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A proportion test is a statistical method used to compare the proportion of a sample to a known population proportion. In our example of Chicken Delight, the company claims that 90% of its orders are delivered within 10 minutes. This claim represents the population proportion, denoted by \( p = 0.90 \).
The proportion test involves taking a sample—in this case, a sample of 100 orders with 82 delivered on time—and checking if this sample proportion is statistically different from the population proportion.
To perform a proportion test, we calculate the sample proportion, \( \hat{p} = \frac{82}{100} = 0.82 \), and use this to compute the test statistic. This statistic helps determine whether the observed sample proportion is significantly different from the hypothesized proportion. In hypothesis testing, calculating the test statistic is essential for making informed conclusions based on data.

In hypothesis testing, the significance level, denoted \( \alpha \), is the probability of rejecting the null hypothesis when it is actually true. It essentially measures the "risk" we are willing to take in making this error, also known as a Type I error.
For Chicken Delight, the significance level is set at \( \alpha = 0.10 \), meaning there's a 10% chance of concluding that less than 90% of orders are delivered within 10 minutes when, in fact, the claim holds true.
Choosing the significance level can have implications for test outcomes. A lower \( \alpha \) is more stringent but reduces the risk of Type I errors. In practice, common significance levels are 0.05, 0.01, or 0.10, depending on the situation and the desirable balance between rigor and risk.
It's crucial to set this threshold before conducting the test to prevent bias and ensure the integrity of conclusions.

The critical value is a point on the scale of the test statistic beyond which we will reject the null hypothesis. It plays a key role in hypothesis testing by setting the decision boundary. For a given significance level in a left-tailed test, the critical value can be found using statistical tables or software.
In our Chicken Delight example, the critical value was determined based on a left-tailed test at \( \alpha = 0.10 \). This meant looking for the point in a standard normal distribution where only 10% of values lie to the left. The critical value for \( \alpha = 0.10 \) in this setting is approximately \( z = -1.28 \).
Once the critical value is known, we compare it with the test statistic. If the test statistic falls beyond the critical value, the null hypothesis is rejected. In this case, since the test statistic \( z = -0.8431 \) did not surpass \( z = -1.28 \), we failed to reject the null hypothesis, indicating insufficient evidence that less than 90% of orders are delivered within the promised time. This comparison highlights whether or not the sample data provides strong enough evidence to support a conclusion.