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When conducting a hypothesis test, the significance level, symbolized as \( \alpha \), is a crucial component. It's essentially the threshold for how much risk of error we're willing to accept. In simpler terms, it's the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true.
In our scenario, the significance level is set at 0.05, or 5%. This means there is a 5% risk that we might incorrectly conclude that the mean price of gasoline in Milwaukee is higher than the national average.
Setting a proper significance level is important. Lower levels mean less risk of a false positive, but also might make it harder to detect a true effect, while higher levels increase risk but may be more sensitive to detecting true differences.

The test statistic is a value that helps us decide whether to reject the null hypothesis. It is calculated from your data and indicates how far, in standard deviation terms, the sample mean is from the population mean.
In this exercise, we calculate the test statistic using the formula: \[z = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \]where:

• \( \bar{x} \) is the sample mean (\\(2.12)
• \( \mu_0 \) is the population mean (\\)2.10)
• \( s \) is the sample standard deviation (\$0.05)
• \( n \) is the sample size (35)

Substituting the values gives us \( z \approx 2.37 \).
This statistic tells us how many standard deviations our sample mean is from the hypothesized mean. A larger absolute test statistic implies stronger evidence against the null hypothesis.

The p-value is a measure that helps you understand the strength of your test results. It represents the probability of obtaining a result that is at least as extreme as the observed result, under the assumption that the null hypothesis is true.
In the test we conducted, a p-value of 0.0089 was found corresponding to our test statistic \( z = 2.37 \).
A low p-value (typically \( \leq 0.05 \)) indicates strong evidence against the null hypothesis, so it is rejected. In this case, our p-value is well below the significance level of 0.05, suggesting that it is very unlikely for the Milwaukee gasoline prices to be the same as the national mean, supporting our alternative hypothesis.

The null hypothesis, denoted as \( H_0 \), is a statement we test, typically asserting no effect or no difference. It provides a baseline against which we compare our samples.
In our example, the null hypothesis is that the mean price of gasoline in Milwaukee is equal to the national mean price, namely, \( H_0: \mu = 2.10 \).
The purpose of the hypothesis test was to determine whether there's sufficient statistical evidence to reject this statement. Once the calculated test statistic exceeded the critical value, and accompanied by a small p-value, we had evidence to reject the null hypothesis.

In hypothesis testing, the alternative hypothesis, represented as \( H_a \), is what you want to prove. It is a statement indicating some type of effect or difference exists.
For our gasoline price example, the alternative hypothesis is that the average price in Milwaukee is higher than the national average: \( H_a: \mu > 2.10 \).
The whole process of hypothesis testing revolves around rejecting the null hypothesis in favor of the alternative hypothesis. Given our test results, which showed a test statistic above the critical value and a low p-value, we were able to conclude that the price in Milwaukee isn't just slightly different, but statistically significantly higher, supporting our alternative hypothesis.