Here are the world record race times for women in the 10,000-meter run over several years. a) Which is the explanatory variable and which is the response variable? Year Race Time (seconds) 1967 2286.4 b) Make a scatterplot of these data. You do not have to draw the scatterplot for me. Describe what you see (form, direction, strength, outliers). 1970 2130.5 1975 2100.4 1975 2041.4 c) Write the equation of the regression line 1977 1995.1 for predicting race time from year. 1979 1972.5 d) Give the meaning of the slope of your line in terms of race time and year. What are the units of the slope in this problem? 1981 1950.8 1981 1937.2 e) What percent of the observed variation in the race times can be explained by your 1982 1895.3 1983 1895.0 model? 1983 1887.6 f) Find the residual for the first data point on 1984 1873.8 the list (the 2286.4 seconds from 1967). 1985 1859.4 8) What does this linear model predict for the race time in the year 2075? Do you think 1986 1813.7 this is reasonable? 1993 1771.8
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
Answer only D, E & F.
![### World Record Race Times for Women in the 10,000-Meter Run
#### Data Table
Here are the world record race times for women in the 10,000-meter run over several years:
| Year | Race Time (seconds) |
|------|---------------------|
| 1967 | 2286.4 |
| 1970 | 2130.5 |
| 1975 | 2100.4 |
| 1975 | 2041.4 |
| 1977 | 1995.1 |
| 1979 | 1972.5 |
| 1981 | 1950.8 |
| 1981 | 1937.2 |
| 1982 | 1895.3 |
| 1983 | 1895.0 |
| 1983 | 1887.6 |
| 1984 | 1873.8 |
| 1985 | 1859.4 |
| 1986 | 1813.7 |
| 1993 | 1771.8 |
#### Questions
a) **Which is the explanatory variable and which is the response variable?**
- **Answer:** The explanatory variable is the year, and the response variable is the race time (in seconds).
b) **Make a scatterplot of these data. You do not have to draw the scatterplot for me. Describe what you see (form, direction, strength, outliers).**
- **Answer:** The scatterplot would likely show a negative linear relationship between the year and race time. As the years progress, the race times decrease. The relationship appears to be strong with a consistent downward trend, and there do not appear to be any significant outliers.
c) **Write the equation of the regression line for predicting race time from year.**
- **Answer:** To find the exact equation of the regression line, one would typically use statistical software. However, the general form would be:
\[
\text{Race Time} = a + b \times (\text{Year})
\]
where \(a\) is the y-intercept and \(b\) is the slope of the line.
d) **Give the meaning of the slope of your line in terms of race time and year. What are the units of the slope in this problem?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85a16b31-7619-4bbb-b006-067a19fea2d2%2F59610c16-0971-4883-a26d-32fd66b1b283%2Fxh5rv8.png&w=3840&q=75)
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