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 a = 0.01. (a) What is the level of significance? State the null and alternate hypotheses О но: ра0.261; Н,: р %3D 0.261 О на: р 3D 0.261; H,: р> 0.261 О на: р 3D 0.261; H,: р <0.261 О на: р 3D 0.261; H,: р+0.261 (b) What sampling distribution will you use? O The standard normal, since n p > 5 and n q> 5. O The Student's t, since n p> 5 and n q 5. O The Student's t, since n-p < 5 and n q < 5. O The standard normal, 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.) P-value P-value P-value -3 -2 - 1 0 1 2 -3 -2 -1 0 2 3 -1 2 3 -2 - 1 0 2

Answer 1

a) The level of significance is a =0.01 b) From the given scenario, the null and alternate hypotheses can be stated as follows: Null hypothesis: The population proportion of the type of five syllable sequence type in which two are short and three are long is equal to 0.261. Ho: P=0.261 Altemate hypothesis: The population proportion of the type of five syllable sequence type in which two are short and three are long is not equal to 0.261. H:P+0.261 b) The sampling distribution is the standard normal, since np > 5 and ng > 5 option (a). The random sample is n = 317 The number of the type of five syllable sequence type in which two are short and three are long is x=61 The sample proportion of the type of five syllable sequence type in which two are short and three are long is p=*=61 0.192 En 317 Calculate the test statistic as follows: Z -P P(1-P) 0.192 -0.261 0.261(1-0.261) 317 -0.069 0.0247 = -2.78

12 = 2.78 c) Calculate the P-value of the test statistic. P=2x P(Z > 2.78) = 21- P(Z < 2.78)] (Normal table value for 2.78 is 0.9973] = 2(1-0.9973) = (2x0.0027) = 0.0054

Distribution Plot Normal, Mean=0. StDev=1 Density 0.002718 0.002718 0.0 -- -2.78 2.78 хо Graph-2 d) At a =0.01 level, we reject the null hypothesis and conclude the data are statistically significant.