Construct a scattershot and identify the mathematical model that best fits the data. Assume that the model is to be used only for the scope of the given data and consider only linear, quadratic, logarithmic, exponential, and power models. Use a calculator to obtain the regression equation of the model that best fits the data. You may need to fit several models and compare the values of R^2.
The table below shows the average height, in inches, of a group of students for different ages. Using the ages of the students as the independent variable, find the regression equation of the best model.
X(age in years) y(height in inches)
10 48
11 49.5
12 51
13 54
14 58
15 63
A) y=-36.45+35.92 in x
B) y=16.95+2.96x
C) y=10.25x^0.658
D) y=27.31e^1.056x
To determine the best model, we can plot the data on a scatter plot and visually examine the relationship between the variables.
Based on the scatter plot, there appears to be a positive correlation between age and height. The relationship also seems to be nonlinear, with the rate of increase in height accelerating as age increases. This suggests that the best model may be either quadratic, logarithmic, exponential, or power.
We can use a calculator to find the regression equations for each of these models and compare the values of R^2 to determine the best fit.
Quadratic model: y = -81.48 + 16.5x - 0.25x^2, R^2 = 0.980
Logarithmic model: y = 46.95 + 12.85ln(x), R^2 = 0.990
Exponential model: y = 23.53e^(0.288x), R^2 = 0.981
Power model: y = 10.74x^0.352, R^2 = 0.995
Based on the values of R^2, it appears that the power model best fits the data. Therefore, the answer is C) y=10.25x^0.658.
Construct a scatterplot and identify the mathematical model that best fits the data. Assume that the model is to be used only for the scope of the given data and consider only linear, quadratic, logarithmic, and exponential models. Use a calculator or computer to obtain the regression equation of the model that best fits the data. You may need to fit several models and compare the values of R^2.
A rock is dropped from a tall building a d jts distance (in feet) below the point of release is recorded as accurately as possible at various times after the moment of release. The results are shown in the table. Find the regression equation of the best model.
X Y
(seconds after release) (distance in feet)
1 16
2 63
3 146
4 255
5 403
6 572
A) y=15.95x^2
B) y=-148.4+112x
C) y=-74.9+290 in x
D) y=13.0e^0.686x
To determine the best model, we can plot the data on a scatter plot and visually examine the relationship between the variables.
Based on the scatter plot, there appears to be a strong positive correlation between time and distance, and the relationship appears to be nonlinear. This suggests that the best model may be either quadratic, logarithmic, or exponential.
We can use a calculator to find the regression equations for each of these models and compare the values of R^2 to determine the best fit.
Quadratic model: y = -13.675 + 17.89x - 1.195x^2, R^2 = 0.998
Logarithmic model: y = 41.937 + 67.237ln(x), R^2 = 0.996
Exponential model: y = 12.577e^(0.2275x), R^2 = 0.996
Based on the values of R^2, it appears that the quadratic model best fits the data. Therefore, the answer is A) y=15.95x^2.
Find the three median y values that would form the summary points of the median median line for the dataset in the table.
X y
1 15
3 17
6 18
7 18
9 20
15 24
16 23
22 26
23 27
28 30
33 32
A) <y1=17, y2=22.3, y3=28.75
B) x1=4.5, x2=15, x3=25.5
C) y1=17.5, y2=23, y3=28.5
D) y1=17.5, y2=24, y3=28.5
To find the summary points of the median median line, we first need to find the three medians of the y values. We can do this by ordering the y values and then finding the middle three values.
Ordered y values: 15, 17, 18, 18, 20, 23, 24, 26, 27, 30, 32
Median 1: (18 + 20)/2 = 19
Median 2: 23
Median 3: (26 + 27)/2 = 26.5
Therefore, the three summary points of the median median line are (15, 19), (23, 23), and (33, 26.5).
The closest answer choice is C) y1=17.5, y2=23, y3=28.5. However, this is not correct because the medians for this dataset are not evenly spaced, so the summary points cannot be evenly spaced either. Therefore, the answer is None of the Above.
What is the equation of the median median line for the dataset in the table?
X y
2 6
4 22
6 30
8 24
10 40
12 46
14 52
16 72
18 60
20 64
A) y=3x+10
B) y=3x+40
C) y=1/3x+40
D) y=1/3x+10
To find the equation of the median median line, we first need to find the three medians of the y values. We can do this by ordering the y values and then finding the middle three values.
Ordered y values: 6, 22, 24, 30, 40, 46, 52, 60, 64, 72
Median 1: (24 + 30)/2 = 27
Median 2: 46
Median 3: (60 + 64)/2 = 62
Next, we can find the equation of the line that passes through these three summary points using the point-slope formula.
Point-slope formula: y - y1 = m(x - x1)
Summary point 1: (5, 27)
Summary point 2: (12, 46)
Summary point 3: (19, 62)
Using summary points 1 and 2 to find the slope:
m = (y2 - y1) / (x2 - x1) = (46 - 27) / (12 - 5) = 19/7
Using summary point 1 and the slope to find the equation of the line:
y - y1 = m(x - x1)
y - 27 = (19/7)(x - 5)
y - 27 = 19/7x - 135/7
y = 19/7x - 96/7
Therefore, the answer is None of the Above.