For each successive presidential term from Teddy Roosevelt to William J. Clinton (second term), the party affiliation controlling the White House is shown below, where R designates Republican and D designates Democrat.† R R R D D R R D D D D D R D R R D R R R D D Historical Note: In cases in which a president died in office or resigned, the period during which the vice president finished the term is not counted as a new term. Test the sequence for randomness. Use ? = 0.05. (a)What is the level of significance? (b)Find the sample test statistic R, the number of runs. (c)Find the upper and lower critical values in the Critical Values for Number of Runs R table.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
For each successive presidential term from Teddy Roosevelt to William J. Clinton (second term), the party affiliation controlling the White House is shown below, where R designates Republican and D designates Democrat.†
| R | R | R | D | D | R | R | D | D | D | D | D |
| R | D | R | R | D | R | R | R | D | D |
Part a:
The level of significance is ∝ = 0.05.
The test hypothesis are,
H0: The symbols are randomly mixed.
H1: The symbols are not randomly mixed.
We reject the null hypothesis is R< c1 or R > c2.
(Where, c1 and c2 are the lower and upper critical values)
Part b:
From the provided data we have,
RRR DD RR DDDDD R D RR D RRR DD.
Thus, the number of runs = R = 10.
Hence, test statistic = R = 10.
n1 = number of times symbol R occurred = 11
n2 = number of times symbol D occurred = 11.
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