-Step Guidance

1. For each variable, ask: Can the values be counted (discrete) or measured on a continuous scale (continuous)?

2. Classify each variable accordingly.

Try solving on your own before revealing the answer!

Q6. Mean of a Discrete Random Variable

Background

Topic: Expected Value (Mean) of a Discrete Random Variable

This question tests your ability to calculate the mean (expected value) of a discrete random variable using its probability distribution.

Key Formula:

• Mean (Expected Value): $\mu = E(X) = \sum x \cdot P(x)$

Step-by-Step Guidance

1. Multiply each value of $x$ by its corresponding probability $P(x)$.

2. Add up all these products to get the mean.

3. Interpret the mean in the context of the game (what does the expected value represent?).

Try solving on your own before revealing the answer!

Q7. Binomial Probability

Background

Topic: Binomial Probability Distribution

This question tests your understanding of the binomial experiment, and how to calculate binomial probabilities for exact, fewer than, and at least outcomes.

Key Terms and Formulas:

• Binomial Probability Formula: $P(X = k) = C(n, k) p^k q^{n-k}$

• $n$ = number of trials, $p$ = probability of success, $q = 1 - p$

• "Fewer than 3" means $X = 0, 1, 2$; "At least 3" means $X \geq 3$

Step-by-Step Guidance

1. For part (a), explain why this is a binomial experiment and identify $n$, $p$, and $q$.

2. For part (b), set up the binomial probability formula for $k = 6$.

3. For part (c), sum the probabilities for $k = 0, 1, 2$.

4. For part (d), use the complement rule: $P(X \geq 3) = 1 - P(X < 3)$.

Try solving on your own before revealing the answer!

Q8. Mean and Standard Deviation of a Binomial Random Variable

Background

Topic: Binomial Distribution Mean and Standard Deviation

This question tests your ability to calculate the mean and standard deviation for a binomial random variable, and interpret the mean in context.

Key Formulas:

• Mean: $\mu = n p$

• Standard Deviation: $\sigma = \sqrt{n p q}$

• $n$ = number of trials, $p$ = probability of success, $q = 1 - p$

Step-by-Step Guidance

1. Identify $n$, $p$, and $q$ from the problem statement.

2. Plug these values into the formulas for mean and standard deviation.

3. Interpret the mean in the context of basketball free throws.

Try solving on your own before revealing the answer!

Q9. Finding Areas Under a Normal Curve

Background

Topic: Normal Distribution, Z-scores, Probability

This question tests your ability to find probabilities for a normal distribution using the mean and standard deviation, and to use Z-scores and normal tables (or calculator functions).

Key Formulas:

• Z-score: $z = \frac{X - \mu}{\sigma}$

• Use the standard normal table or calculator to find probabilities corresponding to Z-scores.

Step-by-Step Guidance

1. For each part, calculate the Z-score for the given value(s) of $X$.

2. Use the Z-score to find the corresponding probability from the standard normal table.

3. For intervals (e.g., between 480 and 560), find the probability for each endpoint and subtract as needed.

Try solving on your own before revealing the answer!

Q10. Finding the Value of a Normal Random Variable

Background

Topic: Normal Distribution, Percentiles, Inverse Normal Calculations

This question tests your ability to find the value of a normal random variable corresponding to a given percentile, and to find values that separate specified areas under the normal curve.

Key Formulas:

• Inverse Z-score: $X = \mu + z \sigma$

• Use the standard normal table or calculator to find the Z-score corresponding to a given percentile.

Step-by-Step Guidance

1. For part (a), find the Z-score that corresponds to the 30th percentile (area = 0.30).

2. Use the formula $X = \mu + z \sigma$ to find the corresponding weight.

3. For part (b), find the Z-scores that correspond to the 10th and 90th percentiles.

4. Use the formula to find the weights that separate the middle 80% from the lowest and highest 10%.

Try solving on your own before revealing the answer!