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When we talk about the null hypothesis, we refer to the initial claim that there is no effect or no difference. In the context of our exercise, the null hypothesis is that the contribution rate to the Paralyzed Veterans of America has not changed from the historical rate of 5%. Formally, this is stated as \(H_0: p = 0.05\). It asserts the idea that any observed difference in sample proportions is due to chance or random sampling error.

To determine if the null hypothesis can be rejected, we would need to calculate the probability of getting a sample proportion as extreme as, or more extreme than the one observed, assuming the null hypothesis is true. This is where statistical significance comes into play.

The alternative hypothesis proposes that there is a real effect or a difference. It is what we seek evidence for in a hypothesis test. For our scenario, the alternative hypothesis is that the contribution rate has indeed decreased due to the redesign of the letter. Mathematically, we express this hypothesis as \(H_1: p < 0.05\), suggesting that the new rate is less than 5%.

If statistical testing shows that the data collected provides substantial evidence to support the alternative hypothesis, we can reject the null hypothesis in favor of this one. It's important to avoid confusion between 'accepting' the alternative hypothesis and simply rejecting the null; we do not prove the alternative hypothesis, we merely acknowledge it as more consistent with the data than the null hypothesis.

The term 'statistical significance' is used to decide if we should attribute our findings to a real effect rather than random chance. It is measured against a pre-set threshold called the significance level, often denoted by \(\text{\(\alpha\)}\). A common practice is to use \(\alpha = 0.05\) or \(\alpha = 0.01\).

In our exercise, we would calculate the probability of observing a sample proportion of 4.78% or less, given that the null hypothesis is true (\(p = 0.05\)). If this probability, known as a p-value, is less than the significance level, the result is deemed statistically significant, and we would reject the null hypothesis. The absence of a stated \(\alpha\) or a p-value in the provided information means we cannot definitively conclude the exercise without further calculations or additional information.

Sample proportion refers to the percentage in a sample that has a certain characteristic—in this case, the proportion of potential donors who actually made a donation. For the Paralyzed Veterans of America, the sample proportion is the 4781 donations out of 100,000 potential donors, which is 4.78%.

This figure is crucial in hypothesis testing as it is compared against the historical or expected proportion to determine if there has been a change. Understanding the sample proportion helps us to analyze and interpret the results of the test, but it's important to note that due to sampling variability, there can be a difference between the sample proportion and the true population proportion. In our exercise, further statistical analysis would be necessary to ascertain if the observed sample proportion of 4.78% is significantly different from the expected 5%.