Step-by-Step Guidance for BUS103 – Business Statistics Sample Final Exam

Q1. What is the probability that a randomly selected shopper either uses a credit card or returns an item on Christmas Day?

Background

Topic: Probability – Addition Rule

This question tests your understanding of the addition rule for probabilities, especially when two events are not mutually exclusive (i.e., they can both occur).

Key Terms and Formulas

• Let C = event that a shopper uses a credit card

• Let R = event that a shopper returns an item

• Given: $P(C) = 0.28$, $P(R) = 0.11$, $P(C \cap R) = 0.07$

Addition Rule for Probability:

$P(C \cup R) = P(C) + P(R) - P(C \cap R)$

Step-by-Step Guidance

1. Write down the addition rule formula for the probability of either event C or event R occurring.

2. Identify and list the given probabilities: $P(C)$, $P(R)$, and $P(C \cap R)$.

3. Substitute the given values into the formula.

4. Perform the addition and subtraction as indicated by the formula, but stop before stating the final result.

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Q2. What is the probability that a randomly selected resident has BOTH health insurance and car insurance?

Background

Topic: Probability – Multiplication Rule (Conditional Probability)

This question tests your ability to use the multiplication rule for conditional probability to find the probability of two events both occurring.

Key Terms and Formulas

• Let H = resident has health insurance

• Let C = resident has car insurance

• Given: $P(H) = 0.60$, $P(C|H) = 0.35$

Multiplication Rule for Conditional Probability:

$P(H \cap C) = P(H) \times P(C|H)$

Step-by-Step Guidance

1. Write the multiplication rule for conditional probability.

2. Identify $P(H)$ and $P(C|H)$ from the information provided.

3. Substitute the values into the formula.

4. Multiply the probabilities, but do not state the final result yet.

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Q3. Conduct a two-tailed Z-test to determine if the sample mean cost of knee replacement surgery is significantly different from the national average of $18,450.

Background

Topic: Hypothesis Testing – Z-test for a Population Mean

This question assesses your ability to perform a hypothesis test for a population mean using the Z-test when the population standard deviation is known.

Key Terms and Formulas

• Null hypothesis ($H_0$): $\mu = 18,450$

• Alternative hypothesis ($H_1$): $\mu \neq 18,450$

• Sample mean ($\bar{x}$): $19,120$

• Population standard deviation ($\sigma$): $2,900$

• Sample size ($n$): $40$

• Significance level ($\alpha$): $0.05$

Z-test formula:

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

Step-by-Step Guidance

1. State the null and alternative hypotheses.

2. Write the Z-test formula and identify all the values to substitute.

3. Substitute the sample mean, population mean, standard deviation, and sample size into the formula.

4. Calculate the denominator ($\sigma / \sqrt{n}$) and the numerator ($\bar{x} - \mu$), but do not compute the final Z value.

5. Look up the critical Z value(s) for a two-tailed test at $\alpha = 0.05$ (typically $\pm 1.96$), and set up the decision rule, but do not make the final decision.

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Q4. For a sample of 100 fire department calls, how many are expected to have a response time between 10 and 20 minutes?

Background

Topic: Normal Distribution – Expected Counts

This question tests your ability to use the normal distribution to find probabilities and expected counts within a given interval.

Key Terms and Formulas

• Population mean ($\mu$): $15$ minutes

• Population standard deviation ($\sigma$): $4$ minutes

• Sample size ($n$): $100$

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

Step-by-Step Guidance

1. Calculate the Z-score for $X = 10$ minutes: $Z_{10} = \frac{10 - 15}{4}$

2. Calculate the Z-score for $X = 20$ minutes: $Z_{20} = \frac{20 - 15}{4}$

3. Use the Z-table to find the probabilities corresponding to $Z_{10}$ and $Z_{20}$.

4. Compute $P(10 < X < 20)$ by subtracting the two probabilities.

5. Multiply the probability by $n = 100$ to find the expected number of calls, but do not state the final count.

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Q5. (a) How many of 400 Australians are expected to drink less than 2 cups of coffee per day? (b) How many are expected to drink between 2 and 3 cups per day?

Background

Topic: Normal Distribution – Probabilities and Expected Counts

This question tests your ability to use the normal distribution to calculate probabilities and expected counts for specified intervals.

Key Terms and Formulas

• Population mean ($\mu$): $2.4$ cups

• Population standard deviation ($\sigma$): $0.35$ cups

• Sample size ($n$): $400$

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

Step-by-Step Guidance

1. For part (a), calculate the Z-score for $X = 2$ cups: $Z_{2} = \frac{2 - 2.4}{0.35}$

2. Use the Z-table to find the probability for $Z_{2}$ (i.e., $P(X < 2)$).

3. Multiply this probability by $n = 400$ to get the expected count for part (a).

4. For part (b), calculate the Z-scores for $X = 2$ and $X = 3$ cups: $Z_{2}$ and $Z_{3} = \frac{3 - 2.4}{0.35}$

5. Use the Z-table to find the probabilities for both Z-scores, then subtract to find $P(2 < X < 3)$.

6. Multiply this probability by $n = 400$ for the expected count in part (b), but do not state the final numbers.

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Q6. Calculate the mean, median, mode, standard deviation, coefficient of variation, and interquartile range for the given earthquake magnitudes: 22.5, 18.9, 5.1, 14.3, 9.7, 12.6, 16.8, 7.2.

