Hypothesis Testing for Population Proportions: One-Sample and Two-Sample Inference

Foundations of Hypothesis Testing

• Definition of Hypothesis Testing: A formal statistical procedure that enables choosing between two competing claims when there is inherent variability in measurements.
• The Four Steps of Hypothesis Testing:
  • Hypothesize: Formulate a specific claim (the alternative hypothesis) to be tested against a neutral, skeptical baseline claim (the null hypothesis).
  • Prepare: Select the significance level ($\alpha$), choose the appropriate test statistic, check that all underlying mathematical conditions and assumptions are satisfied, and confirm that the sample size is sufficiently large to minimize decision errors.
  • Compare and Compute: Gather sample data, compare the observed sample value to the null expected value, compute the test statistic, and calculate the p-value to quantify the degree of surprise under the null hypothesis.
  • Interpret: Decide whether to reject or fail to reject the null hypothesis by comparing the p-value to $\alpha$, and state a clear conclusion within the real-world context of the data.
• Introductory Coin-Spinning Example:
  • Hypothesis: A coin is spun on a flat surface. It is hypothesized that the coin is more likely to land on heads because the heads side bulges out slightly relative to the tails side.
  • Data Collection: The coin is spun $20$ times, resulting in heads on all $20$ spins.
  • Step-by-Step Breakdown:
    • Hypothesize: Test the claim that heads is more likely ($p > 0.50$) against the neutral hypothesis that the coin is fair ($p = 0.50$).
    • Prepare: Use the sample proportion of heads across $n = 20$ spins as the decision metric.
    • Compare and Compute: If the coin were fair, approximately $50\%$ of spins ($10$ heads) would be expected. Obtaining $20$ heads in $20$ spins is exceptionally far above expectation.
    • Interpret: Obtaining $20$ heads in $20$ spins is extremely unusual under the assumption of a fair coin. Therefore, conclude that the coin is genuinely more likely to land on heads when spun.

Formulating Null and Alternative Hypotheses

• The Null Hypothesis ($H_0$):
  • Represents the neutral, status quo, or skeptical statement about a population parameter.
  • Always written with an equality sign ($=$).
  • Receives the benefit of the doubt throughout the entire testing procedure.
  • Is only rejected when the observed sample data would be extraordinarily rare or unusual if $H_0$ were true.
  • Legal Analogy: Analogous to a criminal jury trial, where a defendant is presumed innocent until proven guilty "beyond a reasonable doubt."
• The Alternative Hypothesis ($H_a$):
  • Represents the research hypothesis or claim of change that the investigation seeks to support.
  • Always formulated using an inequality symbol: strictly less than ($<$), strictly greater than ($>$), or not equal to ($\neq$).
• Structure of One-Sided and Two-Sided Hypotheses:
  • One-Sided (Left-Tailed):
    • $H_0: p = p_0$
    • $H_a: p < p_0$
  • One-Sided (Right-Tailed):
    • $H_0: p = p_0$
    • $H_a: p > p_0$
  • Two-Sided:
    • $H_0: p = p_0$
    • $H_a: p \neq p_0$
• Retail Internet Sales Case Study:
  • Context: In the previous fiscal year, $30\%$ ($p = 0.30$) of a retailer's total sales were generated online. The retailer subsequently invested in social media advertising and sampled sales across $20$ random days during the current fiscal year.
  • Two-Sided Question: "Has there been a change in the percentage of online sales?"
    • $H_0: p = 0.30$
    • $H_a: p \neq 0.30$
• One-Sided Question: "Have sales increased?"
  • $H_0: p = 0.30$
  • $H_a: p > 0.30$

The Test Statistic and Standard Error for Proportions

• Conceptual Role of a Test Statistic:
  • Quantifies the difference between the observed sample outcome and the value expected under the null hypothesis.
  • Standardizes this discrepancy by expressing it in terms of standard deviations (standard errors).
  • When the test statistic lies far from zero in the direction specified by the alternative hypothesis, it provides evidence to reject $H_0$.
• General Formula:
  • $\text{Test Statistic} = \frac{\text{Observed Value} - \text{Null Value}}{\text{Standard Error}}$
• One-Sample Proportion Formula ($z$-Statistic):
  • $z = \frac{\hat{p} - p_0}{SE}$
  • $SE = \sqrt{\frac{p_0(1 - p_0)}{n}}$
  • $\hat{p}$: The observed sample proportion, calculated as $\hat{p} = \frac{x}{n}$, where $x$ is the number of successes and $n$ is the total number of observations.
  • $p_0$: The hypothesized population proportion from the null hypothesis ($H_0$).
  • $SE$: The standard error of the sample proportion computed under the assumption that $H_0$ is true.

