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The term "incidence rate" refers to the frequency at which new cases of a particular condition occur within a specific population over a given time period. In the context of our study, we're looking at how often cataracts develop among a group of people. Here, the incidence rate tells us the proportion of individuals who developed cataracts over one year.

To calculate the incidence rate, we use the formula: \[\text{Incidence Rate} = \frac{\text{Number of new cases}}{\text{Total population at risk}}\]For our study, 4 people out of 200 developed cataracts, leading to an incidence rate of 2%. This calculation is crucial as it allows us to determine whether the observed rate stands out compared to general expectations, which in this case is the general population's 1% rate.

In statistics, the population proportion, denoted often by \( p_0 \), represents the expected percentage of a particular outcome within a larger population. When comparing two proportions, like those in this cataracts study, knowing both the sample and population proportions is key.

For our exercise, the population proportion is the expected incidence rate of cataracts among all people aged 65-69, which is 1% over one year. This serves as our benchmark during testing. Contrastingly, the observed sample proportion \( \hat{p} \) was calculated as 2% within our specific subgroup with high sunlight exposure. By comparing these two proportions, we can assess whether the sunlight exposure significantly affects the development of cataracts.

The one-sample z-test is a statistical method used to determine if there is a significant difference between an observed sample statistic and a known population parameter. In our context, we want to see if the sample incidence rate of cataracts differs from the expected rate of 1% in the general population.

To perform a one-sample z-test for a proportion, use the formula:\[z = \frac{\hat{p} - p_0}{\sqrt{ \frac{p_0 (1-p_0)}{n} }}\]Where:

• \( \hat{p} \) is the observed sample proportion (2% for our study),
• \( p_0 \) is the population proportion (1%),
• \( n \) is the sample size (200).

The outcome, the z-score, tells us how many standard deviations away our sample's incidence rate is from the population rate. By comparing this z-score to critical values from the z-table, we decide whether there's a statistically significant difference.

In hypothesis testing, the null hypothesis \( H_0 \) is a statement asserting that any observed effect is due to chance. It often resembles a statement of no effect or no difference. Here, our null hypothesis stipulates that the incidence rate of cataracts in people with excessive sun exposure is the same as that of the general population.

We mathematically express the null hypothesis as:\[H_0: \hat{p} = p_0\]where \( \hat{p} \) is the sample proportion and \( p_0 \) is the population proportion.

To test this, we calculate the z-score and reference it against a critical value determined by our chosen significance level (e.g., 0.05). If the calculated z-score exceeds the critical range, we reject the null hypothesis, suggesting a notable difference between the sample incidence rate and the population proportion. Rejecting \( H_0 \) implies that there is significant evidence that the added sunlight exposure alters cataract incidence rates.