Background

Topic: Descriptive Statistics

This question tests your ability to compute key descriptive statistics for a small dataset.

Key Terms and Formulas

• Mean ($\bar{x}$): $\bar{x} = \frac{\sum x_i}{n}$

• Median: Middle value when data is sorted

• Mode: Most frequent value

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

• Coefficient of Variation (CV): $CV = \frac{s}{\bar{x}} \times 100\%$

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

Step-by-Step Guidance

1. List the data in ascending order.

2. Calculate the mean by summing all values and dividing by $n = 8$.

3. Find the median by identifying the middle value(s) in the sorted list.

4. Check for the mode (value that appears most often).

5. Compute the standard deviation using the formula above (find each $x_i - \bar{x}$, square, sum, divide by $n-1$, then take the square root).

6. Calculate the coefficient of variation using the mean and standard deviation.

7. Find $Q_1$ and $Q_3$ (the first and third quartiles) and compute the IQR, but do not state the final values.

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Q7. (a) Construct a 95% confidence interval for the true mean daily social media use based on a sample of 60 Australians (mean = 2.9 hours, σ = 0.8 hours). (b) Comment on whether the sample evidence is consistent with the national average of 3.2 hours.

Background

Topic: Confidence Intervals for the Mean (Known σ)

This question tests your ability to construct and interpret a confidence interval for a population mean when the population standard deviation is known.

Key Terms and Formulas

• Sample mean ($\bar{x}$): $2.9$ hours

• Population standard deviation ($\sigma$): $0.8$ hours

• Sample size ($n$): $60$

• Confidence level: $95\%$ (Z* ≈ 1.96)

• Confidence interval formula: $CI = \bar{x} \pm z^* \left( \frac{\sigma}{\sqrt{n}} \right)$

Step-by-Step Guidance

1. Write the confidence interval formula and identify all values to substitute.

2. Calculate the standard error: $\frac{\sigma}{\sqrt{n}}$.

3. Multiply the standard error by the Z* value for 95% confidence.

4. Add and subtract this margin of error from the sample mean to get the interval, but do not state the final interval.

5. For part (b), compare the national average (3.2 hours) to your interval and discuss whether it falls inside or outside the interval, but do not state the conclusion.

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Q8. (a) Identify two ethical concerns with RetailMax's data practices and explain why each is problematic. (b) Critically evaluate the company's argument about implicit consent and program benefits. (c) Recommend one change to improve ethical data use and justify it.

Background

Topic: Business Ethics in Data Use

This question tests your ability to identify and analyze ethical issues in business data practices, evaluate arguments, and make justified recommendations.

Key Concepts

• Informed consent

• Privacy and data protection

• Transparency

• Ethical justification of business practices

Step-by-Step Guidance

1. For (a), identify two specific ethical concerns (e.g., lack of explicit consent, potential misuse of personal data) and explain why each is problematic.

2. For (b), analyze the company's argument about implicit consent and increased sales, considering ethical principles such as autonomy and fairness.

3. For (c), suggest a concrete change (e.g., implement explicit opt-in consent) and justify how it would address the ethical issues identified.

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Q9. (a) Write the regression equation and identify the dependent and independent variables. (b) Interpret the slope coefficient. (c) Predict the productivity score for 7 hours of sleep. (d) Explain the meaning of R² = 0.68. (e) What does p < 0.01 indicate about statistical significance?

Background

Topic: Regression Analysis

This question tests your understanding of simple linear regression output, interpretation of coefficients, prediction, and statistical significance.

Key Terms and Formulas

• Regression equation: $\hat{Y} = a + bX$

• Dependent variable (Y): Productivity score

• Independent variable (X): Hours of sleep

• Slope (b): Change in Y for a one-unit increase in X

• Coefficient of determination ($R^2$): Proportion of variance in Y explained by X

• p-value: Probability of observing the data if the null hypothesis is true

Step-by-Step Guidance

1. For (a), write out the regression equation and clearly label Y and X.

2. For (b), interpret the meaning of the slope coefficient in the context of sleep and productivity.

3. For (c), substitute X = 7 into the regression equation to set up the calculation for predicted Y, but do not compute the final value.

4. For (d), explain what $R^2 = 0.68$ means in terms of explanatory power.

5. For (e), discuss what a p-value less than 0.01 implies about the statistical significance of the relationship, referencing hypothesis testing concepts.

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