1.1 Starting with a Question

In business, we often need to answer yes-or-no questions using data:

• Are bags of chips consistently lighter than the 50g printed on the packet?
• Did students score at least 60% on their statistics exam on average?
• Has a new training program actually improved employee performance?

Last week, we used confidence intervals to estimate a range for a population parameter. This week, we use a different approach: we start with a specific claim and ask whether the data provides enough evidence to reject it.

1.2 Confidence Intervals vs. Hypothesis Tests

Both tools help us make decisions from sample data, but they ask different questions:

1.3 The Courtroom Analogy

Hypothesis testing works exactly like a courtroom trial:

1.4 The Null Hypothesis (\(H_0\))

Think of \(H_0\) as the "nothing to see here" hypothesis. It represents the manufacturer's claim, the company's target, or the existing benchmark.

1.5 The Alternative Hypothesis (\(H_1\))

1.6 One-Tailed vs. Two-Tailed Tests

The direction of \(H_1\) determines whether we use a one-tailed or two-tailed test:

1.7 Visualising the Tails

1.8 The Significance Level (\(\alpha\))

Common choices:

Knowledge Check — Section 1

2.1 The Five Steps — Overview

Every hypothesis test follows these five steps, regardless of what you are testing. Let's walk through each one.

2.2 Step 1 — State the Hypotheses

Write down \(H_0\) and \(H_1\) based on the business question.

2.3 Step 2 — Choose the Significance Level

Select your significance level \(\alpha\) based on how much risk of a wrong rejection you can tolerate.

2.4 Step 3 — Compute the Test Statistic

The test statistic measures how far our sample result is from what \(H_0\) claims, in standardised units.

Where \(\bar{X}\) is the sample mean, \(\mu_0\) is the claimed value in \(H_0\), \(\sigma\) (or \(s\)) is the standard deviation, and \(n\) is the sample size.

2.5 Step 4 — Make the Decision (Critical Value Approach)

Compare the test statistic to a critical value from the Z or t table.

Decision rule: If the test statistic falls in the rejection region, reject \(H_0\). Otherwise, do not reject.

2.6 Step 4 — The p-Value Approach

Think of it as: "If the null hypothesis were true, how surprising is our sample result?"

2.7 Understanding the p-Value Visually

2.8 Step 5 — State the Conclusion

Translate the statistical decision back into plain English, in the context of the original business question.

Knowledge Check — Section 2

3.1 Two Ways to Be Wrong

No matter how careful we are, there is always a chance of making the wrong decision:

3.2 Errors in the Courtroom

3.3 Which Error Is Worse?

It depends on the context. Consider these scenarios:

Knowledge Check — Section 3

4.1 Z-Test vs. t-Test — When to Use Which

Both test whether a sample mean differs from a hypothesised value, but the choice depends on what you know:

4.2 Worked Example: Chip Bag Weights (Setup)

Step 1: \(H_0: \mu = 50\)    \(H_1: \mu < 50\)   (left-tailed)

Step 2: \(\alpha = 0.05\)

Step 3: Since \(n = 36 \geq 30\) and \(\sigma\) is unknown, we use the t-test (or Z approximation):

4.3 Worked Example: Chip Bag Weights (Decision)

Step 4 (Critical Value): For a left-tailed test at \(\alpha = 0.05\) with \(df = 35\):

The critical value from the t-table is approximately \(t_{0.05, 35} = -1.690\).

Our test statistic \(t = -2.571\) is more extreme (further left) than \(-1.690\).

Step 5 (Conclusion):

4.4 Worked Example: Student Exam Marks (Setup)

Step 1: \(H_0: \mu \geq 60\)    \(H_1: \mu < 60\)   (left-tailed)

Step 2: \(\alpha = 0.05\)

Step 3: Since \(n = 25 < 30\) and \(\sigma\) unknown, we must use the t-test:

4.5 Worked Example: Student Exam Marks (Decision)

Step 4: Critical value for left-tailed test, \(\alpha = 0.05\), \(df = 24\):

\(t_{0.05, 24} = -1.711\)

Our test statistic \(t = -1.667\) is not more extreme than \(-1.711\). It does not fall in the rejection region.

Step 5 (Conclusion):

Note: This does NOT prove the mean is 60% — only that we lack enough evidence to say it's lower.

4.6 Two-Tailed Test Example (Setup)

Step 1: \(H_0: \mu = 40\)    \(H_1: \mu \neq 40\)   (two-tailed)

Step 2: \(\alpha = 0.05\)

Step 3:

4.7 Two-Tailed Test Example (Decision)

Step 4: For a two-tailed test at \(\alpha = 0.05\), we split the significance between both tails: \(\alpha/2 = 0.025\) each.

Critical values with \(df = 49\): \(\pm t_{0.025, 49} \approx \pm 2.010\)

Our \(t = 2.666\) exceeds the upper critical value of \(+2.010\).

Step 5:

Knowledge Check — Section 4

5.1 Testing a Proportion

Sometimes we are not testing a mean but a proportion — a percentage or fraction.

5.2 Worked Example: Invoice Error Rate (Setup)

Step 1: \(H_0: p \leq 0.02\)    \(H_1: p > 0.02\)   (right-tailed)

Step 2: \(\alpha = 0.05\)

Step 3: Sample proportion: \(\hat{p} = 18/500 = 0.036\)

Check conditions: \(np_0 = 500 \times 0.02 = 10 \geq 5\) and \(n(1-p_0) = 490 \geq 5\). Conditions met.

5.3 Worked Example: Invoice Error Rate (Decision)

Step 4: For a right-tailed test at \(\alpha = 0.05\), the critical value is \(Z_{0.05} = 1.645\).

Our \(Z = 2.555 > 1.645\), so the test statistic falls in the rejection region.

p-Value approach: \(P(Z > 2.555) \approx 0.0053\). Since \(0.0053 < 0.05\), we reject \(H_0\).

Step 5 (Conclusion):

Notice how the conclusion connects the statistical result to a business action. Always do this.

Knowledge Check — Section 5

6.1 Ethical Pitfalls

Hypothesis testing is a powerful tool, but it can be misused — intentionally or accidentally:

6.2 Responsible Use of Hypothesis Tests

As future accountants and business professionals, keep these principles in mind:

• Pre-register your hypothesis and \(\alpha\) before collecting data
• Report all tests, not just the significant ones
• Consider the effect size — is the difference large enough to matter in practice?
• Use appropriate sample sizes — too small risks Type II errors; too large detects trivial differences
• Be transparent about assumptions (normality, independence, sample selection)

7.1 Week 8 Summary