Chapter Notes: Concluding a Test for a Population Proportion

Wrapping Up Your Hypothesis Test

After formulating hypotheses, verifying conditions, and computing your test statistics, the final step is to interpret the results. What do these numbers tell us? The conclusion of a hypothesis test for a population proportion boils down to one key decision: reject or fail to reject the null hypothesis, based on the probability associated with your test statistic.

Understanding the Decision Process

In a hypothesis test, we collect data and calculate a test statistic to find the probability (p-value) of observing our sample results, or something more extreme, assuming the null hypothesis is true. This p-value is compared to a predetermined significance level, known as alpha (α), typically set at 0.05. Here's how it works:

• If the p-value ≤ α (e.g., 0.05), we reject the null hypothesis, suggesting evidence supports the alternative hypothesis.
• If the p-value > α, we fail to reject the null hypothesis, indicating insufficient evidence to support the alternative.

Important: Never say you "accept" the null or alternative hypothesis! Always use "reject" or "fail to reject." Failing to reject the null doesn't prove it's true-it simply means the evidence isn't strong enough to dismiss it.

Using the P-Value to Make Decisions

The p-value represents the likelihood of obtaining your sample results if the null hypothesis is true and the sampling distribution is normal, centered at the null value. Compare the p-value to the significance level (α, usually 0.05):

• Low p-value (≤ α): The results are unlikely to occur by chance, providing evidence to reject the null hypothesis in favor of the alternative.
• High p-value (> α): The results are not unusual under the null hypothesis, so we fail to reject it. This doesn't confirm the null hypothesis as true, only that there's not enough evidence to reject it.

A p-value below 0.05 (5%) is typically considered statistically significant, meaning the results are unlikely under the null hypothesis, supporting the alternative. A higher p-value suggests the observed data aligns with the null hypothesis, offering no strong evidence for the alternative.

Using Z-Scores for Conclusions

Alternatively, you can use the z-score, which measures how many standard deviations your test statistic is from the mean of the sampling distribution. Since we've verified the normal condition, we can apply the Empirical Rule (68-95-99.7 rule), which states that 95% of data in a normal distribution lies within two standard deviations of the mean.

• A z-score greater than 2 or less than -2 (outside the 95% range) indicates a low probability (likely < 0.05) of occurring by chance, supporting rejection of the null hypothesis.
• A z-score between -2 and 2 suggests the result is not unusual, so we typically fail to reject the null hypothesis.

Large z-scores (e.g., 4 or -4) make rejecting the null hypothesis straightforward, as they indicate highly unlikely outcomes under the null.

Conclusion Template for a One-Proportion Z-Test

Use this structure to write a clear conclusion for a one-proportion z-test:
"Since the p-value [insert p-value] is [less than/greater than] the significance level (α = [insert α]), we [reject/fail to reject] the null hypothesis. We [have/do not have] sufficient evidence to suggest [state the alternative hypothesis in context]."

Three Key Elements for a Strong Conclusion

To ensure a complete conclusion (and maximize credit on exams), include these three components:

1. Compare the p-value to the significance level (α).
2. State whether you reject or fail to reject the null hypothesis.
3. Provide context by linking the conclusion to the true population proportion.

Practice Problem: Advertising Campaign Effectiveness

A survey tests whether a new advertising campaign boosts brand awareness. The null hypothesis states the campaign has no effect, while the alternative hypothesis claims it increases awareness. A sample of 500 people is split evenly: 250 see the campaign, and 250 do not. The proportion aware of the brand is 0.7 in the campaign group and 0.5 in the control group. The test statistic is z = 2.8, with a significance level of α = 0.05.
Questions:

• What is the p-value?
• What is the conclusion about the null hypothesis?

Answer: The p-value is 0.0026, indicating a 0.26% chance of observing a test statistic as extreme as 2.8 if the null hypothesis is true. Since 0.0026 < 0.05, we reject the null hypothesis. This provides evidence that the advertising campaign effectively increases brand awareness.

Key Terms to Understand