Glossary
Confidence Interval
A range of values, calculated from sample data, that is likely to contain the true value of an unknown population parameter with a certain level of confidence.
Example:
A 95% confidence interval for the mean difference in test scores between two teaching methods might be (2.5, 7.1) points, suggesting one method is likely better.
Context (in conclusions)
The specific real-world details and variables of the problem that must be included when interpreting statistical results to make them meaningful and relevant.
Example:
When interpreting a confidence interval for mean commute times, you must include the context by mentioning 'the true difference in mean commute times for employees in City A versus City B'.
Degrees of Freedom
A parameter that describes the shape of the t-distribution, typically related to the sample size(s) and indicating the number of independent pieces of information available to estimate a parameter.
Example:
When performing a two-sample t-test, the degrees of freedom are calculated using a complex formula, often approximated by the smaller of (n1-1) or (n2-1), which influences the t-critical value.
Interpreting the Interval (containing 0)
If a confidence interval for the difference between two means contains 0, it suggests there is no statistically significant difference between the two population means at the given confidence level.
Example:
If a confidence interval for the difference in mean reaction times between two energy drinks is (-0.5, 0.3) seconds, then interpreting the interval means we don't have evidence that the drinks have different effects on reaction time because 0 is included.
Interpreting the Interval (not containing 0)
If a confidence interval for the difference between two means does not contain 0, it suggests there is a statistically significant difference between the two population means at the given confidence level.
Example:
If a confidence interval for the difference in mean battery life between two phone brands is (1.2, 2.8) hours, then interpreting the interval means we have significant evidence that one brand's battery life is longer than the other because 0 is not included.
Inverse Relationship (Sample Size & CI Width)
A relationship where as one variable increases, the other variable decreases; specifically, as sample size increases, the width of the confidence interval tends to decrease.
Example:
Because of the inverse relationship between sample size and confidence interval width, a study with 500 participants will yield a narrower interval for the mean height than a study with only 50 participants.
Margin of Error
The maximum expected difference between the true population parameter and the sample estimate, which decreases as sample size increases.
Example:
A poll with a margin of error of ±3% means the true proportion of voters supporting a candidate is likely within 3 percentage points of the reported sample proportion.
Null Hypothesis
The starting assumption in hypothesis testing, typically stating that there is no difference or no relationship between populations or variables.
Example:
When comparing two fertilizers, the null hypothesis would be that there is no difference in the mean plant growth between them.
Sample Size
The number of observations or individuals included in a sample from a population, which directly impacts the precision of statistical estimates.
Example:
Increasing the sample size from 30 to 100 students when estimating average study hours will likely lead to a more precise estimate.
Standard Error
The standard deviation of the sampling distribution of a statistic, indicating how much a sample statistic is expected to vary from the true population parameter.
Example:
A smaller standard error for the mean indicates that sample means are typically closer to the true population mean, leading to more precise estimates.
Statistical Claim
A statement or assertion about a population, often used as the basis for statistical testing to determine if there is enough evidence to support or refute it.
Example:
A researcher might make the statistical claim that the average height of students in two different schools is the same.
Width of the Confidence Interval
The range covered by a confidence interval, calculated as the upper bound minus the lower bound, which indicates the precision of the estimate.
Example:
A width of the confidence interval of 0.5 units is more precise than a width of 2.0 units, meaning the estimate is tighter.
t-critical value
A value from the t-distribution used in constructing confidence intervals or performing hypothesis tests when the population standard deviation is unknown.
Example:
For a 95% confidence interval with 20 degrees of freedom, you would look up the appropriate t-critical value (e.g., 2.086) to calculate the margin of error.