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When we conduct hypothesis testing involving proportions, one important concept is the population proportion, denoted as \( \pi \). The population proportion \( \pi \) represents the fraction of individuals in a population that hold a particular trait or characteristic. In our exercise, the hypothesis is focused on whether the proportion \( \pi \) is less than or equal to 0.70. This means we are assessing if a significant number of individuals in the sample population possess or exhibit this characteristic to a degree greater than 70%.

Understanding population proportion through a sample helps us draw conclusions about a larger group based on observed data. In our case, the sample statistic \( \hat{p} = 0.75 \) indicates the observed proportion of individuals that have the characteristic in a sample of size 100. Such measurements help form the basis for decisions in hypothesis testing.

The test statistic is a crucial calculation in hypothesis testing. It helps us measure how far our sample statistic is from the hypothesized population parameter. For population proportions, the z-test is typically used.

In the context of the exercise, the test statistic formula is: \[z = \frac{{\hat{p} - \pi_0}}{\sqrt{\frac{\pi_0(1-\pi_0)}{n}}}\]
Where:

• \( \hat{p} \) is the sample proportion.
• \( \pi_0 \) is the population proportion in the null hypothesis.
• n is the sample size.

In this specific case, our sample test provided a z-score of approximately 1.091 after applying the values (\( \hat{p} = 0.75 \), \( \pi_0 = 0.70 \), and \( n = 100 \)) into the formula. The z-value tells us how many standard deviations away the sample proportion is from the null hypothesis' proportion.

The significance level, denoted as \( \alpha \), is a threshold that helps us decide whether to reject the null hypothesis. It is usually set by the researcher before analyzing the data. This level represents the probability of rejecting the null hypothesis when it is actually true, often set at 0.05 (5%).

In the exercise, the significance level is 0.05, indicating a 5% risk of concluding that the population proportion is different from what is stated in the null hypothesis when in fact it is not. Using this significance level allows for a clear decision rule. It balances the risk of making a Type I error (incorrectly rejecting a true null hypothesis) with the power to detect a real effect.

The decision rule is a structured plan for deciding whether to reject the null hypothesis. It depends heavily on the calculated test statistic and the predetermined significance level. In one-tailed tests, like the one in our example, the decision rule is based on whether the test statistic falls in a critical region.

For our exercise, with a significance level \( \alpha = 0.05 \), the critical z-value from the z-table is 1.645 for a one-tailed test. Therefore, the decision rule is:

• Reject the null hypothesis if the calculated z-value is greater than 1.645.
• Do not reject the null hypothesis if the z-value is less or equal to 1.645.

Since the calculated z-value (1.091) is not greater than the critical value, we stick to the null hypothesis. The decision rule provides a straightforward guide on which conclusion to reach based on your computations.