Study Notes on Statistical Tests and Hypothesis Testing

The discussion revolves around various statistical tests (t-test, z-test, chi-square test) and their applications, particularly in analyzing hypotheses concerning student behavior and academic performance.

Null Hypothesis (H0): A statement that there is no effect or no difference, often denoted as H0.
Alternative Hypothesis (H1 or HA): The opposite of the null hypothesis, indicating an effect or difference.
Alpha (α): The significance level, often set at 0.1, 0.05, or 0.01, representing the probability of rejecting the null hypothesis when it is true.
Mean (BC): Represents the average in the context of hypotheses.
Standard Deviation (s or A3): A measure of the amount of variation or dispersion in a set of values.

Used when the population standard deviation is unknown and the sample size is small (n < 30).
Involves the calculation of the test statistic:
$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$
Degrees of Freedom (df) for t-tests: $n - 1$ where n is the sample size.
Conclusion from t-tests: If calculated t falls in the rejection region (e.g., beyond cutoff value from t-distribution table), then H0 is rejected.
It was noted that the t-test focuses on means.

Used when population standard deviation is known, or sample size is large (n ≥ 30).
Employs the formula:
$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$
Degrees of Freedom: Not applicable in the same sense as t-tests, but assumed z-distribution applies to larger samples.
Chi-square test applies when dealing with proportions.

Used to determine if there is a significant association between categorical variables.
Goodness-of-fit and independence tests are both variations of the chi-square test.
Important formulas:
$ext{Chi-Squared} = \sum \frac{(O - E)^2}{E}$
Where O = observed frequency, E = expected frequency.
Degrees of Freedom (df) for chi-square tests: $k - 1$ , where k is the number of categories.

Analyzing how many students spend less than 60 minutes on TikTok and their impact on GPAs.
Discussed finding means, sample standard deviation, and rejecting or failing to reject the null hypothesis based on test statistics.

A confidence interval of 90% results in an alpha level of 0.1.
Important to understand how alpha affects the model and set cutoff values for rejecting the null hypothesis.

When calculating t-values, the formula indicates how far the sample mean is from the population mean in terms of the standard error.
Example calculations resulted in t-values illustrating deviations from the mean:
- When confronted with a specified t-value, corresponding tables are consulted to identify whether the null hypothesis is rejected.
- Importance of degrees of freedom in determining the shape of the t-distribution used for critical values.

Commonly questioned practical scenarios:
- Relationship between social media usage (TikTok) and academic performance measured by GPAs.
- Framework for independence tests to see if increased use of TikTok results in decreased GPAs.

Discussed implications of setting confidence levels for tests:
- Higher confidence levels (e.g., 99%) yield lower alpha levels, reducing chances of Type I errors (incorrectly rejecting H0).
- Lower confidence levels (e.g., 90%) could allow for more Type I errors, providing a larger rejection region.

The session emphasizes the importance of understanding the conditions under which different statistical tests apply, especially in the context of hypothesis testing involving behavioral statistics of students about their social media usage and its potential influence on academic performance. Engaging with practical data improves comprehension of theoretical statistical principles, matching real-life scenarios with mathematical understanding.