91Ó°ÊÓ

In hypothesis testing, the critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. It helps to establish a boundary for making decisions about whether to accept or reject a hypothesis.
The critical value is determined by the significance level, denoted by \( \alpha \), which represents the probability of rejecting the null hypothesis when it is actually true (Type I error). For a significance level of 0.05 in a two-tailed test, the critical values are typically obtained using statistical tables or software.
In the context of the exercise with animal bites, we have a two-tailed test with \( \alpha = 0.05 \), giving us critical \( z \)-values of approximately \( \pm 1.96 \). This means any test statistic beyond these values would lead us to reject the null hypothesis at the 5% significance level.

A confidence interval provides a range of values, derived from the sample, that is likely to contain the population parameter. Confidence intervals offer an estimation of the unknown parameter and a measure of the reliability of this estimation.
The width of the confidence interval depends on the standard error and the critical value for the desired level of confidence. For a 95% confidence interval, as used in the exercise, the critical value is \( 1.96 \) (from a standard normal distribution), coupled with the calculated standard error.
In our problem, we use the formula for the difference in proportions:

• \( CI = (\hat{p}_1 - \hat{p}_2) \pm z \cdot SE \)

Substituting the values, we find:

• \( CI = (0.137 - 0.2) \pm 1.96 \cdot SE \)

This results in an interval of \((-0.218, 0.092)\), indicating the potential range for the difference in proportions.

The standard error (SE) measures the variability of a sample statistic from the actual population parameter. It is a critical component in constructing confidence intervals and calculating test statistics.
In the context of proportions, the standard error is calculated using the pooled sample proportion and involves the sizes of the two independent samples.
We use the formula for proportions:

• \[ SE = \sqrt{\hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } \]

Where \( \hat{p} \) is the pooled proportion and \( n_1 \), \( n_2 \) are the sample sizes for each group.
In this exercise, our calculations yield \( SE \approx 0.064 \), indicating the expected variation in the sample proportion difference if the null hypothesis is true.

The test statistic is a standardized value that is calculated from sample data during a hypothesis test. It is used to determine whether to reject the null hypothesis by comparing it with the critical value.
For proportion differences, the test statistic formula is:

• \[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \]

Where \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions, and \( SE \) is the standard error calculated earlier.
In our problem, the computed test statistic is approximately \( z = -0.986 \). This value is used to compare against our critical \( z \)-values to derive the conclusion of the hypothesis test. Because it does not exceed the critical values of \( \pm 1.96 \), the null hypothesis is not rejected, indicating no significant difference between the two samples.