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

• Addition Rule for Probability:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• $P(A \cup B)$: Probability that event A or event B occurs

• $P(A)$: Probability of event A

• $P(B)$: Probability of event B

• $P(A \cap B)$: Probability that both A and B occur

Step-by-Step Guidance

1. Identify the events: Let C = "shopper uses a credit card" and R = "shopper returns an item."

2. Write down the given probabilities: $P(C) = 0.28$, $P(R) = 0.11$, $P(C \cap R) = 0.07$.

3. State the addition rule formula: $P(C \cup R) = P(C) + P(R) - P(C \cap R)$.

4. Substitute the given values into the formula.

Try solving on your own before revealing the answer!

Final Answer: 0.32

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

This means there is a 32% chance that a randomly selected shopper either uses a credit card or returns an item on Christmas Day.

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

• Multiplication Rule for Conditional Probability:

$P(A \cap B) = P(A) \times P(B|A)$

• $P(A)$: Probability of event A

• $P(B|A)$: Probability of event B given that A has occurred

• $P(A \cap B)$: Probability that both A and B occur

Step-by-Step Guidance

1. Identify the events: H = "has health insurance", C = "has car insurance".

2. Write down the given probabilities: $P(H) = 0.60$, $P(C|H) = 0.35$.

3. State the multiplication rule: $P(H \cap C) = P(H) \times P(C|H)$.

4. Substitute the given values into the formula.

Try solving on your own before revealing the answer!

Final Answer: 0.21

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

This means that 21% of residents are expected to have both health and car insurance.

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 tests 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

• Z-test formula:

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

• $\bar{x}$: Sample mean

• $\mu$: Population mean (under $H_0$)

• $\sigma$: Population standard deviation

• $n$: Sample size

Step-by-Step Guidance

1. State the null and alternative hypotheses:

   • $H_0: \mu = 18,450$

   • $H_1: \mu \neq 18,450$

2. Identify the sample statistics: $\bar{x} = 19,120$, $n = 40$, $\sigma = 2,900$.

3. Calculate the test statistic using the Z-formula:

   • First, compute the standard error: $SE = \frac{\sigma}{\sqrt{n}}$

   • Then, plug values into the Z-formula: $Z = \frac{19,120 - 18,450}{SE}$

4. Determine the critical Z-value(s) for a two-tailed test at $\alpha = 0.05$ (commonly $\pm 1.96$).

Try solving on your own before revealing the answer!

Final Answer:

The calculated Z-value is approximately 1.54, which does not exceed the critical value of 1.96. Therefore, we fail to reject the null hypothesis. There is not enough evidence to conclude that the average cost is different from the national average at the 5% significance level.

Q4. How many calls from a sample of 100 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

• Z-score formula:

$Z = \frac{X - \mu}{\sigma}$

• $X$: Value of interest

• $\mu$: Mean

• $\sigma$: Standard deviation

• Expected count:

$\text{Expected count} = P(10 < X < 20) \times n$

Step-by-Step Guidance

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

2. Calculate the Z-score for 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 lower probability from the higher one.

5. Multiply the probability by $n = 100$ to get the expected number of calls.

Try solving on your own before revealing the answer!

Final Answer: 86

After calculating the Z-scores and using the Z-table, you find $P(10 < X < 20) \approx 0.86$. Multiplying by 100 gives an expected count of 86 calls.

Q5. (a) How many of the 400 people surveyed are expected to drink LESS THAN 2 cups 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 find probabilities for intervals and convert them to expected counts.

Key Terms and Formulas

• Z-score formula:

$Z = \frac{X - \mu}{\sigma}$

• $X$: Value of interest

• $\mu$: Mean

• $\sigma$: Standard deviation

• Expected count:

$\text{Expected count} = P(\text{interval}) \times n$

Step-by-Step Guidance

1. For each part, calculate the Z-score(s):

   • For 2 cups: $Z_2 = \frac{2 - 2.4}{0.35}$

   • For 3 cups: $Z_3 = \frac{3 - 2.4}{0.35}$

2. Use the Z-table to find the probabilities for each Z-score.

3. For part (a): $P(X < 2)$ is the probability for Z less than $Z_2$.

4. For part (b): $P(2 < X < 3) = P(X < 3) - P(X < 2)$.

5. Multiply each probability by $n = 400$ to get the expected counts.

Try solving on your own before revealing the answer!

Final Answer:

(a) Approximately 46 people are expected to drink less than 2 cups per day.

(b) Approximately 353 people are expected to drink between 2 and 3 cups per day.

Q6. Calculate the mean, median, mode, standard deviation, coefficient of variation, and interquartile range for the given earthquake magnitudes.

Background

Topic: Descriptive Statistics

This question tests your ability to compute and interpret various descriptive statistics for a small dataset.

