Children's height as a function of their age has been researched so extensively that we can consider known results to describe the relationship for all children in the United States. For instance, between the ages of 13 and 15, population mean height for teenage males (in inches) satisfies µy= 22 + 3x, where x is age in years. Spread about the line is 3.1 inches.
a. Notice that the slope of the regression line for the population is b1 3. If we were to take repeated random samples of 25 males between the ages of 13 and 15 and regress their heights on their ages, then the slopes b1 would vary from sample to sample. At what slope value would their distribution be centered?
b. If we were to take repeated random samples of males between the ages of 13 and 15 and regress their heights on their ages, then the slopes b1 would vary from sample to sample. In which case would the distribution of sample slopes have less spread: when sampling 10 males or 25 males?
c. What notation is used for the spread about the line (3.1 inches): s or s?
d. On average, how much shorter do you predict a 13-year-old to be compared to a 15-year-old?
e. The linear regression model does a good job of summarizing the relationship between height and age for males in a particular age range, such as between 13 and 15 years old. Which two conditions would not be met if we attempted to perform inference about the height/age relationship based on a random sample of 250 males all the way from newborn to 25 years old?
1. Scatterplot should appear linear.
2. Sample size should be large enough to offset non-normality in responses.
3. Spread of responses should appear fairly constant over the range of explanatory values.
4. Explanatory/response values should constitute a random sample of independent pairs.