document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165132>Question 1165132</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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.\r<BR>\n" );
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document.write( "What is the value of the sample test statistic? (Round your answer to two decimal places.) \r<BR>\n" );
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document.write( "(c) Find the P-value of the test statistic.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853114 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=853114\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a hypothesis test for a single population proportion ($p$).\r<BR>\n" );
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document.write( "### 1. State the Hypotheses\r<BR>\n" );
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document.write( "The goal is to determine if the proportion in the new manuscript is **different** (either way) from Plato's *Republic*.\r<BR>\n" );
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document.write( "* **Null Hypothesis ($H_0$):** The population proportion of the two short and three long (S-S-L-L-L) sequence type in the new manuscript is the same as Plato's Republic ($p = 0.261$).<BR>\n" );
document.write( "    $$H_0: p = 0.261$$<BR>\n" );
document.write( "* **Alternative Hypothesis ($H_1$):** The population proportion is different from Plato's Republic ($p \neq 0.261$). (This is a two-tailed test.)<BR>\n" );
document.write( "    $$H_1: p \neq 0.261$$\r<BR>\n" );
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document.write( "### 2. Identify the Given Data\r<BR>\n" );
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document.write( "* **Hypothesized Population Proportion ($p_0$):** $0.261$<BR>\n" );
document.write( "* **Sample Size ($n$):** $317$<BR>\n" );
document.write( "* **Number of Successes ($x$):** $61$<BR>\n" );
document.write( "* **Significance Level ($\alpha$):** $0.01$\r<BR>\n" );
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document.write( "### 3. Calculate the Sample Proportion ($\hat{p}$)\r<BR>\n" );
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document.write( "$$\hat{p} = \frac{x}{n} = \frac{61}{317} \approx 0.1924$$\r<BR>\n" );
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document.write( "### 4. Calculate the Test Statistic ($Z_{\text{test}}$)\r<BR>\n" );
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document.write( "The formula for the $Z$-test statistic for a proportion is:<BR>\n" );
document.write( "$$Z_{\text{test}} = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$$\r<BR>\n" );
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document.write( "1.  **Calculate the Standard Error (SE) of the Proportion:**<BR>\n" );
document.write( "    $$\text{SE} = \sqrt{\frac{0.261(1 - 0.261)}{317}} = \sqrt{\frac{0.261 \times 0.739}{317}} \approx \sqrt{0.00060907}$$<BR>\n" );
document.write( "    $$\text{SE} \approx 0.02468$$\r<BR>\n" );
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document.write( "2.  **Calculate the $Z$-score:**<BR>\n" );
document.write( "    $$Z_{\text{test}} = \frac{0.1924 - 0.261}{0.02468} = \frac{-0.0686}{0.02468}$$<BR>\n" );
document.write( "    $$Z_{\text{test}} \approx -2.7796$$\r<BR>\n" );
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document.write( "Rounding the answer to two decimal places:<BR>\n" );
document.write( "$$\mathbf{Z_{\text{test}} \approx -2.78}$$\r<BR>\n" );
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document.write( "### 5. Find the $P$-value of the test statistic.\r<BR>\n" );
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document.write( "Since this is a two-tailed test, the $P$-value is twice the area in the tail defined by the test statistic.<BR>\n" );
document.write( "$$P\text{-value} = 2 \times P(Z \le -|Z_{\text{test}}|) = 2 \times P(Z \le -2.7796)$$\r<BR>\n" );
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document.write( "Using a standard normal distribution table or calculator:<BR>\n" );
document.write( "$$P(Z \le -2.7796) \approx 0.00273$$\r<BR>\n" );
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document.write( "$$P\text{-value} = 2 \times 0.00273 \approx 0.00546$$\r<BR>\n" );
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document.write( "Rounding the answer to four decimal places:<BR>\n" );
document.write( "$$\mathbf{P\text{-value} \approx 0.0055}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Conclusion (Optional)<BR>\n" );
document.write( "Since the $P$-value $(0.0055)$ is less than the significance level $\alpha$ $(0.01)$, we **reject the null hypothesis ($H_0$)**. The data indicate that the population proportion of the two short and three long five-syllable sequence type in the newly discovered manuscript is significantly **different** from the text of Plato's *Republic*. </SPAN>\n" );
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