H2 Maths Notes (JC 1-2): 6.5) Hypothesis Testing

Before you revise\ Prepare a quick-reference sheet: null/alternative templates, common critical values, and GC commands for z, t, and binomial tests. Always practise writing the full conclusion sentence—MOE wants context, not just “reject \( H_0 \)”.

Hypothesis Testing Framework

1. State hypotheses: \( H_0 \) (status quo) and \( H_1 \) (claim), referencing the parameter (\( \mu \), \( p \), etc.).
2. Choose test statistic and significance level \( \alpha \).
3. Calculate the observed statistic and either the p-value or compare with critical values.
4. Decision: reject or fail to reject \( H_0 \) based on the evidence.
5. Conclusion: interpret in context, noting the significance level.

z-tests for Means (Known \( \sigma \) or Large \( n \))

• Test statistic: \( Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}} \).
• Critical regions for two-tailed tests at \( \alpha \): reject if \( |Z| > z_{1 - \alpha/2} \).

Example -- JC lecture attendance

Historical mean attendance is 520 students. After a scheduling change, a sample of \( n = 64 \) lectures gives \( \bar{X} = 508 \), \( \sigma = 40 \). Test at 5% whether attendance decreased.

1. \( H_0: \mu = 520 \), \( H_1: \mu < 520 \).
2. \( Z = \frac{508 - 520}{40 / \sqrt{64}} = -2.40 \).
3. Critical value: \( z_{0.05} = -1.645 \). Since \( -2.40 < -1.645 \), reject \( H_0 \).
4. Conclusion: There is sufficient evidence at the 5% level that average attendance fell below 520.

t-tests for Means (Unknown \( \sigma \), Small Samples)

• Statistic: \( T = \frac{\bar{X} - \mu_0}{S / \sqrt{n}} \) with \( n - 1 \) degrees of freedom.
• Use t-distribution tables or GC tTest functions.

Example -- Tutorial completion time

A new tutorial format should reduce the mean completion time of \( 85 \) minutes. A sample of \( n = 12 \) classes yields \( \bar{X} = 81.3 \) minutes, \( s = 6.5 \). Test at 10% significance.

1. \( H_0: \mu = 85 \), \( H_1: \mu < 85 \).
2. \( t = \frac{81.3 - 85}{6.5 / \sqrt{12}} = -1.99 \).
3. Critical value: \( t_{0.10, 11} = -1.363 \). Since \( -1.99 < -1.363 \), reject \( H_0 \).
4. Conclude the average completion time has decreased at the 10% level.

Proportion Tests

• Statistic: \( Z = \frac{\hat{p} - p_0}{\sqrt{p_0(1 - p_0)/n}} \).
• Ensure \( np_0 \) and \( n(1 - p_0) \) are both \( \geq 5 \).

Example -- Device adoption

Suppose previously \( 60% \) of students used graphing calculators daily. A survey of \( n = 150 \) finds \( 105 \) users. Test at 1% if usage increased.

1. \( H_0: p = 0.6 \), \( H_1: p > 0.6 \).
2. \( \hat{p} = 0.7 \), \( Z = \frac{0.7 - 0.6}{\sqrt{0.6 \times 0.4 / 150}} = 2.74 \).
3. Critical value \( z_{0.99} = 2.326 \). Since \( 2.74 > 2.326 \), reject \( H_0 \).
4. Evidence suggests the proportion of daily users exceeded 60%.

Binomial Tests (Small n)

• When \( n \) is small, evaluate exact probabilities using \( \operatorname{Bin}(n, p_0) \).

Example -- Quality audit

A production line historically has defect rate \(5\%\). In \( n = 20 \) items, \( 3 \) are defective. Test at \(5\%\) if the defect rate increased.

1. \( H_0: p = 0.05 \), \( H_1: p > 0.05 \).
2. Compute \( P(X \geq 3) = 1 - [P\( X = 0 \) + P\( X = 1 \) + P\( X = 2 \)] \).
3. Using GC binomcdf(20, 0.05, 2) = 0.924; hence \( P(X \geq 3) = 0.076 \).
4. Since 0.076 > 0.05, fail to reject \( H_0 \); insufficient evidence that defects increased.

P-values vs Critical Regions

• P-value: probability, under \( H_0 \), of observing a statistic at least as extreme as the sample.
• If p-value \( < \alpha \), reject \( H_0 \); otherwise do not reject.
• Mention explicitly when reporting: “p-value = 0.032 < 0.05, so reject \( H_0 \).”

Calculator Workflows

• TI: ZTest, TTest, 1-PropZTest, 2-SampTTest; record the inputs and outputs (test statistic, p-value).
• Casio: Use STAT > DIST > NORM / t / BINM; for binomial tail probabilities, Bcd command handles cumulative sums.
• Always double-check the tail (left/right/two) before pressing ENTER.

Exam Watch Points

• Hypotheses must refer to parameters (\( \mu \), \( p \)), not statistics (\( \bar{X} \)).
• Quote the significance level and justify one- vs two-tailed tests from the context wording (“increase”, “different”).
• Round the test statistic to 3 decimal places and p-values to 3 significant figures.
• Ensure the final conclusion mentions the context and the significance level (e.g. “At the 5% level…”).
• When using approximations, verify assumptions (normality, sample size).

Quick Revision Checklist

• [ ] Set up \( H_0 \), \( H_1 \) correctly and select the appropriate test statistic.
• [ ] Carry out \(z\), \(t\), and binomial tests with calculator support while writing full working.
• [ ] Interpret p-values versus critical regions and articulate conclusions in context.
• [ ] Explain possible Type I / Type II errors qualitatively when asked.

Next steps: Pair hypothesis tests with regression outputs (Topic 6.6) and past-year structured questions combining sampling and inference.