The p-Value and the Significance Level

• The p-Value as a Metric of Surprise:
  • Definition: The probability of obtaining a test statistic that is as extreme as or more extreme than the observed value, assuming the null hypothesis is true.
  • A small p-value indicates that the observed sample data would be exceptionally unusual under $H_0$, casting strong doubt on the null hypothesis.
  • As the magnitude of the test statistic ($|z|$) increases, the tail area beyond it decreases, resulting in a smaller p-value.
• Determining p-Values from the Alternative Hypothesis:
  • Two-Sided Test ($H_a: p \neq p_0$): The p-value equals the combined area of both tails beyond $\pm |z|$.
  • 
  • Right-Tailed Test ($H_a: p > p_0$): The p-value equals the area under the standard normal curve to the right of the test statistic ($z$).
  • 
  • Left-Tailed Test ($H_a: p < p_0$): The p-value equals the area under the standard normal curve to the left of the test statistic ($z$).
• The Significance Level ($\alpha$):
  • The predetermined threshold representing the maximum probability tolerated of incorrectly rejecting a true null hypothesis (Type I error).
  • Standard thresholds are commonly $\alpha = 0.05$, $\alpha = 0.01$, or $\alpha = 0.10$.
• Decision Rules:
  • If $p < \alpha$: Reject $H_0$. The evidence supporting the alternative hypothesis is statistically significant.
  • If $p \ge \alpha$: Fail to reject $H_0$. There is insufficient evidence to conclude that $H_0$ is false.

Required Assumptions and Conditions for One-Sample Proportion Tests

• Random Sampling Condition: The sample must be selected using a random sampling method (such as a simple random sample) to represent the target population without bias.
• Large Sample Size (Success/Failure Condition):
  • The expected number of successes must be at least $10$: $n p_0 \ge 10$.
  • The expected number of failures must be at least $10$: $n(1 - p_0) \ge 10$.
  • This condition ensures that the sampling distribution of $\hat{p}$ is approximately normal.
• Large Population Condition (10% Condition): When sampling without replacement, the total population size ($N$) must be at least $10$ times larger than the sample size ($n$):
  • $N \ge 10n$
• Independence Condition: Each observed individual or trial must have no influence on any other observation in the sample.

