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Understanding null and alternative hypotheses is the foundation of hypothesis testing. The null hypothesis (\(H_0\) ) represents the status quo or a position of skepticism that there is no effect or no difference.
For our candy company example, the null hypothesis states that half or fewer people will prefer candy bar I - formally written as
\[\begin{equation}H_0: p \<= 0.5.\end{equation}\]The alternative hypothesis (\(H_a\) or \(H_1\)), on the other hand, denotes what the researcher is seeking evidence for. It challenges the status quo.In this case, the alternative hypothesis avers that more than half of the population prefers candy bar I, expressed as\[\begin{equation}H_a: p > 0.5.\end{equation}\]It's important to note that the null hypothesis is assumed true until evidence suggests otherwise. The aim of hypothesis testing is to determine whether there is enough statistical evidence in the sample data to reject the null hypothesis in favor of the alternative. If the evidence suggests the contrary, the null hypothesis cannot be rejected.

The concept of a critical value is related to the level of significance of a test, usually denoted by \(\alpha\) . It serves as a threshold, which determines whether the observed test statistic is extreme enough to reject the null hypothesis.In hypothesis testing, we compare the test statistic to this critical value. If the test statistic falls within critical regions (beyond the critical value in the direction of the alternative hypothesis), it suggests that the null hypothesis is unlikely.For the given candy bar preference test at a 0.05 significance level (\(\alpha = 0.05\) ), the critical value is the point in the distribution that has 5% of the observations above it in a one-tailed test. For \(\alpha = 0.05\) in a one-tailed Z-distribution, the critical value is approximately 1.645.This means if our calculated Z-statistic is greater than 1.645, we have sufficient evidence at the 5% level of significance to reject the null hypothesis.

The Z-statistic is a type of standard score that indicates how many standard deviations an element is from the mean. In hypothesis testing, the Z-statistic is used to measure the difference between the sample statistic and the null hypothesis. It's calculated using the following formula:\[\begin{equation}Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\end{equation}\]where \(\hat{p}\) is the sample proportion, \(p_0\) is the hypothesized value of population proportion under the null hypothesis, and \(n\) is the sample size.In our candy bar scenario, the Z-statistic was calculated to be 2.449 which is more about what happens when \(\alpha = 0.05\) for a one-tailed test. Since 2.449 exceeds the critical value of 1.645, the null hypothesis is rejected. This means we have found significant evidence to suggest that more than half of the public may indeed prefer candy bar I.