The problems on this page concern the data set
S = \{ 0.28, 0.2, 0.01, 0.80, 0.1 \} .
Let x_ i denote the i'th element of the data set S.
Consider the QQ-plot of the data set S against the distribution \text {Unif}(0,1). (You may graph the plot using computational tools.)
How many points in the QQ-plot lie above the line y = x?
To determine the number of points in the QQ-plot that lie above the line y = x, we can compare the data set S to the uniform distribution \text {Unif}(0,1). A QQ-plot compares the quantiles of the two distributions.
Step 1: Sort the data set S in ascending order.
The sorted data set S is: \{ 0.01, 0.1, 0.2, 0.28, 0.8 \}
Step 2: Calculate the theoretical quantiles of the uniform distribution \text {Unif}(0,1) for the given probabilities.
The probabilities are evenly distributed from 1/(n+1) to n/(n+1), where n is the number of data points. In this case, n = 5.
So the probabilities are: 1/6, 1/3, 1/2, 2/3, 5/6.
To calculate the theoretical quantiles, use the formula: x_p = F^−1(p), where F^−1(.) is the inverse cumulative distribution function of the uniform distribution.
For p = 1/6, x_p = F^−1(1/6) = (1/6 - 0)/(1 - 0) = 1/6
For p = 1/3, x_p = F^−1(1/3) = (1/3 - 0)/(1 - 0) = 1/3
For p = 1/2, x_p = F^−1(1/2) = (1/2 - 0)/(1 - 0) = 1/2
For p = 2/3, x_p = F^−1(2/3) = (2/3 - 0)/(1 - 0) = 2/3
For p = 5/6, x_p = F^−1(5/6) = (5/6 - 0)/(1 - 0) = 5/6
The calculated theoretical quantiles are: \{ 1/6, 1/3, 1/2, 2/3, 5/6 \}
Step 3: Plot the QQ-plot by plotting the sorted data set on the x-axis and the theoretical quantiles on the y-axis.
Using computational tools or graphing software, plot the points (0.01, 1/6), (0.1, 1/3), (0.2, 1/2), (0.28, 2/3), (0.8, 5/6) on the graph.
Step 4: Count the number of points that lie above the line y = x.
In the QQ-plot, count the number of points that are above the line y = x. In this case, it can be observed that all the points lie above the line y = x.
Therefore, the number of points in the QQ-plot that lie above the line y = x is 5.
To determine how many points lie above the line y = x on the QQ-plot, we need to compare the ordered values of the data set S with the ordered quantiles of the uniform distribution.
First, let's sort the values in S in ascending order:
S = {0.01, 0.1, 0.2, 0.28, 0.8}
Now, let's determine the quantiles of the uniform distribution. Since we are dealing with five values, we can assign each value to a quantile based on its position:
0.01 -> 0.2
0.1 -> 0.4
0.2 -> 0.6
0.28 -> 0.8
0.8 -> 1.0
Now, plotting the points (x_i, quantile) on the QQ-plot, we find:
(0.01, 0.2)
(0.1, 0.4)
(0.2, 0.6)
(0.28, 0.8)
(0.8, 1.0)
Examining the points on the plot, we can see that all five points lie above the line y = x. Therefore, the answer is that all points in the QQ-plot lie above the line y = x.