Glossary
Alternative Hypothesis ($H_a$)
A statement that contradicts the null hypothesis, proposing that there is an effect, a difference, or a relationship. It is what the researcher is trying to find evidence for.
Example:
If the null hypothesis states a drug has no effect, the alternative hypothesis might state that the drug reduces blood pressure.
Assumptions (for t-tests)
Conditions that must be met for the results of a t-test to be valid and reliable. These typically include randomness, independence, and approximate normality of the sampling distribution.
Example:
Before performing a t-test on student test scores, one must check the assumptions like whether the sample was randomly selected and if the population of scores is approximately normal.
Context (in conclusions)
The practice of interpreting statistical results and conclusions within the real-world scenario or problem being investigated. It ensures that the statistical findings are meaningful and relevant.
Example:
When concluding a t-test about battery life, stating 'We have convincing evidence that the average battery life is less than 50 hours' is providing the conclusion in context.
Degrees of Freedom (df)
A value that indicates the number of independent pieces of information available to estimate a parameter. For a one-sample t-test, it's typically calculated as sample size minus one ($n-1$).
Example:
In a study with 20 participants, the degrees of freedom for a one-sample t-test would be 19, influencing the shape of the t-distribution.
Fail to Reject the Null Hypothesis
The decision made when the p-value is greater than the significance level, indicating insufficient statistical evidence to conclude that the alternative hypothesis is true. This does not mean the null hypothesis is proven true.
Example:
If a p-value is 0.15 and , one would fail to reject the null hypothesis, meaning there isn't enough evidence to support the alternative claim.
Hypothesized Population Mean ($\mu$)
The specific value for the population mean that is assumed to be true under the null hypothesis. It's the value against which the sample mean is compared.
Example:
A company claims their light bulbs last 1000 hours, so 1000 hours would be the hypothesized population mean in a test of their claim.
Null Hypothesis ($H_0$)
A statement of no effect, no difference, or no relationship, which is assumed to be true until evidence suggests otherwise. It typically includes an equality.
Example:
The null hypothesis for a study on a new teaching method might be that the average test scores of students using the new method are equal to those using the old method.
One-Tailed Test
A hypothesis test where the alternative hypothesis specifies a directional difference (e.g., greater than or less than). The p-value is calculated from only one tail of the distribution.
Example:
If a researcher wants to know if a new drug increases reaction time, they would use a one-tailed test.
P-Value
The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. It helps determine the strength of evidence against the null hypothesis.
Example:
A p-value of 0.03 means there's a 3% chance of seeing results as extreme as the sample if the null hypothesis were true, suggesting strong evidence against it.
Reject the Null Hypothesis
The decision made when the p-value is less than or equal to the significance level, indicating sufficient statistical evidence to conclude that the alternative hypothesis is true.
Example:
If a study finds a p-value of 0.01 and , the researchers would reject the null hypothesis, concluding there's a significant effect.
Sample Mean ($\bar{x}$)
The average value calculated from a collected sample of data. It serves as an estimate for the unknown population mean.
Example:
After measuring the heights of 50 randomly selected high school students, the calculated average height of 67 inches is the sample mean.
Sample Size ($n$)
The number of observations or individuals included in a statistical sample. It influences the degrees of freedom and the precision of estimates.
Example:
If a survey collects responses from 200 people, then the sample size is 200.
Sample Standard Deviation ($s$)
A measure of the typical spread or variability of data points around the sample mean. It is used to estimate the unknown population standard deviation.
Example:
If a class's test scores have a sample standard deviation of 10 points, it indicates that scores typically vary by about 10 points from the average score.
Significance Level ($\alpha$)
The predetermined threshold for the p-value, below which the null hypothesis is rejected. It represents the maximum probability of making a Type I error (rejecting a true null hypothesis).
Example:
Setting a significance level of 0.05 means that if the p-value is less than 0.05, the results are considered statistically significant.
T-Score
A test statistic used in t-tests that measures how many standard errors a sample mean is away from the hypothesized population mean. A larger absolute t-score indicates a greater difference.
Example:
If a student calculates a t-score of 2.5 for their experiment, it means their sample mean is 2.5 standard errors away from the hypothesized population mean, suggesting an unusual result.
T-Tests
Statistical hypothesis tests used to compare means, typically when the population standard deviation is unknown. They help determine if the difference between a sample mean and a hypothesized population mean is statistically significant.
Example:
A researcher uses a t-test to see if a new fertilizer significantly increases the average yield of corn per acre compared to the standard yield.
Two-Tailed Test
A hypothesis test where the alternative hypothesis specifies a non-directional difference (e.g., not equal to). The p-value is calculated from both tails of the distribution.
Example:
If a scientist wants to know if a new manufacturing process changes the average weight of a product (either increases or decreases), they would use a two-tailed test.