An energy analyst wants to test if U.S. oil production is random over time. The analyst has monthly production values for the two years. The analyst finds 12 months are above the median, 12 months are below the median, six runs are below the median, and five runs are above the median. Assume the R follows a normal distribution. The value of the test statistic is ________.
A) -2.231
B) -6.868
C) 6.868
D) 2.231

Question

Quizplus · Accepted Answer

To test if U.S. oil production is random over time, we can use a runs test. The test statistic for a runs test is given by:
Z = (R - mu_R) / sigma_R
where R is the number of runs, mu_R is the expected number of runs, and sigma_R is the standard deviation of the number of runs. For a two-tailed test at the 5% level of significance, the critical value of Z is ±1.96.
The expected number of runs can be calculated as:
mu_R = (2n1n2) / (n1 + n2) + 1
where n1 and n2 are the sample sizes of the two groups (above and below the median) and n1 + n2 = n, the total sample size.
In this case, n1 = n2 = 12 and n = 24. Therefore, mu_R = (2*12*12) / (12 + 12) + 1 = 13. The standard deviation of the number of runs can be calculated as:
sigma_R = sqrt{ [2n1n2(2n1n2-n1-n2)] / [(n1+n2)^2 (n1+n2-1)] }
= sqrt{ [2*12*12*(2*12*12-12-12)] / [(12+12)^2 (12+12-1)] }
= sqrt(148.800) = 12.187
The observed number of runs is R = 6 + 5 = 11. Therefore, the test statistic is:
Z = (11 - 13) / 12.187 = -0.1646
Since the absolute value of Z is less than the critical value of 1.96, we fail to reject the null hypothesis that U.S. oil production is random over time.

An Energy Analyst Wants to Test If U