Q: Bright Life Insurance Company finds that 60% of residents have health insurance, and among those, 35% also have car insurance. What is the probability that a randomly selected resident has both health and car insurance? (Use the Multiplication Rule for Conditional Probability.)

$$P(H \cap C) = P(H) \times P(C|H) = 0.60 \times 0.35 = 0.21$$

Q: A sample of 60 Australians has a mean daily social media use of 2.9 hours ($$\sigma = 0.8). $$Construct a 95% confidence interval for the true mean. (Use $$z^* = 1.96.)$$

$$CI = 2.9 \pm 1.96 \times \frac{0.8}{\sqrt{60}} = 2.9 \pm 0.202 = (2.70, 3.10)$$

Q: If a new employee sleeps 7 hours per night, what is their predicted productivity score using the regression equation $$\hat{Y} = 40 + 8X$$?

$$\hat{Y} = 40 + 8 \times 7 = 96$$

Question 1

A retail store records the following probabilities on Christmas Day: the probability a shopper uses a credit card is 0.28, the probability a shopper returns an item is 0.11, and the probability a shopper does both is 0.07. What is the probability that a randomly selected shopper either uses a credit card or returns an item? (Use the Addition Rule of Probability.)

Accepted Answer

$$P(C \cup R) = P(C) + P(R) - P(C \cap R) = 0.28 + 0.11 - 0.07 = 0.32$$

Question 2

Bright Life Insurance Company finds that 60% of residents have health insurance, and among those, 35% also have car insurance. What is the probability that a randomly selected resident has both health and car insurance? (Use the Multiplication Rule for Conditional Probability.)

Accepted Answer

$$P(H \cap C) = P(H) 	imes P(C|H) = 0.60 	imes 0.35 = 0.21$$

Question 3

A health economist wants to test if the average cost of knee replacement surgery has changed from $$18,450. A sample of 40 patients has a mean of 19$$,120 and a population standard deviation of $$2,900. At $$\alpha = 0.05$, what is the value of the test statistic for a two-tailed Z-test?

Accepted Answer

$$Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} = \frac{19120 - 18450}{2900 / \sqrt{40}} \approx 1.54$$

Question 4

The Fire Department’s response times are normally distributed with mean $$\mu = 15$$ minutes and standard deviation $$\sigma = 4$$ minutes. Out of 100 calls, how many are expected to have response times between 10 and 20 minutes?

Accepted Answer

About 82 calls, since $$P(10 \u003c X \u003c 20) \approx 0.82$$ and $$0.82 	imes 100 = 82$$

Question 5

Australians drink an average of 2.4 cups of coffee per day ($$\mu = 2.4$$, $$\sigma = 0.35). In a $$sample of 400, how many are expected to drink less than 2 cups per day?

Accepted Answer

About 51 people, since $$Z = \frac{2 - 2.4}{0.35} \approx -1.14$$, $$P(Z \u003c -1.14) \approx 0.127$$, $$0.127 	imes 400 \approx 51$$

Question 6

Given the earthquake magnitudes: 22.5, 18.9, 5.1, 14.3, 9.7, 12.6, 16.8, 7.2, what is the mean magnitude?

Accepted Answer

$$\bar{x} = \frac{22.5 + 18.9 + 5.1 + 14.3 + 9.7 + 12.6 + 16.8 + 7.2}{8} = 13.39$$

Question 7

A sample of 60 Australians has a mean daily social media use of 2.9 hours ($$\sigma = 0.8). $$Construct a 95% confidence interval for the true mean. (Use $$z^* = 1.96.)$$

Accepted Answer

$$CI = 2.9 \pm 1.96 	imes \frac{0.8}{\sqrt{60}} = 2.9 \pm 0.202 = (2.70, 3.10)$$

Question 8

Which of the following is an ethical concern with RetailMax’s use of customer data for targeted marketing?

Accepted Answer

Lack of explicit customer consent for data use beyond general terms and conditions

Question 9

In the regression equation $$\hat{Y} = 40 + 8X$$, where $$Y is $$productivity score and $$X is $$hours of sleep, what does the slope coefficient of 8 mean?

Accepted Answer

For each additional hour of sleep, the predicted productivity score increases by 8 points.

Question 10

If a new employee sleeps 7 hours per night, what is their predicted productivity score using the regression equation $$\hat{Y} = 40 + 8X$$?

Accepted Answer

$$\hat{Y} = 40 + 8 	imes 7 = 96$$

Question 11

The coefficient of determination $$R^2$$ for a regression model is 0.68. What does this mean?

Accepted Answer

68% of the variation in productivity scores is explained by hours of sleep.

Question 12

A regression analysis finds a p-value less than 0.01 for the relationship between sleep and productivity. What is the correct interpretation?

Accepted Answer

There is strong evidence to reject the null hypothesis; sleep and productivity are significantly related.

Question 13

Which formula is used to calculate the standard deviation $$s of a $$sample?

Accepted Answer

$$s = \sqrt{\frac{\sum (x_i - \bar{x})^2$}{n-1}}$$

Question 14

Which of the following best describes the coefficient of variation (CV)?

Accepted Answer

$$CV = \frac{s}{\bar{x}} 	imes 100\%$$; it measures relative variability as a percentage of the mean.

Question 15

A 95% confidence interval for the mean daily social media use is (2.70, 3.10) hours. The national average is 3.2 hours. What can you conclude?

Accepted Answer

The sample evidence is not consistent with the national average, since 3.2 is outside the confidence interval.

Question 16

Which of the following is NOT a recommended ethical improvement for RetailMax’s data practices?

Accepted Answer

Continue using customer data without explicit consent as long as sales increase.

Question 17

If the interquartile range (IQR) is defined as $$Q_3$$ - $$Q_1$$, what does it represent?

Accepted Answer

The range of the middle 50% of the data.

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