One-Sample Proportion Examples and Case Studies

• Global Warming Cool Skeptics Example (Gallup 2001 vs. 2014):
  • Context: In $2001$, Gallup reported that $12\%$ ($p_0 = 0.12$) of Americans were "cool skeptics" who worried "a little or not at all" about global warming. In $2014$, a poll of $n = 500$ Americans found $25\%$ ($\hat{p} = 0.25$) identifying as cool skeptics.
  • Hypotheses: $H_0: p = 0.12$ versus $H_a: p > 0.12$ (right-tailed).
  • Count of Successes: $x = n \times \hat{p} = 500 \times 0.25 = 125$.
  • Standard Error: $SE = \sqrt{\frac{0.12(1 - 0.12)}{500}} = \sqrt{\frac{0.12 \times 0.88}{500}} \approx 0.014532722$.
  • Test Statistic Calculation:
    • $z = \frac{0.25 - 0.12}{\sqrt{\frac{0.12(1 - 0.12)}{500}}} = \frac{0.13}{0.014532722} \approx 8.9453306 \approx 8.95$
    • The sample proportion lies $8.95$ standard errors above the hypothesized null value.
  • p-Value and Interpretation:
    • Using the standard normal distribution, the probability of obtaining a $z$-score greater than or equal to $8.95$ is $P(X \ge 8.95) < 0.0001$ (StatCrunch output displays $0$).
    • 
    • Conclusion: Because $p < 0.05$, reject $H_0$. There is sufficient evidence to conclude that the percentage of American cool skeptics has increased since $2001$.
  • StatCrunch Procedure: Navigate to Stat > Proportion Stats > One Sample > With Summary. Input $125$ successes, $500$ observations, set null proportion to $0.12$, and select $>$ for $H_A$.
• College Student Loan Dropout Rate Example:
  • Context: Time Magazine (June 11, 2012) reported that the dropout rate for all college students with loans is $30\%$. A random sample of $n = 200$ college students with loans reveals $65$ dropouts.
  • Hypotheses: $H_0: p = 0.30$ versus $H_a: p \neq 0.30$ (two-sided).
  • TI-84 1-PropZTest Results:
    • Test statistic: $z = 0.7715$
    • p-value: $p = 0.4404$
    • 
  • Conclusion: The p-value of $0.4404$ is substantially larger than $\alpha = 0.05$. Fail to reject $H_0$. There is insufficient evidence that the dropout rate differs from $30\%$.
• Support for the Death Penalty Example (Declining Trend):
  • Context: In $2009$, $63\%$ of Americans supported the death penalty. In $2014$, a Rasmussen Poll of $n = 943$ Americans found $538$ supported it.
  • Hypotheses: Testing for a decline gives $H_0: p = 0.63$ versus $H_a: p < 0.63$.
  • Test Metrics:
    • Sample proportion: $\hat{p} = \frac{538}{943} \approx 0.57051962$
    • Standard error: $SE \approx 0.015722271$
    • Test statistic: $z = -3.7831926$
    • p-value: $p < 0.0001$
  • Conclusion: Reject $H_0$. Sufficient evidence indicates that support for the death penalty among Americans has declined since $2009$.
• Public Library Funding Example:
  • Context: A random poll of $n = 250$ voters in a city found $140$ favoring increased funding for public libraries. Test whether a majority (more than $50\%$) favors funding at $\alpha = 0.01$.
  • Hypotheses: $H_0: p = 0.50$ versus $H_a: p > 0.50$.
  • Test Metrics:
    • Count: $140$, Total: $250$
    • Sample proportion: $\hat{p} = \frac{140}{250} = 0.56$
    • Standard error: $SE = \sqrt{\frac{0.50 \times 0.50}{250}} \approx 0.031622777$
    • Test statistic: $z = 1.8973666 \approx 1.90$
    • p-value: $p = 0.0289$
  • Conclusion: Compare $p = 0.0289$ to $\alpha = 0.01$. Because $0.0289 \not< 0.01$, fail to reject $H_0$. The evidence is not strong enough at the $1\%$ significance level to conclude that a majority of city voters favor increased funding.
• Sleepwalking Prevalence Example:
  • Context: Time Magazine (May 28, 2012) reported that $33\%$ of Americans have sleepwalked at least once. A sample of $n = 200$ people found $42$ who reported sleepwalking. Test whether the proportion differs from $0.33$ at $\alpha = 0.05$.
  • Hypotheses: $H_0: p = 0.33$ versus $H_a: p \neq 0.33$.
  • Test Metrics:
    • Count: $42$, Total: $200$
    • Sample proportion: $\hat{p} = \frac{42}{200} = 0.21$
    • Standard error: $SE = 0.03324906$
    • Test statistic: $z = -3.6091246$
    • p-value: $p = 0.0003$
  • Conclusion: Since $p = 0.0003 < 0.05$, reject $H_0$. The proportion of Americans who have sleepwalked is significantly different from $0.33$.
• Global Warming Belief Example (Gallup 2010):
  • Context: Historically (2001 to 2005), about $58\%$ ($p_0 = 0.58$) of Americans believed Earth's temperature was rising. A March $2010$ Gallup poll of $n = 1014$ adults found $527$ believing global warming was real. Test whether this proportion changed at $\alpha = 0.05$.
  • Hypotheses: $H_0: p = 0.58$ versus $H_a: p \neq 0.58$.
  • Sample Proportion: $\hat{p} = \frac{527}{1014} \approx 0.51972387$ ($51.97\%$), which is less than the historical $58\%$.
  • Test Metrics:
    • Standard error: $SE = 0.015499571$
    • Test statistic: $z = -3.8888906$
    • p-value: $p = 0.0001$
  • Conclusion: Reject $H_0$. In $2010$, the percentage of Americans who believe global warming is real changed significantly from the historical baseline of $58\%$.
• Taxes on the Wealthy Example:
  • Context: A poll of $n = 500$ randomly selected independent respondents found $281$ who believe the rich should pay more taxes. Test whether a majority favors higher taxes at $\alpha = 0.05$.
  • Hypotheses: $H_0: p = 0.50$ versus $H_a: p > 0.50$.
  • Test Metrics:
    • Sample proportion: $\hat{p} = \frac{281}{500} = 0.562$
    • Standard error: $SE = 0.02236068$
    • Test statistic: $z = 2.7727243$
    • p-value: $p = 0.0028$
    • 
  • Conclusion: Because $p = 0.0028 < 0.05$, reject $H_0$. There is sufficient evidence that a majority of the population believes the rich should pay more taxes.

