The students in Mr. Shenk's class measured the arm spans and...

The students in Mr. Shenk's class measured the arm spans and heights ininches) of a random sample of 18 students from their large high school. Same computer output from a least-squares regression analysis on these data is shown below. Construct and interpret a $90 \%$ confidence interval for the slope of the population repression line. Assume that the conditions for performing inference are met. (FIGURE CANNOT COPY)

Key Concepts

t-Distribution

The t-distribution is used in the construction of confidence intervals and hypothesis tests when the sample size is small and the population standard deviation is unknown. It accounts for additional variability due to the estimation of the standard deviation, and its specific shape is determined by the degrees of freedom, typically (n - 2) in regression.

Interpreting the Slope

In regression analysis, the slope represents the estimated change in the dependent variable for each one-unit increase in the independent variable. The confidence interval for the slope indicates the range of plausible values for this effect in the population, which helps in understanding the strength and direction of the relationship.

Least Squares Regression Analysis

This is a statistical method used to model the relationship between two quantitative variables by fitting a linear equation to observed data. The method minimizes the sum of the squared differences between the observed values and those predicted by the line, allowing for estimation of parameters such as the slope and intercept of the regression line.

Confidence Interval

A confidence interval is a range of values, derived from the sample data, that is believed to contain the true population parameter with a certain level of confidence, such as 90%. In the context of regression, it provides a range for the slope that reflects the uncertainty associated with its estimation.

Conditions for Inference

These are the assumptions that must be satisfied in order for the results of the regression analysis, including confidence intervals, to be valid. They typically include linearity, independence of observations, normally distributed residuals, and constant variance (homoscedasticity) of the errors.

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