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In statistics, **proportions** help us understand the part of a whole that meets a particular criterion. For example, in surveys or polls, proportions allow us to express results as percentages or fractions.
In the exercise, proportions were calculated to compare behaviors between two groups: teens and parents. To find the proportion, we simply take the number of individuals meeting the criterion and divide it by the total number in the sample:

• For teens: 220 out of 1000 said they check social sites frequently, giving a proportion of \( p_1 = \frac{220}{1000} = 0.22 \).
• For parents: 40 out of 1000 thought their teens checked frequently, giving a proportion of \( p_2 = \frac{40}{1000} = 0.04 \).

Understanding these proportions is vital, as it sets the baseline for comparison and further statistical testing. Proportions such as these highlight the extent to which different groups engage in or perceive a specific behavior.

When conducting research, the **significance level** is crucial as it signifies the risk of rejecting a true null hypothesis. For this exercise, the significance level is set at 0.01. This means we are willing to accept a 1% chance of incorrectly rejecting the null hypothesis, implying that we could falsely find a difference that doesn't actually exist.
The implication here is caution; the smaller the significance level, the stronger the evidence must be to reject the null hypothesis. Choosing a significance level is an important step because it guides the interpretation of results:

• If the calculated p-value is less than the significance level (0.01 in this situation), we reject the null hypothesis.
• It's common to use levels like 0.01, 0.05, or 0.10, but the choice depends on the context and consequences of potential errors.

A significance level of 0.01 indicates the need for strong evidence to assert that parents perceive their teens check social networking less frequently than the teens actually do.

A **Z-Test** is a statistical method used to determine if there is a significant difference between sample means or proportions. It's applicable when dealing with large sample sizes and known population standard deviations.In the scenario provided, a Z-Test for two proportions is appropriate since we want to compare two independent sample proportions:

• Null Hypothesis \( H_0: p1 = p2 \)
• Alternative Hypothesis \( H_a: p1 < p2 \)

Using the formula:\[z = \frac{p1 - p2}{\sqrt{p(1-p)\left(\frac{1}{n1} + \frac{1}{n2}\right)}}\]We calculate the test statistic, where \( p \) represents the pooled proportion. The pooled proportion accounts for combined data from both groups, leading to more reliable statistical outcomes. In this exercise, the Z-Test helped demonstrate whether the proportion of perceived behaviors differs significantly between the groups.
Given the calculated z-value of 15.94, it's clear that there's a substantial difference, leading to the rejection of the null hypothesis, as the p-value is effectively 0, less than our significance level of 0.01.