Step-by-Step Guidance for Warranty Cost Model and Statistical Analysis

Q1. Create a probability table or tree diagram using the manufacturer's assumptions.

Background

Topic: Probability Models (Joint, Marginal, Conditional Probability)

This question tests your ability to organize and analyze probabilities for overlapping events, using either a joint probability table or a tree diagram. You'll use the given probabilities to deduce probabilities for combinations of events (e.g., both transmission and engine replacement).

Key Terms and Formulas:

• Joint Probability: $P(A \cap B)$

• Marginal Probability: $P(A)$ or $P(B)$

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

• Independence: $P(A \cap B) = P(A)P(B)$ if A and B are independent

Step-by-Step Guidance

1. List the events: T = transmission replacement, E = engine replacement, N = neither, B = both.

2. Write the given probabilities: $P(T) = 0.065$, $P(E) = 0.017$, $P(T \cup E) = 0.078$.

3. Recall that $P(T \cup E) = P(T) + P(E) - P(T \cap E)$, and solve for $P(T \cap E)$ (probability both replacements are needed).

4. Use the joint probability to fill in the table or tree diagram, and find the probabilities for each possible outcome (only T, only E, both, neither).

Try solving on your own before revealing the answer!

Q2. Analyze the distribution of the random variable X (cost to manufacturer).

Background

Topic: Discrete Probability Distributions, Expected Value, Standard Deviation

This question asks you to enumerate all possible warranty outcomes, assign costs and probabilities, and then compute the expected value and standard deviation of the cost.

Key Terms and Formulas:

• Random Variable X: Cost to manufacturer for a warranty

• Expected Value: $E[X] = \sum x_i P(x_i)$

• Standard Deviation: $\sigma_X = \sqrt{\sum (x_i - E[X])^2 P(x_i)}$

Step-by-Step Guidance

1. List all possible outcomes for a warranty (neither, only T, only E, both), and assign the corresponding cost to each outcome.

2. Assign the probability to each outcome using your results from Q1.

3. Set up the expected value formula using the costs and probabilities for each outcome.

4. Set up the standard deviation formula using the same outcomes, costs, and probabilities.

Try solving on your own before revealing the answer!

Q3. Hypothesis testing for model assumptions using warranty data (transmission, engine, both, average costs).

Background

Topic: Hypothesis Testing for Proportions and Means

This question asks you to test whether the observed data matches the manufacturer's assumptions for various rates and averages, using appropriate statistical tests.

Key Terms and Formulas:

• Null Hypothesis ($H_0$): The manufacturer's assumed rate or mean is correct.

• Alternative Hypothesis ($H_A$): The manufacturer's assumed rate or mean is incorrect.

• Test Statistic for Proportion: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• Test Statistic for Mean: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

• p-value: Probability of observing a test statistic as extreme as, or more extreme than, the observed value under $H_0$.

Step-by-Step Guidance

1. For each parameter (transmission rate, engine rate, both, average costs), state the null and alternative hypotheses.

2. Check assumptions for each test (e.g., sample size, normality, independence).

3. Calculate the sample statistic (e.g., sample proportion or mean) from the data.

4. Set up the formula for the test statistic (z or t) using the sample and assumed values.

Try solving on your own before revealing the answer!

Q4. Update inconsistent parameters using confidence intervals.

Background

Topic: Confidence Intervals for Proportions and Means

This question asks you to construct confidence intervals for parameters that failed the hypothesis test, and update the model using the upper bound to avoid underestimating costs.

Key Terms and Formulas:

• Confidence Interval for Proportion: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Confidence Interval for Mean: $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

Step-by-Step Guidance

1. Identify which parameters need updating based on hypothesis test results.

2. Calculate the sample statistic (proportion or mean) and standard error.

3. Set up the confidence interval formula using the appropriate critical value ($z^*$ or $t^*$).

4. Use the upper bound of the interval to update the parameter in the model for X.

Try solving on your own before revealing the answer!

Q5. Analyze the total cost for 2500 warranties (expected value, standard deviation, simulation, pricing).

Background

Topic: Sums of Random Variables, Central Limit Theorem, Simulation, Pricing for Profit

This question asks you to compute the expected value and standard deviation for the total cost of 2500 warranties, simulate the sampling distribution, and determine a pricing strategy to ensure profitability with high probability.

Key Terms and Formulas:

• Let $Y$ be the total cost for 2500 warranties: $Y = X_1 + X_2 + ... + X_{2500}$

• Expected Value: $E[Y] = 2500 \times E[X]$

• Standard Deviation: $\sigma_Y = \sqrt{2500} \times \sigma_X$ (if independent)

• Central Limit Theorem: For large n, $Y$ is approximately normal.

• Pricing: Find the price per warranty so that $P(\text{Total Revenue} > Y) \geq 0.99$

Step-by-Step Guidance

1. Calculate the expected value and standard deviation for Y using your updated model for X.

2. Simulate the sampling distribution for Y (e.g., using StatCrunch or another tool) and verify normality.

3. Set up the inequality for pricing: $P(\text{Total Revenue} > Y) \geq 0.99$, and use the normal distribution to solve for the required price per warranty.

Try solving on your own before revealing the answer!