stats exam 2

Definition:
- Null Hypothesis (H0): A statement asserting that there is no effect or no difference; it serves as the default assumption.
- Alternative Hypothesis (H1 or Ha): A statement that contradicts the null hypothesis, indicating the presence of an effect or difference.
Example: In a clinical trial testing a new drug, the null hypothesis might state that the drug has no effect on patients' recovery rates (H0: μ1 = μ2), while the alternative hypothesis would state that the drug does improve recovery rates (H1: μ1 ≠ μ2).

Definition:
- Narrative: Clearly states the research question and hypotheses.
- Mathematical: Typically expressed in symbols (e.g., H0: μ1 = μ2 vs. H1: μ1 ≠ μ2).
Example: A narrative hypothesis might be "Regular exercise improves mental health," while the mathematical form would be H0: μ_exercise = μ_no_exercise vs. H1: μ_exercise > μ_no_exercise.

Definition:
- Two-tailed test: Tests for any significant difference (H1: μ ≠ μ0).
- One-tailed test: Tests for a specific direction of difference (H1: μ > μ0 or H1: μ < μ0).
Example: In testing a new teaching method, a two-tailed test would assess if there’s any difference in student performance (better or worse), whereas a one-tailed test might only check if the new method leads to improved performance.

Definition:
- Type I Error (False Positive): Rejecting the null hypothesis when it is true.
- Type II Error (False Negative): Failing to reject the null hypothesis when it is false.
Example: In a medical study, a Type I error would mean concluding that a drug works when it actually does not, while a Type II error would mean concluding it does not work when it actually does.

Definition:
- t Statistic: Used when the sample size is small (n < 30) or when the population standard deviation is unknown.
- z-Score: Used when the sample size is large (n ≥ 30) or when the population standard deviation is known.
Example: A researcher studying the effect of a new teaching method on a class of 25 students would use a t statistic, while a national survey of 1,000 students would use a z-score.

Definition:
- Statistical Significance: Indicates whether the results are likely due to chance (often p < .05).
- Practical Significance: Refers to the real-world relevance or impact of the findings, even if they are statistically significant.
Example: A new teaching method might show statistically significant improvement in test scores (p < .05), but if the actual improvement is only 1 point, it may lack practical significance.

Definition: The t distribution is symmetric and bell-shaped, similar to the normal distribution but has heavier tails, which account for increased variability with smaller sample sizes.
Example: Used in small sample studies where the population standard deviation is unknown, such as a study comparing two groups of less than 30 participants.

Definition: A one-sample t test compares the mean of a single sample to a known value or population mean.
Example: Testing whether the average height of a sample of students is different from the national average height.

Definition: The value specified in the null hypothesis that the sample mean is compared against.
Example: H0: μ = 70 inches, where 70 inches is the known population mean height.

Definition: In repeated measures, the null hypothesis typically asserts that there is no difference between paired observations.
Example: H0: μ_d = 0, indicating that the mean difference between pre-test and post-test scores is zero.

Definition: A design where the same subjects are tested multiple times under different conditions. It controls for individual differences.
Example: Measuring the same group of students' test scores before and after implementing a new teaching method.

Definition: Involves comparing two different groups that are not related.
Example: Comparing test scores of two different classes taught by different instructors.

Definition: Assumes normality, homogeneity of variance, and independence of observations.
Example: When comparing two independent groups, both groups should be normally distributed, and their variances should be similar.

Definition: Assumes that the differences between pairs are normally distributed.
Example: When testing the same group before and after treatment, the differences in scores should follow a normal distribution.

Definition: If the p-value is less than the significance level (commonly set at .05), reject the null hypothesis.
Example: If a t-test results in a p-value of .03, the null hypothesis would be rejected, indicating a statistically significant difference.

Definition: APA format requires specific structure and citation styles for reporting statistical results.
Example: “A paired-samples t-test revealed a significant difference in scores (M = 75.6, SD = 10.2) before (M = 70.0, SD = 9.8) and after treatment (M = 75.6, SD = 10.2), t(29) = 3.02, p = .005.”

Definition: Effect size quantifies the magnitude of a difference, often reported using Cohen's d.
Example: If Cohen's d = 0.8, it indicates a large effect size, suggesting that the intervention had a substantial impact.

Effect Size: Cohen’s d can be calculated using the formula d=M1−M2SDpooledd=SDpooledM1−M2.
Degrees of Freedom:
- Paired Samples: df=n−1df=n−1
- Independent Samples: df=n1+n2−2df=n1+n2−2
One-Sample t Test: t=M−μSDnt=nSDM−μ
Paired-Sample t Test: t=MDSDDnt=nSDDMD
Mean Differences (𝑋D): 𝑋D=Mbefore−MafterXD=Mbefore−Mafter
Standard Deviation of the Differences (sD): Calculate the standard deviation of the differences between paired observations.
Independent-Samples t Test: t=M1−M2(SD12n1)+(SD22n2)t=(n1SD12)+(n2SD22)M1−M2
Pooled Variance: SDpooled=(n1−1)SD12+(n2−1)SD22n1+n2−2SDpooled=n1+n2−2(n1−1)SD12+(n2−1)SD22