Unit 6: Inference for Categorical Data: Proportions

Constructing and Interpreting a Confidence Interval for a Population Proportion

A confidence interval estimates the true population proportion \( p \) using sample data.

Conditions:
Random: Data must come from a random sample or randomized experiment.
10% Condition: The sample size should be less than 10% of the population: \( n \leq 0.10N \).
Large Counts: Both \( np \geq 10 \) and \( n(1 - p) \geq 10 \) must be true.

Formula: The confidence interval is given by: \[ \hat{p} \pm z^* \sqrt{ \frac{\hat{p}(1 - \hat{p})}{n} } \] where \( \hat{p} \) is the sample proportion, \( z^* \) is the critical value based on the confidence level, and \( n \) is the sample size.

Interpretation: "We are C% confident that the true population proportion lies between [lower bound] and [upper bound]."

Tip: Use the 1-PropZInt function on your calculator (TI-84).

Setting Up and Carrying Out a Test for a Population Proportion

Significance tests assess whether sample results provide convincing evidence against a claim about a population proportion.

Step 1: Hypotheses
Null Hypothesis: \( H_0: p = p_0 \)
Alternative Hypothesis: \( H_a: p < p_0 \), \( p > p_0 \), or \( p \ne p_0 \)
Step 2: Conditions (same as confidence intervals)
Step 3: Test Statistic \[ z = \frac{ \hat{p} - p_0 }{ \sqrt{ \frac{p_0(1 - p_0)}{n} } } \]
Step 4: P-value (use normal distribution to find area under curve)
Step 5: Conclusion (Compare p-value to \( \alpha \))

Tip: Use the 1-PropZTest function on your calculator.

Interpreting a P-value and Justifying a Claim

P-value: The probability, assuming \( H_0 \) is true, that the sample results are as extreme or more extreme than what was observed.
Low p-value (≤ α): Reject \( H_0 \). There is convincing evidence for \( H_a \).
High p-value (> α): Fail to reject \( H_0 \). There is not convincing evidence for \( H_a \).
Justification: Link your conclusion to context and whether evidence supports or does not support the claim.

Example: If p-value = 0.03 and α = 0.05, conclude "Since 0.03 < 0.05, we reject the null hypothesis. There is convincing evidence that..."

Hypothesis Test for a Single Proportion

State Hypotheses:
\(H_0: p = p_0\)
\(H_a: p <, \ne, \text{or } > p_0\)

Conditions: random, independent, and \(np_0\ge10\), \(n(1-p_0)\ge10\)
Test Statistic:

\[ z = \frac{p̂ - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

p‑value: Use normal table — area beyond |z| (two‑sided) or tail in direction of \(H_a\).
Decision: If p‑value ≤ α, reject \(H_0\); else fail to reject.
Conclusion: Contextualize in terms of original claim.

Example:
Claim: 60% of students like Stats → \(H_0: p=0.60\); Sample: 128/200 → \(p̂=0.64\).
Check: \(n p_0 = 200·0.60 = 120\ge10\); OK.
SE = \(\sqrt{0.60·0.40/200} ≈ 0.0346\).
z = \((0.64−0.60)/0.0346 ≈ 1.16\).
normalcdf(-1.16,1.16,0,1)= 0.751786
1-0.751786
Two‑tailed p ≈ 0.248 → fail to reject at α=0.05; insufficient evidence that proportion ≠ 0.60.

Type I and Type II Errors

Type I Error: Rejecting \( H_0 \) when it's actually true.
Type II Error: Failing to reject \( H_0 \) when it's actually false.
α (significance level): Probability of making a Type I error.
Power of a test: The probability of correctly rejecting \( H_0 \) (1 - probability of Type II error).

Tip: Think of courtroom analogy — Type I is convicting an innocent person, Type II is letting a guilty person go free.

Inference for the Difference of Two Proportions

This is used to compare two population proportions \( p_1 \) and \( p_2 \) using samples.

Conditions:
Both samples must be random and independent.
10% condition for both samples.
Large counts: \( n_1\hat{p}_1, n_1(1 - \hat{p}_1), n_2\hat{p}_2, n_2(1 - \hat{p}_2) \geq 10 \)

Confidence Interval: \[ (\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{ \frac{ \hat{p}_1(1 - \hat{p}_1) }{ n_1 } + \frac{ \hat{p}_2(1 - \hat{p}_2) }{ n_2 } } \] Significance Test (pooled):

Pooled proportion: \[ \hat{p}_{\text{pooled}} = \frac{ x_1 + x_2 }{ n_1 + n_2 } \]

Test statistic: \[ z = \frac{ (\hat{p}_1 - \hat{p}_2) }{ \sqrt{ \hat{p}_{\text{pooled}} (1 - \hat{p}_{\text{pooled}}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } } \]

Calculator Tip: Use 2-PropZInt for confidence intervals and 2-PropZTest for significance tests.

Summary Table

Topic	Calculator Function	Key Formula
1-Proportion Confidence Interval	1-PropZInt	\( \hat{p} \pm z^* \sqrt{ \frac{ \hat{p}(1 - \hat{p}) }{ n } } \)
1-Proportion z-Test	1-PropZTest	\( z = \frac{ \hat{p} - p_0 }{ \sqrt{ \frac{p_0(1 - p_0)}{n} } } \)
2-Proportion Confidence Interval	2-PropZInt	\( (\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{ \frac{ \hat{p}_1(1 - \hat{p}_1) }{ n_1 } + \frac{ \hat{p}_2(1 - \hat{p}_2) }{ n_2 } } \)
2-Proportion z-Test	2-PropZTest	\( z = \frac{ (\hat{p}_1 - \hat{p}_2) }{ \sqrt{ \hat{p}_{\text{pooled}} (1 - \hat{p}_{\text{pooled}}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } } \)

Tips & Common Mistakes

-Always check conditions before computing unless the question states "All conditions have been met"
-Use pooled vs. unpooled SE correctly for tests vs. CIs.
-Confidence level \(C\) relates to \(z^*\). For example:
90%: \(z^* = 1.645\)
95%: \(z^* = 1.96\)
99%: \(z^* = 2.575\)

CI measures precision; hypothesis test assesses evidence against \(H_0\).
Don’t confuse margin of error (\(z^*·SE\)) with \(SE\) itself.