For each successive presidential term from Franklin Pierce (the 14th president, elected in 1853) to George W. Bush (43rd president), the party affi­liation controlling the White House follow, where R designates Republican and D designates Democrat

Test the sequence for randomness at the 5% level of signi­ficance. Use the following outline.

(a) State the null and alternate hypotheses.

(b) Find the number of runs R, n₁, and n₂. Let n₁ = number of Republicans and n₂ = number of Democrats.

(c) In this case, n₁ = 21, so we cannot use Table 10 of Appendix II to ­find the critical values. Whenever either n₁ or n₂ exceeds 20, the number of runs R has a distribution that is approximately normal, with

We convert the number of runs R to a z value, and then use the normal distribution to find the critical values. Convert the sample test statistic R to z using the formula

(d) The critical values of a normal distribution for a two-tailed test with level of signifi­cance a = 0.05 are ‑1.96 and 1.96 (see Table 5(c) of Appendix II). Reject H₀ if the sample test statistic z ≤ –1.96 or if the sample test statistic z ≤ –1.96. Otherwise, do not reject H₀.

Using this decision process, do you reject or fail to reject H₀ at the 5% level of signifi­cance? What is the P-value for this two-tailed test? At the 5% level of signifi­cance, do you reach the same conclusion using the P-value that you reach using critical values? Explain.

(e) Interpret your results in the context of the application.

Transcribed Image Text:

D DRRD R R R R DRD RR R RDDR R D D D DDRRDD R R D RRR D DR Znin2 Sample z s-1.96 -1.96 < Sample z < 1.96 | Sample Z21.96 Reject Ho Reject Ho Fail to reject H