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

Q1. Sample Space and Classical Probability

Background

Topic: Probability – Sample Space, Classical Probability, Counting Outcomes

This question tests your understanding of how to list all possible outcomes (sample space) for repeated experiments (coin flips), and how to calculate probabilities for specific events using classical probability.

Key Terms and Formulas:

• Sample Space (S): The set of all possible outcomes.

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

• "Exactly 2 heads" means you want outcomes with 2 heads and 1 tail.

• "At least 1 tail" means any outcome with one or more tails.

Step-by-Step Guidance

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

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

3. For part (b), identify all outcomes with exactly 2 heads. Count how many there are.

4. Use the classical probability formula to set up the probability for exactly 2 heads.

5. For part (c), determine how many outcomes have at least 1 tail (hint: consider the complement of "no tails").

6. Set up the probability for at least 1 tail using the classical probability formula.

Try solving on your own before revealing the answer!

Q2. Complement Rule and Multiplication Rule

Background

Topic: Probability – Complement Rule, Multiplication Rule, Sampling Without Replacement

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

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), determine how many monitors are not defective and how many total monitors there are.

2. Calculate the probability that the first monitor selected is not defective.

3. Given the first was not defective, calculate the probability that the second is also not defective (remember, no replacement).

4. Multiply these probabilities to set up the probability that both are not defective.

5. For part (b), use the complement rule: "at least one defective" is the complement of "none defective".

6. Set up the probability using $P(\text{at least one defective}) = 1 - P(\text{none defective})$.

Try solving on your own before revealing the answer!

Q3. Contingency Table, Conditional Probability, and Independence

Background

Topic: Probability – Contingency Tables, Conditional Probability, Independence

This question tests your ability to read and interpret contingency tables, calculate marginal and joint probabilities, conditional probabilities, and test for independence of events.

Key Terms and Formulas:

• Marginal Probability: Probability of a single event (e.g., being female).

• Joint Probability: Probability of two events both occurring (e.g., widowed and female).

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

• Independence: Events A and B are independent if $P(A|B) = P(A)$ or $P(A \text{ and } B) = P(A) \times P(B)$

Step-by-Step Guidance

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

2. Set up the probability as a fraction: $P(E) = \frac{\text{number of favorable cases}}{\text{total cases}}$

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

4. For independence, compare $P(A|B)$ and $P(A)$, or check if $P(A \text{ and } B) = P(A) \times P(B)$.

5. Show your calculations for each probability, but stop before the final numeric result.

Try solving on your own before revealing the answer!

Q4. Counting Techniques: Permutations and Combinations

Background

Topic: Counting – Permutations and 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: $C(n, r) = \frac{n!}{r!(n - r)!}$

• ! (factorial): $n! = n \times (n-1) \times \ldots \times 1$

Step-by-Step Guidance

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

2. Set up the permutation formula with $n = 12$ and $r = 3$.

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

4. Set up the combination formula with $n = 12$ and $r = 3$.

5. Write out the factorial expressions for each formula, but do not compute the final value yet.

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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 and examples.

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. Consider whether the variable can take on fractional values (continuous) or only whole numbers (discrete).

3. Classify each variable accordingly, but do not write the final classification yet.

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, and to interpret the result in context.

Key Formula:

• $\mu = E(X) = \sum x \cdot P(x)$

Step-by-Step Guidance

1. List each value of $x$ and its corresponding probability $P(x)$.

2. Multiply each $x$ by its probability $P(x)$ to get $x \cdot P(x)$ for each outcome.

3. Add up all the $x \cdot P(x)$ values to set up the sum for the mean.

4. For part (b), think about what the mean represents in the context of the game (e.g., average profit per game in the long run).

5. Write a sentence interpreting the mean, but do not state the final value yet.

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, identifying parameters, and calculating binomial probabilities for exact, fewer than, and at least outcomes.

Key Terms and Formulas:

• Binomial Experiment: Fixed number of independent trials, two outcomes (success/failure), constant probability.

• Parameters: $n$ (number of trials), $p$ (probability of success), $q = 1 - p$ (probability of failure)

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

• Cumulative Probability: For "fewer than" or "at least" use sums of binomial probabilities.

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 formula for $P(X = 6)$ using the binomial probability formula.

3. For part (c), set up the sum for $P(X < 3)$, i.e., $P(X = 0) + P(X = 1) + P(X = 2)$.

4. For part (d), set up the sum for $P(X \geq 3)$, or use the complement rule: $1 - P(X < 3)$.

5. Write out the expressions, but do not compute the final probabilities yet.

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 to interpret the mean in context.

Key Formulas:

• Mean: $\mu = n \cdot p$

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

• Where: $n$ = 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. Set up the formula for the mean using the identified values.

3. Set up the formula for the standard deviation using the identified values.

4. For part (c), interpret what the mean represents in the context of basketball free throws.

5. Do not compute the final values yet.

Try solving on your own before revealing the answer!

Q9. Finding Areas Under a Normal Curve

Background

Topic: Normal Distribution – Finding Probabilities (Areas)

This question tests your ability to standardize values (find z-scores) and use the standard normal table to find probabilities for normal distributions.

Key Terms and Formulas:

• Normal Distribution: $X \sim N(\mu, \sigma)$

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

• Standard Normal Table: Used to find probabilities for z-scores.

Step-by-Step Guidance

1. For each part, identify the value(s) of $X$ you are interested in (e.g., $X = 620$, $X = 450$, $X = 480$ and $X = 560$).

2. Calculate the z-score for each value using $z = \frac{X - \mu}{\sigma}$, with $\mu = 500$ and $\sigma = 100$.

3. For part (a), set up the probability $P(X < 620)$ as $P(z < z_{620})$.

4. For part (b), set up the probability $P(X > 450)$ as $1 - P(z < z_{450})$ or directly as $P(z > z_{450})$.

5. For part (c), set up the probability $P(480 < X < 560)$ as $P(z_{480} < z < z_{560})$.

6. Do not look up or calculate the final probabilities yet.

Try solving on your own before revealing the answer!

Q10. Finding the Value of a Normal Random Variable

Background

Topic: Normal Distribution – Finding Percentiles and Cutoff Values

This question tests your ability to use the standard normal table in reverse: given a percentile, find the corresponding value of the random variable.

Key Terms and Formulas:

• Percentile: The value below which a given percentage of observations fall.

• Z-score for a percentile: Use the standard normal table to find $z$ corresponding to the desired percentile.

• Value of X: $X = \mu + z \cdot \sigma$

Step-by-Step Guidance

1. For part (a), identify the percentile (30th) and use the standard normal table to find the corresponding $z$-score.

2. Set up the formula $X = \mu + z \cdot \sigma$ with $\mu = 7.4$ and $\sigma = 1.2$.

3. For part (b), recognize that the middle 80% means you need the 10th and 90th percentiles (cutoffs for the lowest and highest 10%).

4. Find the $z$-scores for the 10th and 90th percentiles using the standard normal table.

5. Set up the formulas for the two cutoff weights using $X = \mu + z \cdot \sigma$ for each percentile.

6. Do not calculate the final weights yet.

Try solving on your own before revealing the answer!