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The null hypothesis, denoted as \(H_0\), is a crucial part of hypothesis testing. It represents a statement or default position that there is no effect or no difference. In the context of our exercise, the null hypothesis asserts that the average daily calorie intake for teenage girls is 2200 calories. Mathematically, it's written as: \[ H_0: \mu = 2200 \]
This step is essential because it sets a baseline against which statistical evidence is compared. The goal is to determine if there is enough evidence to reject this null hypothesis, suggesting an effect or difference exists. The null hypothesis is assumed true until evidence suggests otherwise. Always clearly state the null hypothesis when setting up your hypothesis test.

The alternative hypothesis, denoted as \(H_a\), is what you want to prove. It represents a new effect or difference that the researcher believes to be true. In our example, the nutritionist believes that teenage girls consume fewer than the recommended 2200 calories daily, likely due to societal pressures from advertising. Mathematically, this is represented as: \[ H_a: \mu < 2200 \]
Unlike the null hypothesis, the alternative hypothesis is what we accept if we find enough evidence against \(H_0\). It is a critical part of hypothesis testing as it embodies the research question. By setting a clear alternative hypothesis, you establish the direction of the test: whether you are looking for something less, greater, or simply different from what is assumed under the null hypothesis.

A test statistic is a standardized value calculated from sample data during a hypothesis test. It assesses how far the sample statistic is from the null hypothesis parameter, considering the variability in the data. In our exercise, the test statistic is calculated using the formula: \[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \] Where:
- \( \bar{x}\) is the sample mean (2150 calories).
- \( \mu\) is the population mean (2200 calories).
- \( \sigma\) is the standard deviation (200 calories).
- \( n\) is the sample size (36 girls).
Plugging in the numbers, the calculation turns out to be: \[ z = \frac{2150 - 2200}{\frac{200}{6}} = -1.5 \]
This z-score tells us how many standard deviations the sample mean is from the population mean. It helps us determine if the observed sample mean is statistically significant.

The significance level, often denoted as \( \alpha \), is the threshold at which we decide whether to reject the null hypothesis. Commonly used significance levels are 0.01, 0.05, and 0.10. In our example, we use \( \alpha = 0.05 \). This means we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis (a Type I error).
It's a critical part of hypothesis testing as it defines the cutoff point for making a decision. The lower the significance level, the stricter the criteria for rejecting the null hypothesis. Importantly, the significance level should be set before analyzing the data to avoid bias. Selecting a significance level involves striking a balance between being too conservative (risking Type II errors) and too lenient (risking Type I errors).

The critical value corresponds to the significance level and helps determine the cutoff point where we reject the null hypothesis. For a one-tailed test with \( \alpha = 0.05 \), the critical value from the z-table is -1.645.
In our example, we compare our test statistic (z = -1.5) to this critical value:
\ -1.5 \> -1.645 \
Since -1.5 is greater than -1.645, it falls outside the rejection region, meaning we fail to reject the null hypothesis.
Critical values are essential for hypothesis testing because they provide a specific point of comparison. If the test statistic lies beyond the critical value, it suggests that the observed effect is significant enough to reject the null hypothesis. Make sure to use the correct table (z-table, t-table, etc.) and consider the test type (one-tailed or two-tailed) when determining the critical value.