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In hypothesis testing, the null hypothesis, often denoted as \(H_0\), serves as a starting point for statistical testing. It represents a baseline assumption that there is no change or no difference in the population parameter we study. In simpler terms, it assumes that whatever effect or change we're looking for is not present.
For the manager's exercise with soda sales, the null hypothesis is set at \(H_0: p = 0.15\). This means the manager assumes that, even after introducing the free refill promotion, the proportion of customers ordering soda remains at 15%.
The purpose of establishing a null hypothesis is to test whether any observed effect (in this case, an increase in soda sales due to free refills) can be considered statistically significant, rather than due to random chance. If evidence shows that the data does not support the null hypothesis, it may be rejected in favor of an alternative explanation.

The alternative hypothesis, symbolized by \(H_a\), proposes what the researcher predicts or wants to prove. It stands in contrast to the null hypothesis by suggesting that there is indeed a significant effect or change in the parameter being observed.
In our soda sales scenario, the alternative hypothesis is formulated as \(H_a: p = 0.17\). This indicates that the manager expects the promotion to have successfully increased the proportion of customers purchasing soda to 17%.

• Unlike the null hypothesis, which suggests no effect, the alternative offers a potential outcome the manager is looking to support.
• A correct alternative hypothesis must reference the population proportion, not the sample proportion.

The importance of the alternative hypothesis lies in its role in hypothesis testing; it provides a guide for determining whether to reject the null hypothesis based on the statistical evidence collected. If the data supports this hypothesis, it implies a successful change in customer behavior due to the promotion.

The sample proportion, denoted as \(\hat{p}\), is a key concept in hypothesis testing and statistics. It represents the fraction or percentage of items in a sample that meet a particular criterion. Essentially, it gives us a snapshot of the sample data.
In the context of the soda sales exercise, \(\hat{p} = 0.17\) denotes that in the observed sample period during the promotion, 17% of customers ordered soda with their meal.
Understanding sample proportion is critical because:

• It is used to make inferences about the larger population from which the sample is drawn.
• This measure helps compare observed effects in the sample against the expectations stated in the null and alternative hypotheses.

While \(\hat{p}\) is invaluable for assessing sample data, hypothesis testing primarily focuses on inferring about the population proportion \(p\). It is important to differentiate these terms to ensure accurate testing and results interpretation.