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In statistics, understanding the population proportion is crucial when dealing with hypothesis testing.The population proportion, denoted as \( p \), is a parameter that represents the fraction of the population that exhibits a certain characteristic. For example, in our duck hunting survey, it represents the proportion of the local residents who strongly oppose hunting on Morro Bay.
To estimate this population proportion, we use a sample proportion, denoted as \( \hat{p} \).This is calculated by dividing the number of favorable outcomes by the total sample size.The sample proportion provides a point estimate of the population proportion and is foundational in constructing and testing hypotheses against it.

• Example: In a sample of 750 residents, 560 expressed strong opposition. So, \( \hat{p} = \frac{560}{750} = 0.747 \).

The Z-score calculation is a fundamental part of hypothesis testing. It helps us determine how far a sample proportion is from the assumed population proportion under the null hypothesis.The formula for the Z-score in the context of testing a proportion is:
\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1-p_0)}{n}}} \]
Where \( \hat{p} \) is the sample proportion, \( p_0 \) is the population proportion under the null hypothesis, and \( n \) is the sample size.
This formula allows researchers to standardize their observations, leading to a comparison on how unusual the sample result is under the null hypothesis.
To facilitate understanding, let's break down each component:

• Numerator \( (\hat{p} - p_0) \): The difference between the sample proportion and assumed population proportion.
• Denominator: Standard error of the proportion, reflecting variability within the data.

Calculating the Z-score gives us an indication of the rarity of our observation under the assumption that the null hypothesis is true.

The significance level \( \alpha \) is a pre-determined threshold used in hypothesis testing to decide whether to accept or reject the null hypothesis.It represents the probability of rejecting the null hypothesis when it is actually true (Type I error).
Typically, significance levels of 0.05, 0.01, or 0.10 are used, but this decision depends on the context and how stringent the criteria for evidence are.
In our example, we use a significance level of \( \alpha = 0.01 \).This means you are quite strict, accepting only 1% risk of concluding that the majority opposes hunting when they don't.
By choosing such a low \( \alpha \), the evidence must be very strong to conclude that there is majority opposition.Hence, a calculated test statistic must be exceptionally large compared to the critical value of the respective distribution.

A right-tailed test is employed when we are interested in determining if the sample statistic is significantly greater than the hypothesized parameter.In the context of testing our hypothesis regarding the population proportion, the alternative hypothesis \( H_1: p > 0.5 \) leads us to a right-tailed test.
Here, we specifically look for evidence that the proportion of residents opposing hunting on Morro Bay is more than 0.5.
This involves observing whether the calculated Z-score exceeds the critical value from the Z-distribution for our chosen significance level (1% in this case).
Using a right-tailed test involves focusing on the area to the right of the critical value in the standard normal distribution.

• Why Right-Tailed? Because our alternative hypothesis states "greater than," which naturally fits the right tail of the distribution.

Hence, if the Z-score we compute is in this right "extreme," we reject the null hypothesis in favor of the alternative.