Math 101 – Chapter 2 – Scatter Diagrams – Mathematical Models
1) A computer science major found that the grade he earned on his programs increased with the time he spent working on them in the computer lab. He made a table of the hours spent in the lab and the corresponding program grades. Let g represent the grade earned when h hours are spent in the lab.
a) Determine whether this relation is a function. Explain why or why not.
b) Sketch a scattergram for the data. Label axes with variables, words and numbers.
c) Sketch a line that comes “close” to the data points in the scattergram (“eyeball” best fit line) This will be your “mathematical model”. Use the graph of that line to estimate the Program grade when 4 hours are spent in the lab. State whether you used interpolation or extrapolation to obtain your result.
c) Find the equation of “your” mathematical model. Then use the equation to estimate the Program grade when 4 hours are spent in the lab.
d) Add a third column to the table given on the first page and use the equation of your model to find the estimated Program grade for each of the x-values given on the table. In each case indicate whether you obtained an overestimate or an underestimate of the actual Program grade?
e) Now we’ll use the calculator to find the “least squares regression line”, or the “line of best fit”.
The steps are outlined on the next page.
f) Find the y-intercept. What does it mean in terms of the situation?
g) What is the slope of the equation? What does the slope represent in terms of the situation?
h) What is the x-intercept? What does it mean in terms of the situation?
i) Find f(3). What does your result mean in terms of the situation?
j) Find x when f(x) = 100. What does your result mean in term of the situation?
2) Although the number of people arrested for arson has remained fairly constant during the 1990s, the percent of arson arrests that have been juveniles has been on the rise. The data is printed in the table below.
(Source: U.S. Justice Department)
Let P = f(t) represent the percent of arson arrests that are juveniles at t years since 1990. The function models the data well.
i) Use the linear model to predict when 75 percent of arson arrests will be juveniles.
ii) Use the linear model to estimate the percent of arson arrests that were juveniles in 1985.
iii) Find the t-intercept. What does it mean in terms of arson arrests?
iv) What is the slope of the equation P = 1.83t + 44.76? What does the slope represent in terms of arson arrests?
v) Find the P-intercept. What does it mean in terms of arson arrests?
vi) Find f(100). What does your result mean in terms of the situation?
vii) Find t when f(t) = 100. What does your result mean in term of the situation?
3) Over the past quarter century, the number of bachelor’s degrees conferred by degree-granting institutions has steadily increased. The following table contains data on the number of bachelor’s degrees (in thousands) earned by women in a given year. The input t represents the number of years since 1977.
year / 1977 / 1981 / 1985 / 1990 / 1995 / 2000t, number of years since 1977
F(t), number of degrees in thousands / 423 / 465 / 492 / 558 / 634 / 708
a) Sketch a scatterplot of the given data. Label axes with variables, words and numbers.
b) Sketch a line that comes close to the data points in the scattergram. Find the equation of this line. This will be your “mathematical model”.
c) Now use your calculator to obtain the scattergram and determine the line of best fit.
d) What is the slope of the line? What is the practical meaning of the slope in this istuation?
e) What is the y-intercept? What is the practical meaning of the y-intercept in this situation?
f) Find f(8). Interpret in terms of the situation. Is the obtained value an underestimate or an overestimate? Compare with the actual value.
g) Use the regression model to predict the number of bachelor’s degrees that will be granted in the year 2007. State whether you used interpolation or extrapolation to obtain your result.
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