Study Cases

Statistical Inferences~ Week 14

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1 Case Study 1

1.1 One-Sample Z-Test (Statistical Hypotheses)

A digital learning platform claims that the average daily study time of its users is 120 minutes. Based on historical records, the population standard deviation is known to be 15 minutes.

A random sample of 64 users shows an average study time of 116 minutes.

\[ \begin{eqnarray*} \mu_0 &=& 120 \\ \sigma &=& 15 \\ n &=& 64 \\ \bar{x} &=& 116 \end{eqnarray*} \]

1.2 Tasks

  1. Formulate the Null Hypothesis (H₀) and Alternative Hypothesis (H₁).
  2. Identify the appropriate statistical test and justify your choice.
  3. Compute the test statistic and p-value using \(\alpha = 0.05\).
  4. State the statistical decision.
  5. Interpret the result in a business analytics context.

2 Case Study 2

2.1 One-Sample T-Test (σ Unknown, Small Sample)

A UX Research Team investigates whether the average task completion time of a new application differs from 10 minutes.

The following data are collected from 10 users:

\[ 9.2,\; 10.5,\; 9.8,\; 10.1,\; 9.6,\; 10.3,\; 9.9,\; 9.7,\; 10.0,\; 9.5 \]

2.2 Tasks

  1. Define H₀ and H₁ (two-tailed).
  2. Determine the appropriate hypothesis test.
  3. Calculate the t-statistic and p-value at \(\alpha = 0.05\).
  4. Make a statistical decision.
  5. Explain how sample size affects inferential reliability.

3 Case Study 3

3.1 Two-Sample T-Test (A/B Testing)

A product analytics team conducts an A/B test to compare the average session duration (minutes) between two versions of a landing page.

Version Sample Size (n) Mean Standard Deviation
A 25 4.8 1.2
B 25 5.4 1.4

3.2 Tasks

  1. Formulate the null and alternative hypotheses.
  2. Identify the type of t-test required.
  3. Compute the test statistic and p-value.
  4. Draw a statistical conclusion at \(\alpha = 0.05\).
  5. Interpret the result for product decision-making.

4 Case Study 4

4.1 Chi-Square Test of Independence

An e-commerce company examines whether device type is associated with payment method preference.

Device / Payment E-Wallet Credit Card Cash on Delivery
Mobile 120 80 50
Desktop 60 90 40

4.2 Tasks

  1. State the Null Hypothesis (H₀) and Alternative Hypothesis (H₁).
  2. Identify the appropriate statistical test.
  3. Compute the Chi-Square statistic (χ²).
  4. Determine the p-value at \(\alpha = 0.05\).
  5. Interpret the results in terms of digital payment strategy.

5 Case Study 5

5.1 Type I and Type II Errors (Conceptual)

A fintech startup tests whether a new fraud detection algorithm reduces fraudulent transactions.

  • H₀: The new algorithm does not reduce fraud.
  • H₁: The new algorithm reduces fraud.

5.2 Tasks

  1. Explain a Type I Error (α) in this context.
  2. Explain a Type II Error (β) in this context.
  3. Identify which error is more costly from a business perspective.
  4. Discuss how sample size affects Type II Error.
  5. Explain the relationship between α, β, and statistical power.

6 Case Study 6

6.1 P-Value and Statistical Decision Making

A churn prediction model evaluation yields the following results:

  • Test statistic = 2.31
  • p-value = 0.021
  • Significance level: \(\alpha = 0.05\)

6.2 Tasks

  1. Explain the meaning of the p-value.
  2. Make a statistical decision.
  3. Translate the decision into non-technical language for management.
  4. Discuss the risk if the sample is not representative.
  5. Explain why the p-value does not measure effect size.
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