Question

(1) Symposium is part of a larger work referred to as Plato's Dialogues. Wishart and Leach† found that about 21.4% of five-syllable sequences in Symposium are of the type in which four are short and one is long. Suppose an antiquities store in Athens has a very old manuscript that the owner claims is part of Plato's Dialogues. A random sample of 488 five-syllable sequences from this manuscript showed that 128 were of the type four short and one long. Do the data indicate that the population proportion of this type of five-syllable sequence is higher than that found in Plato's Symposium? Use α = 0.01.

(a) What is the level of significance? _________

(b) 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.) ________

(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 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? _____________

(b) 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.)

Answer 1

1)

a) level of significance = 0.05

b) Ho :   p =    0.214                  
H1 :   p >   0.214       (Right tail test)          
                          
Level of Significance,   α =    0.01                  
Number of Items of Interest,   x =   128                  
Sample Size,   n =    488                  
                          
Sample Proportion ,    p̂ = x/n =    0.2623                  
                          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0186                  
Z Test Statistic = ( p̂-p)/SE = (   0.2623   -   0.214   ) /   0.0186   =   2.60

c) p-Value   =   0.0046   [Excel function =NORMSDIST(-z)
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2)

a)  the level of significance =0.01

b) Ho :   p =    0.261                  
H1 :   p ╪   0.261       (Two tail test)          
                          
Level of Significance,   α =    0.01                  
Number of Items of Interest,   x =   61                  
Sample Size,   n =    316                  
                          
Sample Proportion ,    p̂ = x/n =    0.1930                  
                          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0247                  
Z Test Statistic = ( p̂-p)/SE = (   0.1930   -   0.261   ) /   0.0247   =   -2.75

c)

p-Value   =   0.0059   [excel formula =2*NORMSDIST(z)]