Glossary
Confidence Interval
A range of plausible values for a population parameter, such as the true slope of a linear regression model, calculated from sample data.
Example:
After analyzing data on study hours and exam scores, we constructed a confidence interval of (2.5, 4.1) for the true increase in score per hour studied, suggesting the real effect is likely within this range.
Confidence Level
The percentage of confidence intervals, constructed from many random samples, that would contain the true population parameter.
Example:
If we set our confidence level at 99%, it means that if we repeated our sampling process many times, about 99% of the intervals we create would capture the true population slope.
Degrees of Freedom (df)
A parameter that specifies the shape of the t-distribution, calculated as n-2 for regression slopes, where n is the sample size.
Example:
For a study with 15 data points examining the relationship between two variables, the degrees of freedom for the regression slope would be 15 - 2 = 13.
Interval Estimate
A range of values, typically a confidence interval, that is used to estimate an unknown population parameter.
Example:
Instead of just saying the average height is 68 inches, an interval estimate might state we are 95% confident the true average height is between 67.5 and 68.5 inches.
Linear Correlation
A statistical measure that describes the strength and direction of a straight-line relationship between two quantitative variables.
Example:
If a confidence interval for the slope of 'temperature vs. ice cream sales' does not contain zero and is entirely positive, it provides strong evidence of a positive linear correlation.
Point Estimate
A single value calculated from sample data that is used to estimate an unknown population parameter.
Example:
If our sample regression line for predicting house prices based on square footage has a slope of 150, then 150 is our point estimate for the true increase in price per square foot.
Sample Size (effect on interval width)
The number of observations in a sample; a larger sample size generally leads to a narrower confidence interval due to reduced standard error.
Example:
Increasing the sample size from 20 to 200 students when studying the effect of tutoring on grades would likely make our confidence interval for the slope much tighter, giving us a more precise estimate.
Slope Estimate (b)
The calculated slope of the least-squares regression line from sample data, serving as the point estimate for the true population slope.
Example:
If our regression analysis shows that for every additional hour of sunshine, plant growth increases by 0.5 cm, then 0.5 cm is our slope estimate.
Standard Error of the Slope (SE of b)
A measure of the variability or precision of the estimated regression slope, indicating how much the sample slope is expected to vary from the true population slope.
Example:
A small standard error of the slope (e.g., 0.01) suggests our estimated slope is a very precise estimate of the true relationship, while a large one (e.g., 0.5) indicates more uncertainty.
Zero in the Interval
Refers to whether the value 0 is contained within a confidence interval for a regression slope, which determines if there is evidence of a linear relationship.
Example:
If our confidence interval for the slope of 'hours studied vs. exam score' is (-0.5, 1.2), the presence of zero in the interval suggests there might be no linear relationship between studying and scores.
t-score
A critical value from the t-distribution used in constructing confidence intervals and performing hypothesis tests when the population standard deviation is unknown.
Example:
To calculate a 95% confidence interval for a slope with 28 degrees of freedom, we'd look up the appropriate t-score (around 2.048) from the t-distribution table.