The following sequence of 22 consecutive data values was collected. This sequence was found to contain ten 0 's and twelve 1 's. Conduct a runs test for randomness for this sequence at the 0.05 significance level, and test the claim that the order of this sequence is not random. 1110011100001110101001 Round your answers to 3 places after the decimal point, if necessary. (a) Find the value of the test statistic. Test statistic's value: (b) Find the critical values. List both critical values in the answer box with a comma between them. Critical values: (c) What is the correct conclusion of this test? There is not sufficient evidence to warrant rejection of the claim that the order of this sequence is not random. There is sufficient evidence to warrant rejection of the claim that the order of this sequence is not random. There is not sufficient evidence to support the claim that the order of this sequence is not random. There is sufficient evidence to support the claim that the order of this sequence is not random.
Solution
Step 1 ：Given the sequence 1110011100001110101001, we have 10 zeros and 12 ones, and a total of 22 observations.
Step 2 ：A run is a consecutive sequence of the same number. Counting the runs in the sequence, we find there are 11 runs.
Step 3 ：We calculate the expected number of runs, E(R), using the formula \(E(R) = \frac{2n1n2}{n} + 1\), where n1 is the number of 0's, n2 is the number of 1's, and n is the total number of observations. Substituting the given values, we find \(E(R) = 11.909090909090908\).
Step 4 ：We calculate the variance of the number of runs, V(R), using the formula \(V(R) = \frac{2n1n2(2n1n2 - n)}{n^2(n - 1)}\). Substituting the given values, we find \(V(R) = 5.147579693034238\).
Step 5 ：We calculate the test statistic, Z, using the formula \(Z = \frac{R - E(R)}{\sqrt{V(R)}}\), where R is the number of runs. Substituting the given values, we find \(Z = -0.40068748260408144\).
Step 6 ：Rounding to three decimal places, the value of the test statistic is \(\boxed{-0.401}\).
