Final answer:
The variability in the estimated slope of a regression line is indeed smaller when the x-values are more spread out because the standard error of the slope estimate decreases. This concept highlights the importance of variability in explanatory variables for the precision of regression estimates and aligns with statistical principles such as the central limit theorem.
Step-by-step explanation:
The statement that the variability in the estimated slope is smaller when the x-values are more spread out is indeed true. This concept is closely related to the principles of regression analysis in statistics. When the x-values in a data set are more spread out, it increases the precision of the estimated regression line, as there is less uncertainty about the slope. This is because the standard error of the slope estimate is inversely related to the spread of the x-values. A larger spread in x-values results in a smaller standard error and thus less variability in the estimated slope.
Analogous to this concept, several statistical distributions and theorems also emphasize the impact of sample size and variability. For example, according to the central limit theorem, the sampling distribution of the sample mean becomes more normally distributed as the sample size increases. This is relevant when discussing the natural variability of sample statistics and the efficiency of estimators in a regression context.
Therefore, when we talk about regression analysis, larger variabilities in explanatory variables (spread of x-values), provided they are not associated with a corresponding increase in the variability of the errors, tend to produce more reliable estimates of the relationship between the variables.