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When performing hypothesis testing, it's crucial to calculate a test statistic that helps us determine whether observed data statistically differ from what we expect under the null hypothesis—that is, the hypothesis of no effect or no difference. Here, we're testing the null hypothesis that the proportion, denoted by `p`, is 0.25. To calculate the test statistic, typically a Z-score in this context, we use a specific formula:

\[ z = \frac{ \hat{p} - p_{0} }{ \sqrt{ \frac{p_{0}(1-p_{0})}{n} } } \]
In this formula, \( \hat{p} \) is the sample proportion, \( p_{0} \) is the hypothesized population proportion, and \( n \) is the sample size. This Z-score represents how many standard deviations our sample proportion is from the hypothesized proportion. The calculated Z-score gives us a standardized value that can be used to directly compare against a distribution that we know, the standard normal distribution. After following the calculations provided in the exercise, we determine the Z-score to be approximately -2.08, providing a basis for further interpretation in the hypothesis test.

After calculating the test statistic, we interpret the p-value to understand the strength of evidence against the null hypothesis. The p-value quantifies the probability of observing a test statistic as extreme as, or more extreme than, the one computed from the sample data, assuming the null hypothesis is true.

In this case, the p-value associated with our Z-score of -2.08 is approximately 0.0188. What does this mean? If the true population proportion were indeed 0.25, there would be about a 1.88% chance of finding a sample proportion as low as 0.16 or lower in a sample of 100 individuals. If the p-value is less than the predetermined significance level—commonly set at 0.05 or 5%—it suggests that such an extreme result is unlikely to occur by chance alone, and hence, we would reject the null hypothesis. Since our computed p-value is indeed less than 0.05, our evidence is strong enough to suggest the population proportion is indeed less than 0.25, supporting the alternative hypothesis.

The Z-score is a powerful tool in statistics that allows the comparison of individual data points to a standard. It is computed taking the value of interest, subtracting the expected value under the null hypothesis, and then dividing this difference by the standard deviation of the expected value.

When the exercises asks us to compute the Z-score, we are essentially standardizing our sample result to see how far off it is from what we would typically expect if the null hypothesis were true. To perform the computation, we follow the steps meticulously. First, we find the difference between our sample proportion, \( \hat{p} \), and the hypothesized proportion, \( p_0 \). Next, we find the standard error of the sample proportion, which is the denominator of our Z-score formula, using the hypothesized proportion and the sample size. Finally, we divide the difference by the standard error to obtain the Z-score.

Correct computation of the Z-score is essential, as it is the linchpin in determining whether the observed data are statistically significantly different from what the null hypothesis proposes.