Decision Errors in Hypothesis Testing

• Definitions of Decision Errors:
  • Type I Error: Rejecting the null hypothesis when $H_0$ is actually true.
    • The probability of committing a Type I error equals the significance level: $P(\text{Type I error}) = \alpha$.
  • Type II Error: Failing to reject the null hypothesis when $H_0$ is actually false.
• Hypothesis Testing Decision Matrix:
  • When $H_0$ is True & Action is Reject $H_0$: Type I error (probability $= \alpha$).
  • When $H_0$ is True & Action is Fail to Reject $H_0$: Good decision (correct inference).
  • When $H_0$ is False & Action is Reject $H_0$: Good decision (correct inference).
  • When $H_0$ is False & Action is Fail to Reject $H_0$: Type II error.
• School Bond Ballot Measure Example:
  • Hypotheses: $H_0: p = 0.50$ versus $H_a: p > 0.50$.
  • Type I Error Formulation: The analyst concludes that more than $50\%$ of voters favor the bond when, in reality, the actual proportion favoring the bond is $50\%$ or less.
  • Type II Error Formulation: The analyst concludes that $50\%$ or less favor the bond when, in reality, more than $50\%$ actually favor it.
  • Significance Level ($\alpha = 0.05$) Meaning: There is exactly a $5\%$ probability of mistakenly concluding that a majority favors the bond when the true support is $50\%$ or less.
• Balancing and Mitigating Errors:
  • Decreasing $\alpha$ lowers the probability of a Type I error, but automatically increases the probability of a Type II error.
  • The only method to simultaneously reduce the likelihood of both Type I and Type II errors is to increase the sample size ($n$), which improves estimator precision.
• Statistical versus Practical Significance:
  • A result is practically significant only if it is both statistically significant and practically meaningful.
  • Cancer Risk Metaphor: Suppose the baseline risk of developing a rare cancer is $1$ in $10\text{ million}$. A study finds that daily cell phone usage doubles this risk to $2$ in $10\text{ million}$. While the doubling of risk may be statistically significant in a massive sample, a risk of $2$ in $10\text{ million}$ is practically negligible and unlikely to alter human behavior.
• Mandatory Language Cautions:
  • Because sample estimates are subject to sampling variability, absolute certainty is impossible.
  • When the p-value is greater than or equal to $\alpha$, never state:
    • "We accept $H_0$."
    • "We have proved $H_0$ is true."
  • Permissible phrasing includes:
    • "We fail to reject $H_0$."
    • "We cannot reject $H_0$."

Confidence Intervals and Two-Sided Hypothesis Tests

• Relationship and Inherent Questions:
  • Confidence Interval Question: "What is the value of this parameter?"
  • Hypothesis Test Question: "Are the data consistent with the parameter being one particular value, or might the parameter be something else?"
• Equivalence for Two-Sided Tests:
  • A two-sided hypothesis test conducted at significance level $\alpha$ reaches conclusions consistent with a $(1 - \alpha) \times 100\%$ confidence interval.
  • If the hypothesized parameter value ($p_0$) falls inside the confidence interval, fail to reject $H_0$.
  • If $p_0$ lies completely outside the confidence interval, reject $H_0$.
• Mapping Confidence Levels to Significance Levels:
  • $90\%$ Confidence Level ($1 - \alpha = 0.90$) corresponds to a Two-Sided ($\neq$) test at $\alpha = 10\%$ ($0.10$).
• $95\%$ Confidence Level ($1 - \alpha = 0.95$) corresponds to a Two-Sided ($\neq$) test at $\alpha = 5\%$ ($0.05$).
• $99\%$ Confidence Level ($1 - \alpha = 0.99$) corresponds to a Two-Sided ($\neq$) test at $\alpha = 1\%$ ($0.01$).
• Death Penalty Example (Two-Sided Synthesis):
  • Hypotheses: $H_0: p = 0.63$ versus $H_a: p \neq 0.63$.
• Sample Data: $n = 943$, $x = 538$, $\hat{p} \approx 0.57051962$.
• Hypothesis Test Output: Test statistic $z = -3.7831926$, p-value $= 0.0002$. Since $p < 0.05$, reject $H_0$.
• 95% Confidence Interval Construction:
  • Using the Standard-Wald method:
  • Lower Limit: $0.53892601$
  • Upper Limit: $0.60211322$
  • Interval: $(0.54, 0.60)$
• Synthesis: The null value $p_0 = 0.63$ is not contained within the $95\%$ confidence interval $(0.54, 0.60)$. This directly confirms the hypothesis test rejection, proving that American support for the death penalty has changed.

