1. A random sample of40binomial trials resulted in16successes. Test the claim that the population proportion of successes does not equal 0.50. Use a level of significance of 0.05.

(a) Can a normal distribution be used for thedistribution? Explain. i. No,npandnqare both less than 5.

ii. No,npis greater than 5, butnqis less than 5.

iii. No,nqis greater than 5, butnpis less than 5.

iv. Yes,npandnqare both less than 5.

v. Yes,npandnqare both greater than 5.

(b) State the hypotheses. i. H₀:p<0.5;H₁:p= 0.5

ii. H₀:p= 0.5;H₁:p>0.5

iii. H₀:p= 0.5;H₁:p0.5

iv. H₀:p= 0.5;H₁:p<0.5

(c) Computep hat. (Enter a number.) Compute the corresponding standardized sample test statistic. (Enter a number. Round your answer to two decimal places.) (d) Find theP-value of the test statistic. (Enter a number. Round your answer to four decimal places.)

2. Prose rhythm is characterized as the occurrence of five-syllable sequences in long passages of text. This characterization may be used to assess the similarity among passages of text and sometimes the identity of authors. The following information is based on an article by D. Wishart and S. V. Leach appearing in Computer Studies of the Humanities and Verbal Behavior (Vol. 3, pp. 90-99). Syllables were categorized as long or short. On analyzing Plato's Republic, Wishart and Leach found that about 26.1% of the five-syllable sequences are of the type in which two are short and three are long. Suppose that Greek archaeologists have found an ancient manuscript dating back to Plato's time (about 427 - 347 B.C.). A random sample of 317 five-syllable sequences from the newly discovered manuscript showed that 61 are of the type two short and three long. Do the data indicate that the population proportion of this type of five syllable sequence is different (either way) from the text of Plato's Republic? Use = 0.01.

(a)What is the level of significance? State the null and alternate hypotheses. i. H₀: p = 0.261; H₁: p < 0.261

ii. H₀: p 0.261; H₁: p = 0.261

iii. H₀: p = 0.261; H₁: p > 0.261

iv. H₀: p = 0.261; H₁: p 0.261

(b)What sampling distribution will you use? i. The Student's t, since np < 5 and nq < 5.

ii. The Student's t, since np > 5 and nq > 5.

iii. The standard normal, since np < 5 and nq < 5.

iv. The standard normal, since np > 5 and nq > 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.) (c)Find the P-value of the test statistic. (Round your answer to four decimal places.)