[!def] Null and Alternative Hypotheses The null hypothesis (\(H_0\)) often represents a skeptical perspective or a claim to be tested. The alternative hypothesis (\(H_A\)) represents an alternative claim under consideration and is often represented by a range of possible parameter values.

Just because we don't find supporting evidence for the null hypothesis doesn't mean that the alternative is true. Think about a court case: even if the jurors leave unconvinced of guilt beyond a reasonable doubt, this does not mean they believe the defendant is innocent.

Example

Question: How many of the world's 1 year old children have been vaccinated against some disease: a. 20% b. 50% c. 80%

We ask this question to randomly selected people with a college degree and would like to determine if their guesses are as accurate as randomly guessing. That is, the proportion of people who pick the correct answer, is about 33.3%. This is our null hypothesis: \(H_0: p = 0.333\) \(H_A: p \ne 0.333\)

We want to make a conclusion about the population parameter \(p\). The value we are comparing the parameter to is called the null value - in this case, 0.333.

We have sample data for 50 sampled adults. Of this group, 24% got the question correct. We want to know if this deviation from 33% is simply due to random chance, or if the data provides strong evidence that the population proportion is different from 33%.

First, we need to check whether it is reasonable to construct a confidence interval for \(p\) using the sample data. If so, we construct a 95% confidence interval.

• Success-failure condition is passed
  • \(\hat{p} = 0.24\)
  • \(n\hat{p} = 12\), \(n(1-\hat{p})=38\)
  • Trials are independent
• To construct the confidence interval, we need the point estimate (0.24), the critical value for the 95% confidence interval (\(z^* = 1.96\)), and the standard error of \(\hat{p}\) which in this case is \(0.060\). We can then construct the confidence interval: We are 95% confident that the proportion of all college-educated adults to answer this question correctly is between 12.2% and 35.8%

Since the null value (0.333) falls within the range of plausible value from the confidence interval, we cannot say the null value is implausible.

[!info] Compare the p-value to \(\alpha\) to evaluate \(H_0\) When the p-value is less than the significance level, \(\alpha\), reject \(H_0\), and report that the data provide strong evidence supporting the alternative hypothesis. When the p-value is greater than \(\alpha\), do not reject \(H_0\), and report that we do not have sufficient evidence to reject the null hypothesis.