Comparing Proportions Across Two Independent Populations

• Assumptions and Conditions for Two-Sample Tests:
  • Large Samples: Both sample sizes must be large enough to ensure normal approximations for each sample proportion.
  • Random Samples: Both samples must be selected randomly.
  • Independent within Samples: Observations within sample 1 must be independent of one another, and observations within sample 2 must be independent of one another.
  • Independent Samples: The two groups being compared must be completely independent of each other.
• Hypothesis Formulations for Two Proportions:
  • $H_0: p_1 = p_2$ (or equivalently, $H_0: p_1 - p_2 = 0$)
  • Alternative options:
    • $H_a: p_1 < p_2$ (or $p_1 - p_2 < 0$)
    • $H_a: p_1 > p_2$ (or $p_1 - p_2 > 0$)
    • $H_a: p_1 \neq p_2$ (or $p_1 - p_2 \neq 0$)
• Economic Confidence Example (Gallup 2014 vs. 2015):
  • Context: In January $2014$, Gallup reported that $45\%$ of Americans felt "pretty good" about their spending money ($n_1 = 3500$). In January $2015$, Gallup reported $49\%$ felt this way ($n_2 = 3500$). Test whether economic confidence improved at $\alpha = 0.05$.
  • Hypotheses: $H_0: p_1 - p_2 = 0$ versus $H_a: p_1 - p_2 < 0$.
  • Success Counts:
    • $x_1 = n_1 p_1 = 3500 \times 0.45 = 1575$
    • $x_2 = n_2 p_2 = 3500 \times 0.49 = 1715$
  • StatCrunch Output Results:
    • Sample difference: $\hat{p}_1 - \hat{p}_2 = -0.04$
    • Standard error: $SE = 0.011930753$
    • Test statistic: $z = -3.3526804 \approx -3.35$
    • p-value: $p = 0.0004$ ($0.00040406$)
    • 
  • Conclusion: Because $p = 0.0004 < 0.05$, reject $H_0$. The population proportion who felt confident was significantly higher in $2015$ than in $2014$.
  • StatCrunch Procedure: Navigate to Stat > Proportion Stats > Two Sample > With Summary. Enter Sample 1 ($1575$ successes, $3500$ observations) and Sample 2 ($1715$ successes, $3500$ observations). Select $<$ under the hypothesis test for $p_1 - p_2$.
• Gender Differences in Texting While Driving Example:
  • Context: A survey investigates whether texting while driving varies by gender. Data is collected from a random sample of $n_1 = 80$ men and $n_2 = 60$ women. Of the men, $x_1 = 38$ texted while driving; of the women, $x_2 = 27$ texted while driving. Test at $\alpha = 0.05$.
  • Hypotheses: $H_0: p_1 - p_2 = 0$ versus $H_a: p_1 - p_2 \neq 0$.
• StatCrunch Output Results:
  • Sample proportions: $\hat{p}_1 = \frac{38}{80} = 0.475$, $\hat{p}_2 = \frac{27}{60} = 0.45$
  • Sample difference: $\hat{p}_1 - \hat{p}_2 = 0.475 - 0.45 = 0.025$ ($2.5\%$)
  • Standard error: $SE = 0.085173143$
  • Test statistic: $z = 0.29351975 \approx 0.29$
  • p-value: $p = 0.7691 \approx 0.77$
• Conclusion: With $p = 0.77 \ge 0.05$, fail to reject $H_0$. An observed sample difference of $2.5\%$ is expected to occur frequently by chance when the true proportions are equal. There is insufficient evidence of a difference in texting while driving rates between men and women.

Conceptual Checks and Review Problems

• Extreme Test Statistics in Left-Tailed Tests:
  • Question: In a left-tailed test, which test statistic provides the strongest evidence against $H_0$: $-2.78$, $-1.64$, $0.001$, or $2.95$?
  • Answer: $-2.78$. The value farthest from $0$ in the negative direction provides the smallest left-tail area and lowest p-value. (In a right-tailed test, $2.95$ would provide the strongest evidence).
• Strong Evidence Metric:
  • In hypothesis testing, strong evidence against the null hypothesis occurs when the p-value is close to $0$.
• Stricter Gun Laws Example:
  • Context: In $2010$, $48\%$ of voters favored stricter gun control. A researcher tested $H_a: p > 0.48$, obtaining $z = 1.27$ and $p = 0.10$.
  • Correct Interpretation: At $\alpha = 0.05$, there is not enough evidence to conclude that the proportion has increased. (It is incorrect to state that "the proportion is still $48\%$" because a hypothesis test never proves $H_0$).
• Public Smiling Frequency by Gender Example:
  • Context: A researcher tests whether men (Population 1) smile less often in public than women (Population 2). Alternative hypothesis: $H_a: p_1 < p_2$. Output yields $z = -6.13$ and $p < 0.0001$.
  • Correct Interpretation: Evidence indicates that men smile less often in public than women.