The least-squares regression line equation is y = -0.28x + 610.31
The scatter plot of the data
To plot the scatter plot of the data, we represent the x-axis with year and we represent the y-axis with time
Next, we plot the scatter plot using the dataset
See attachment for the scatter plot
Describe the relationship between the variables in the graph.
From the attached scatter plot, we can see that the relationship between the variables in the graph is approximately linear.
This is so because the points appear to be on a straight line
Find the correlation coefficient.
To do this, we make use of a graphing calculator, with the following summary:
X Values
- ∑ = 21608
- Mean = 1964.364
- ∑(X - Mx)2 = SSx = 10062.545
Y Values
- ∑ = 686.41
- Mean = 62.401
- ∑(Y - My)2 = SSy = 842.83
X and Y Combined
- N = 11
- ∑(X - Mx)(Y - My) = -2806.684
R Calculation
r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))
r = -2806.684 / √((10062.545)(842.83)) = -0.9638
Hence, the correlation coefficient is -0.9638
Find the least-squares regression line.
To do this, we make use of a graphing calculator, with the following summary:
- Sum of X = 21608
- Sum of Y = 686.41
- Mean X = 1964.3636
- Mean Y = 62.4009
- Sum of squares (SSX) = 10062.5455
- Sum of products (SP) = -2806.6836
The least-squares regression line equation is
y = bx + a
Where
b = SP/SSX = -2806.68/10062.55 = -0.27892
a = MY - bMX = 62.4 - (-0.28*1964.36) = 610.30872
So, we have
y = -0.27892x + 610.30872
Approximate
y = -0.28x + 610.31
Hence, the least-squares regression line equation is y = -0.28x + 610.31
The estimated gold medal time for 1924 and for 1976.
We have:
y = -0.28x + 610.31
Substitute 1924 and 1976 for x
y = -0.28 * 1924 + 610.31 = 71.59
y = -0.28 * 1976 + 610.31 = 57.03
Are these estimates reliable?
Yes, the estimates are reliable
Are they reasonable?
Yes, the estimates are reasonable
Why are your answers in Question 5 different than the times given in the table?
The results are different because the answers in 5 are estimated from the formula while the table are the actual values
Find the estimated gold medal time for 2022 and for 2200.
We have:
y = -0.28x + 610.31
Substitute 2022 and 2200 for x
y = -0.28 * 2022 + 610.31 = 44.15
y = -0.28 * 2200 + 610.31 = -5.69
Are these estimates reliable?
No, the estimates are not reliable
Are they reasonable?
The estimate in 2022 is reasonable, while the estimate in 2200 is not reasonable
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