91Ó°ÊÓ

In hypothesis testing, proportions represent the parts of a whole that exhibit a certain characteristic or outcome. For instance, the problem at hand compares two proportions. Here, the proportions describe the percentage of people who reported using Facebook in two different years: 2012 and 2018.
To calculate these proportions, we simply divide the number of successes (people in the survey who use Facebook) by the total number of individuals surveyed. For 2012, the proportion is represented mathematically as \( P_1 = 0.66 \), meaning 66% of the sample was using Facebook. Similarly, for 2018, the proportion is \( P_2 = 0.76 \), implying a 76% usage rate.
In comparing these proportions, it's essential to determine whether or not there is a significant difference between them. This is what the hypothesis test aims to resolve.

Sample size is critical in hypothesis testing because it affects the accuracy and reliability of the results. In the exercise, two different sample sizes are considered: 100 and 1500.
A larger sample size tends to give more reliable estimates of the population parameter, which reduces variability and provides a clearer distinction between proportions if a real difference exists. For a sample of 100, estimate variability is relatively high, which can sometimes lead to less confidence in claiming a real difference between the population proportions.
Conversely, when the sample size is increased to 1500, the standard error decreases. This means the calculated test statistic becomes more stable, yielding a more precise p-value. - Greater sample sizes reduce the margin of error. - Larger samples give more power to detect a true effect. - In this problem, the shift from 100 to 1500 samples significantly alters the conclusions you might draw from the data.

The p-value is central in deciding whether to reject or fail to reject a null hypothesis. It indicates the probability that the observed data, or something more extreme, would occur if the null hypothesis were true.
In our scenario, the test statistic derived from the proportions is used to calculate the p-value. A smaller p-value signifies that, if the null hypothesis were true, the observed difference in proportions is unlikely. For smaller samples, this p-value is generally larger because of higher variability and less certainty.
The computation of a p-value also considers the type of test being performed. A two-tailed test, for example, captures the probability of a difference in either direction, doubling the p-value from a one-tailed test.
- If the p-value is less than the significance level, reject the null hypothesis. - A small p-value (<0.01) usually indicates strong evidence against the null hypothesis.

The significance level, often denoted by \( \alpha \), represents the threshold for determining whether the p-value indicates enough evidence to reject the null hypothesis. In this problem, a 0.01 significance level is used.
Choosing a significance level is crucial because it balances the risk of making a Type I error (incorrectly rejecting the null hypothesis) with the ability to detect true differences. Common significance levels are 0.05, 0.01, and 0.10, with a lower significance level indicating more stringent criteria for rejecting the null hypothesis.
Here, no statistically significant difference between the two proportions is found unless the p-value falls below 0.01. This level of significance implies a 1% risk of concluding that a difference exists when there isn't one, ensuring a high level of confidence in the test's conclusions.