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Sample proportions are calculated by dividing the number of favorable outcomes by the total number of observations in a sample. In hypothesis testing, understanding sample proportions is crucial because they help assess the behavior of a population based on a smaller subset.

In our example, we have two groups: single individuals and married individuals. For the single individuals, 120 out of 400 policies involved accidents. Thus, the sample proportion for singles is calculated as \[ \hat{p}_1 = \frac{120}{400} = 0.30 \]

For the married individuals, 150 out of 600 policies involved accidents. This yields a sample proportion of \[ \hat{p}_2 = \frac{150}{600} = 0.25 \]

These proportions represent the observed likelihood of accidents happening in each group over the given period. They are key inputs for further analysis in hypothesis testing.

The pooled proportion is an average proportion that weights both sample proportions by their respective sample sizes. It is used when testing for differences between two population proportions.

To find the pooled proportion, we sum the number of favorable outcomes from both samples and divide by the total number of observations across both samples. In the problem, we have:\[ \hat{p} = \frac{120 + 150}{400 + 600} = \frac{270}{1000} = 0.27 \]

This pooled proportion (\( 0.27 \)) is crucial since it provides a common reference point for determining the variability when comparing two proportions. It is especially important when using formulas to calculate the test statistic for differences between proportions.

A test statistic in hypothesis testing compares the observed data to what is expected under the null hypothesis. It helps to determine whether the results are statistically significant.

In tests involving proportions, the formula for the test statistic (\( z \) value) considers the difference between sample proportions, the pooled proportion, and the sample sizes. For our example:\[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p} (1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}} \]

Plugging in our values (\( n_1 = 400 \), \( n_2 = 600 \)), we compute:\[ z = \frac{0.30 - 0.25}{\sqrt{0.27 \times 0.73 \times \left( \frac{1}{400} + \frac{1}{600} \right)}} \]

This test statistic helps determine if the difference in proportions is significant enough to reject the null hypothesis.

In hypothesis testing, a critical value defines the threshold beyond which we would reject the null hypothesis. It is derived from the desired level of significance, commonly denoted by \( \alpha \). A high significance level means strong evidence is needed to reject the null hypothesis.

For a two-tailed test at the 0.05 significance level, the critical z-values are -1.96 and 1.96. These values define the rejection regions in a standard normal distribution.

In our exercise, the calculated z-score of approximately 1.78 does not exceed the boundaries set by our critical values. Since it lies within these bounds, we fail to reject the null hypothesis. This implies that the observed difference in accident proportions between single and married individuals is not statistically significant at the 5% level.

Understanding critical values is vital as they tell us whether the effect observed in the sample is likely to exist in the population.