Question

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 α = 0.01.

(a) What is the level of significance? ________

State the null and alternate hypotheses.

___H₀: p ≠ 0.261; H₁: p = 0.261

___H₀: p = 0.261; H₁: p ≠ 0.261     

___H₀: p = 0.261; H₁:  p > 0.261

___H₀: p = 0.261; H₁:  p < 0.261

(b) What sampling distribution will you use?

___The standard normal, since n·p > 5 and n·q > 5.

___The standard normal, since n·p < 5 and n·q < 5.     

___The Student's t, since n·p < 5 and n·q < 5.

___The Student's t, since n·p > 5 and n·q > 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.) ______

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

___At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

___At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.    

___At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

___At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

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

___There is sufficient evidence at the 0.01 level to conclude that the true proportion of the five-syllable sequence differs from that of the text of Plato's Republic.

___There is insufficient evidence at the 0.01 level to conclude that the true proportion of the five-syllable sequence differs from that of the text of Plato's Republic.    

Answer 1

a)

H0: p = 0.261; H1: p ≠ 0.261   

b)
The standard normal, since n·p > 5 and n·q > 5.

c)

phat = 61/317 = 0.192

z = (phat - p)/sqrt(p*(1-p)/n)
= (0.192 - 0.261)/sqrt(0.261*(1-0.261)/317)
= -2.80

c)

p value = 0.0052

d)

_At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

e)

There is sufficient evidence at the 0.01 level to conclude that the true proportion of the five-syllable sequence differs from that of the text of Plato's Republic.