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 316 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 Student's t, since np > 5 and nq > 5.The Student's t, since np < 5 and nq < 5.    The standard normal, since np > 5 and nq > 5.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.)


Sketch the sampling distribution and show the area corresponding to the P-value.

(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

Bolded text are the answers.

Here we have

(a) What is the level of significance?

it is given in the problem.

State the null and alternate hypotheses.

H0: p = 0.261; H1: p ≠ 0.261 (since the words "either way" is given in the problem)

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

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

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

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.)

Test statistic:

we have

  

Therefore, the  value of the sample test statistic =-2.75

(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)

We shall look into the EXCEL function 2*NORM.S.DIST(-2.75,TRUE) for which is the p-value of the statistic. =2*0.00298=0.00596. (The probability that we get is the left tail one and that's why we need to multiply it by 2.

The P-value of the test statistic=0.006

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.