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 = 0.261H₀: p = 0.261; H₁: p > 0.261    H₀: p = 0.261; H₁: p ≠ 0.261H₀: 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.)


Sketch the sampling distribution and show the area corresponding to the P-value. (Select the correct graph.)

(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

Sol:

Let be the the proportion of five-syllable five-syllable sequences are of the type in which two are short and three are long.

Now we have a hypothesized value of by wishart and leach. Based on our sample we have check whether the newly discovered manuscript matches with plato's republic or not.

To do so we set our hypotheses

a).

The level of the significance is

b).

The test statistic is

Where

Hence

c).

Since this is a two tailed test

P-value is

Since this is <0.01. We conclude that the proportion found in plato's republic and the newly discovered manuscript are different. Hence the data indicate that the population proportion of this type of five syllable sequence is different.

d) and e)

the data do indicate that the population proportion of this type of five syllable sequence is different (either way) from the text of Plato's Republic