Q: Bright Life Insurance Company finds that 60% of residents have health insurance (H), and of those, 35% also have car insurance (C). 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 health economist wants to test if the average cost of knee replacement surgery differs from $$18,450. A sample of 40 patients has a mean of 19$$,120 and population standard deviation $$2,900. At $$\alpha = 0.05$$, what is the correct null hypothesis (H_0$) and alternative hypothesis (H_1$) for a two-tailed Z-test?

$$H_0$$: \mu = 18,450;\quad $$H_1$$: \mu \neq 18,450$$

Q: A researcher collects a sample of 60 Australians with a mean daily social media use of 2.9 hours and population standard deviation 0.8 hours. What is the 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: A regression analysis yields the equation $$\hat{Y} = 40 + 8X. $$What is the predicted productivity score for an employee who sleeps 7 hours per night?

$$\hat{Y} = 40 + 8 \times 7 = 40 + 56 = 96$$

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 correct addition rule formula.

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 (H), and of those, 35% also have car insurance (C). 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 differs from $$18,450. A sample of 40 patients has a mean of 19$$,120 and population standard deviation $$2,900. At $$\alpha = 0.05$$, what is the correct null hypothesis (H_0$) and alternative hypothesis (H_1$) for a two-tailed Z-test?

Accepted Answer

$$H_0$$: \mu = 18,450;\quad $$H_1$$: \mu 
eq 18,450$$

Question 4

Given a normal distribution with mean $$\mu = 15$$ minutes and standard deviation $$\sigma = 4$$ minutes, what is the Z-score for a response time of 10 minutes?

Accepted Answer

$$Z = \frac{10 - 15}{4} = -1.25$$

Question 5

Australians drink an average of 2.4 cups of coffee per day ($$\mu = 2.4$$), with $$\sigma = 0.35. In a $$sample of 400, how many are expected to drink less than 2 cups per day? (Use $$Z = \frac{2 - 2.4}{0.35}$$ and standard normal probabilities.)

Accepted Answer

$$Z = \frac{2 - 2.4}{0.35} = -1.14$$;\quad P(Z \u003c -1.14) \approx 0.1271;\quad 0.1271 	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 ($$\bar{x}) of $$this dataset?

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 researcher collects a sample of 60 Australians with a mean daily social media use of 2.9 hours and population standard deviation 0.8 hours. What is the 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 best describes the ethical concern of 'informed consent' in the context of RetailMax's data practices?

Accepted Answer

Customers were not explicitly informed or asked for permission to use their data for targeted marketing, violating the principle of informed consent.

Question 9

A regression analysis yields the equation $$\hat{Y} = 40 + 8X. $$What is the predicted productivity score for an employee who sleeps 7 hours per night?

Accepted Answer

$$\hat{Y} = 40 + 8 	imes 7 = 40 + 56 = 96$$

Question 10

In the regression model $$\hat{Y} = 40 + 8X$$ with $$R^2 = 0.68$, what does R^2$ = 0.68$$ mean in practical terms?

Accepted Answer

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

Question 11

A p-value of less than 0.01 is reported for the slope in a regression analysis. What does this indicate about the relationship between sleep and productivity?

Accepted Answer

There is strong evidence to reject the null hypothesis and conclude that sleep is significantly related to productivity.

Question 12

Which formula is used to calculate the coefficient of variation (CV) for a dataset?

Accepted Answer

$$CV = \frac{s}{\bar{x}} 	imes 100\%$$

Question 13

If a 95% confidence interval for the mean daily social media use is (2.70, 3.10) hours, what can you conclude about the national average of 3.2 hours?

Accepted Answer

The national average of 3.2 hours is not within the confidence interval, so the sample evidence suggests the true mean may be lower than the national average.

Question 14

Which of the following is the correct formula for the standard deviation $$s of a $$sample?

Accepted Answer

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

Question 15

A city fire department's response times are normally distributed with $$\mu = 15$$ minutes and $$\sigma = 4$$ minutes. In a sample of 100 calls, what is the expected number of calls with response times between 10 and 20 minutes? (Use $$Z-$$scores and standard normal probabilities.)

Accepted Answer

$$Z_{10} = -1.25,\ Z_{20} = 1.25;\ P(10 \u003c X \u003c 20) = P(Z \u003c 1.25) - P(Z \u003c -1.25) \approx 0.8944 - 0.1056 = 0.7888;\ 0.7888 	imes 100 \approx 79$$

Question 16

Which of the following is NOT a valid ethical concern in the RetailMax scenario?

Accepted Answer

RetailMax's use of customer data increased sales, so there is no ethical issue.

Question 17

In hypothesis testing, what is the correct formula for the Z-test statistic for a sample mean?

Accepted Answer

$$Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$$

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