To calculate the Kolmogorov-Smirnov (KS) test statistic, we first need to sort the data set in ascending order:
S = {0.01, 0.1, 0.2, 0.28, 0.8}
Next, we calculate the cumulative distribution function (CDF) for the uniform distribution on the interval (0, 1):
CDF(x) = (x - 0)/(1 - 0) = x
Now, we calculate the empirical distribution function (EDF) for the data set S:
EDF(x) = (number of elements in S ≤ x) / (total number of elements in S)
For each data point x_i in S, we calculate EDF(x_i). In this case, since all elements in S are unique, the EDF values will be 1/5, 2/5, 3/5, 4/5, and 5/5 for the corresponding data points.
Finally, we calculate the KS test statistic as the maximum absolute difference between the CDF and EDF:
KS statistic = max|CDF(x_i) - EDF(x_i)|
Using the sorted data set S and the calculated EDF values, we can calculate the KS statistic.
KS statistic = max|CDF(0.01) - EDF(0.01)|, |CDF(0.1) - EDF(0.1)|, |CDF(0.2) - EDF(0.2)|, |CDF(0.28) - EDF(0.28)|, |CDF(0.8) - EDF(0.8)|
Calculating each difference:
|0.01 - 0.2| = 0.19
|0.1 - 0.4| = 0.3
|0.2 - 0.6| = 0.4
|0.28 - 0.8| = 0.52
|0.8 - 1| = 0.2
The maximum absolute difference is 0.52, so the KS statistic is 0.52.
Therefore, T_5^{\text {KS}}/\sqrt{5} = 0.52/\sqrt{5} = 0.2328.
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.
In this problem, you will test the null and alternative hypotheses
\displaystyle H_0 \displaystyle = \text {the data set is distributed as} \, \text {Unif}(0,1)
\displaystyle H_1 \displaystyle = \text {the data set is not distributed as} \, \text {Unif}(0,1) .
What is the value of the Kolmogorov-Smirnov test statistic on the data set S? Enter \, T_5^{\text {KS}}/\sqrt{5},\, the KS statistic without the factor of \sqrt{n}, below.
T_5^{\text {KS}}/\sqrt{5}=\quad
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