Q: Are transmission replacements and engine replacements independent events? Use the manufacturer’s assumptions to justify your answer.

No, because $$P(T \cap E) \neq P(T) \times P(E)$$

Q: Using the manufacturer’s assumptions, what is the expected value $$E[X] of $$the cost per warranty?

$$E[X] = 2500 \times 0.0601 + 6000 \times 0.0121 + 8500 \times 0.0049$$

Q: Suppose the manufacturer wants to test if the transmission replacement rate is consistent with their assumption of 6.5%. Which is the correct null and alternative hypothesis for this test?

$$H_0$$: p = 0.065$$; H_a$$: p \neq 0.065$$

Q: After updating the model parameters, the manufacturer wants to determine the expected value and standard deviation for $$Y$$, the total cost of 2500 warranties. Assuming independence, which formulas should be used?

$$E[Y] = 2500 \times E[X]$$ and $$\sigma_Y = \sqrt{2500 \times \sigma_X^2$}$$

Q: Suppose the manufacturer wants to price each warranty so that the probability the total revenue from 2500 warranties exceeds the total cost is at least 99%. If $$Y is $$approximately normal with mean $$\mu_Y$$ and standard deviation $$\sigma_Y$$, which equation should be used to determine the minimum price per warranty?

$$\text{Price} = \frac{\mu_Y + z_{0.99} \sigma_Y}{2500}$$

Question 1

A car manufacturer models the cost of a 7-year extended warranty using the following assumptions: 6.5% of cars require a transmission replacement ($$2500), 1.7% require an engine replacement (6000$$), and 7.8% require at least one of these replacements. What is the probability that a car requires both a transmission and an engine replacement?

Accepted Answer

$$0.0049$$

Question 2

Given that a car requires a transmission replacement, what is the probability it also requires an engine replacement? Use the manufacturer’s assumptions.

Accepted Answer

$$\frac{0.0049}{0.065} \approx 0.075$$

Question 3

Are transmission replacements and engine replacements independent events? Use the manufacturer’s assumptions to justify your answer.

Accepted Answer

No, because $$P(T \cap E) 
eq P(T) 	imes P(E)$$

Question 4

Which of the following best describes the sample space for the random variable $$X$$, the cost to the manufacturer for a single warranty?

Accepted Answer

$$\{0, 2500, 6000, 8500\}$$

Question 5

Using the manufacturer’s assumptions, what is the expected value $$E[X] of $$the cost per warranty?

Accepted Answer

$$E[X] = 2500 	imes 0.0601 + 6000 	imes 0.0121 + 8500 	imes 0.0049$$

Question 6

What is the practical significance of the expected value $$E[X] to $$the manufacturer?

Accepted Answer

It represents the average cost per warranty the manufacturer should expect to pay over many warranties.

Question 7

Suppose the manufacturer wants to test if the transmission replacement rate is consistent with their assumption of 6.5%. Which is the correct null and alternative hypothesis for this test?

Accepted Answer

$$H_0$$: p = 0.065$$; H_a$$: p 
eq 0.065$$

Question 8

When performing a hypothesis test for the engine replacement rate using a sample of 2500 warranties, which test statistic should be used?

Accepted Answer

$$z = \frac{\hat{p} - p_0$}{\sqrt{\frac{p_0$(1-p_0$)}{n}}}$$

Question 9

If the manufacturer finds that the average cost of a transmission replacement is higher than assumed, how should they update their model?

Accepted Answer

Use the upper bound of the confidence interval for the average cost to update the model.

Question 10

After updating the model parameters, the manufacturer wants to determine the expected value and standard deviation for $$Y$$, the total cost of 2500 warranties. Assuming independence, which formulas should be used?

Accepted Answer

$$E[Y] = 2500 	imes E[X]$$ and $$\sigma_Y = \sqrt{2500 	imes \sigma_X^2$}$$

Question 11

The Central Limit Theorem (CLT) is used to justify that the sampling distribution of $$Y is $$approximately normal. Which condition is most important for the CLT to apply in this scenario?

Accepted Answer

The claims on individual warranties are independent.

Question 12

Suppose the manufacturer wants to price each warranty so that the probability the total revenue from 2500 warranties exceeds the total cost is at least 99%. If $$Y is $$approximately normal with mean $$\mu_Y$$ and standard deviation $$\sigma_Y$$, which equation should be used to determine the minimum price per warranty?

Accepted Answer

$$	ext{Price} = \frac{\mu_Y + z_{0.99} \sigma_Y}{2500}$$

Question 13

If the manufacturer’s data shows a statistically significant difference from the assumed engine replacement rate, which step should be taken next?

Accepted Answer

Construct a confidence interval for the engine replacement rate and update the model using the upper bound.

Question 14

A warranty claim can result in a transmission replacement, an engine replacement, both, or neither. If the probability of both is $$0.0049$$, transmission only is $$0.0601$$, engine only is $$0.0121$$, and neither is $$0.923$$, what is the probability distribution for $$X$$?

Accepted Answer

$$P(X=0)=0.923$$, $$P(X=2500)=0.0601$$, $$P(X=6000)=0.0121$$, $$P(X=8500)=0.0049$$

Question 15

If the manufacturer wants to test whether the average cost of an engine replacement is $$6000$$, which statistical test should be used?

Accepted Answer

A one-sample $$t-$$test for the mean

Question 16

When constructing a confidence interval for the proportion of warranties resulting in both engine and transmission replacements, which formula should be used?

Accepted Answer

$$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 17

If the manufacturer uses the upper bound of a confidence interval to update a parameter in their model, what is the likely effect on the expected value $$E[X]$$?

Accepted Answer

The expected value $$E[X]$$ will increase.

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

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