Key Terms and Formulas

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

• Median: Middle value when data is sorted

• Mode: Most frequent value

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

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

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

Step-by-Step Guidance

1. List the data in order: 5.1, 7.2, 9.7, 12.6, 14.3, 16.8, 18.9, 22.5.

2. Calculate the mean: Add all values and divide by 8.

3. Find the median: For 8 values, average the 4th and 5th values in the sorted list.

4. Determine the mode: Look for any repeated values (if none, state "no mode").

5. Calculate the standard deviation using the formula above.

6. Compute the coefficient of variation: $CV = \frac{s}{\bar{x}} \times 100\%$.

7. Find $Q_1$ (25th percentile) and $Q_3$ (75th percentile), then calculate $IQR = Q_3 - Q_1$.

Try solving on your own before revealing the answer!

Final Answer:

Mean: 13.39, Median: 13.45, Mode: No mode, Standard Deviation: 5.77, Coefficient of Variation: 43.09%, IQR: 9.65

All calculations are rounded to two decimal places as required.

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

Background

Topic: Confidence Intervals

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

• Confidence Interval formula:

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

• $\bar{x}$: Sample mean

• $z^*$: Z-value for the desired confidence level (for 95%, $z^* \approx 1.96$)

• $\sigma$: Population standard deviation

• $n$: Sample size

Step-by-Step Guidance

1. Identify the sample statistics: $\bar{x} = 2.9$, $n = 60$, $\sigma = 0.8$.

2. Find the Z-value for a 95% confidence interval ($z^* = 1.96$).

3. Calculate the standard error: $SE = \frac{0.8}{\sqrt{60}}$.

4. Compute the margin of error: $ME = 1.96 \times SE$.

5. Construct the confidence interval: $2.9 \pm ME$.

6. For part (b), compare the interval to the national average of 3.2 hours and comment on consistency.

Try solving on your own before revealing the answer!

Final Answer:

(a) The 95% confidence interval is approximately (2.69, 3.11) hours.

(b) Since 3.2 is not within the interval, the sample evidence is not consistent with the national average. This suggests that the true mean may be lower than the reported national average.

Q8. (a) Identify TWO ethical concerns raised by RetailMax's data practices. (b) Critically evaluate the company's argument about implicit consent and program benefits. (c) Recommend ONE specific change to improve ethical standing.

Background

Topic: Business Ethics in Data Use

This question tests your ability to identify and evaluate ethical issues in business data practices, and to recommend improvements.

Key Terms and Concepts

• Informed Consent: Customers should be clearly informed and give explicit permission for how their data is used.

• Privacy: Respecting individuals' rights to control their personal information.

• Ethical Justification: Evaluating whether business practices are fair, transparent, and respect stakeholders.

Step-by-Step Guidance

1. For (a), identify two concerns such as lack of explicit consent and potential misuse of sensitive data. Briefly explain why each is problematic.

2. For (b), analyze the company's argument: Does accepting terms and conditions count as true consent? Does increased sales justify the practice ethically?

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

Try answering on your own before revealing the answer!

Final Answer:

(a) 1. Lack of explicit consent: Customers were not clearly informed or asked for permission to use their data for targeted marketing, which violates principles of informed consent. 2. Privacy concerns: Collecting and using detailed personal data without clear consent can be seen as an invasion of privacy.

(b) Implicit consent via terms and conditions is not sufficient for ethical data use, as most customers do not read or fully understand these documents. Increased sales do not justify practices that compromise customer autonomy and privacy.

(c) RetailMax should implement an explicit opt-in process for marketing data use, ensuring customers are fully informed and can make a clear choice. This would improve transparency and respect for customer autonomy.

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

Background

Topic: Regression Analysis

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

Key Terms and Formulas

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

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

• R² (coefficient of determination): 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. (a) Write the regression equation: $\hat{Y} = 40 + 8X$. Identify Y (productivity score) as the dependent variable and X (hours of sleep) as the independent variable.

2. (b) Interpret the slope: For each additional hour of sleep, productivity increases by 8 points.

3. (c) Predict for X = 7: Substitute X = 7 into the equation: $\hat{Y} = 40 + 8 \times 7$.

4. (d) Explain R²: 0.68 means 68% of the variation in productivity is explained by hours of sleep.

5. (e) p < 0.01: The relationship is statistically significant at the 1% level, so we reject the null hypothesis that the slope is zero.

Try answering on your own before revealing the answer!

Final Answer:

(a) $\hat{Y} = 40 + 8X$; Y is productivity, X is hours of sleep.

(b) Each extra hour of sleep increases productivity by 8 points.

(c) Predicted productivity for 7 hours: $\hat{Y} = 40 + 8 \times 7 = 96$.

(d) R² = 0.68: 68% of productivity variation is explained by sleep hours.

(e) p < 0.01: The relationship is statistically significant; we reject the null hypothesis.