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A z-test is a statistical method used to determine if there is a significant difference between two population proportions. In this exercise, we want to see if a smaller percentage of adults aged 33 to 49 rate landline phones highly compared to those aged 50 to 68. By collecting sample data and calculating the difference between two proportions, we use the z-test to check for significance.
In a z-test, the difference between sample means or proportions is measured against what we would expect due to random chance alone. For this, we calculate a z-score through the formula:

• \[z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}\]

Here, \(\hat{p}_1\) and \(\hat{p}_2\) are the sample proportions, and \(\hat{p}\) is the pooled proportion. The resulting z-score tells us how many standard deviations the observed difference is from the expected difference of zero. If the z-score is large enough, it indicates a significant effect, guiding us to potential conclusions.

The significance level, often denoted by \(\alpha\), is a benchmark in hypothesis testing that helps decide if a result is statistically significant. In this problem, the chosen significance level is 0.05, or 5%. This means there is a 5% risk of concluding that there is a difference when there isn't one.
Setting this level helps control for Type I errors – false positives.

• If the p-value from the test is less than \(\alpha\), we reject the null hypothesis, suggesting the observed effect is real.
• If the p-value is greater than \(\alpha\), we fail to reject the null, meaning we don’t have enough evidence to support an effect or difference.

Choosing an appropriate significance level is vital as it reflects the risk you are willing to take in making a wrong inference. In scientific research, a 5% level is standard, balancing risk and sensitivity.

The p-value is a crucial part of hypothesis testing, representing the probability of obtaining the test results under the null hypothesis. This allows for understanding if the evidence is strong enough to reject the null hypothesis in favor of the alternative.
In the context of a z-test:

• We compute \(p\text{-value} = P(Z < z)\), especially useful since our alternative hypothesis involves checking if something is significantly smaller.
• A small p-value (typically less than 0.05) indicates strong evidence against the null hypothesis, suggesting a true effect.

Consider the p-value as a measure of the strength of evidence. It's not about the magnitude but about where the sample statistic falls within the sampling distribution under the null hypothesis. By comparing this with the significance level, researchers make informed decisions.

Formulating null \((H_0)\) and alternative \((H_a)\) hypotheses is the foundation of hypothesis testing. These hypotheses provide a framework to test claims using statistical evidence. In this exercise, let \(p_1\) be the proportion for ages 33 to 49, and \(p_2\) for ages 50 to 68.

• **Null Hypothesis \((H_0)\):** \(p_1 - p_2 \geq 0\)
  This suggests that the proportion of people in the younger age group rating landlines as top is not less than that of the older group.
• **Alternative Hypothesis \((H_a)\):** \(p_1 - p_2 < 0\)
  This predicts that the younger group is significantly less likely to rate landlines as top than the older group.

Testing hypotheses involves collecting data and using statistical techniques to see if the evidence supports the claim made by the alternative hypothesis over the null. It is the structure around which the entire testing process revolves.