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Plato's Republic: Syllable Patterns Prose rhythm is characterized by 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 \(\alpha=0.01.\)

Step by step solution

01

State the Hypotheses

We want to test if the proportion of five-syllable sequences with two short and three long syllables in the new manuscript is different from that of Plato's Republic. Thus, we set up the null hypothesis as \( H_0: p = 0.261 \) and the alternative hypothesis as \( H_a: p eq 0.261 \). Here, \( p \) is the population proportion in the ancient manuscript.

02

Determine the Test Statistic

We will use a z-test for proportions. The test statistic is calculated by the formula \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \), where \( \hat{p} = \frac{x}{n} = \frac{61}{317} \approx 0.1922 \) is the sample proportion, \( p_0 = 0.261 \) is the hypothesized proportion, and \( n = 317 \) is the sample size.

03

Perform the Calculations

Substitute the known values into the z-test formula: \[ z = \frac{0.1922 - 0.261}{\sqrt{\frac{0.261(1-0.261)}{317}}} \] \[ = \frac{-0.0688}{\sqrt{\frac{0.261 \times 0.739}{317}}} \] \[ \approx \frac{-0.0688}{0.0245} \approx -2.807 \].

04

Make the Decision

We compare the test statistic to the critical value for a two-tailed test at \( \alpha = 0.01 \). The critical z-values are approximately \( \pm 2.576 \). Since \( z = -2.807 \) is less than \( -2.576 \), we reject the null hypothesis \( H_0 \).

05

Conclusion

The data provide sufficient evidence at the \( \alpha = 0.01 \) level to conclude that the population proportion of five-syllable sequences in the ancient manuscript is different from that in Plato's Republic.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Z-test for proportions

A Z-test for proportions is a statistical method used to determine if there is a significant difference between the proportion in a sample and a known population proportion. In essence, it tells us whether our observations, like those in the new manuscript, are consistent with what's expected based on prior data, such as Plato's Republic.
If the calculated Z-value from your sample is beyond a critical threshold, it suggests the sample does not match the historical proportion, and hence, could be significantly different.
The formula for calculating the test statistic is:

• Let \( \hat{p} \) be the sample proportion, \( p_0 \) be the population proportion under the null hypothesis, and \( n \) be the sample size.
• The formula is: \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \) This quantifies the difference between your data and the known standard in terms of standard error.

This test is particularly useful when dealing with proportions, provided that the sample size is large enough to justify the use of normal approximation, as was the case in analyzing the five-syllable sequences.

Null hypothesis

The null hypothesis is the foundational idea that acts as the starting point for statistical hypothesis testing. It represents a baseline assumption of "no effect" or "no difference." In our exercise, the null hypothesis assumes that the proportion of five-syllable sequences in the ancient manuscript is the same as in Plato's Republic.
Symbolically, this is represented as:
\( H_0: p = 0.261 \)
The essence of the null hypothesis is its role as a challenge to disprove. When performing hypothesis testing, we use data to decide whether to reject the null hypothesis in favor of an alternative hypothesis, which suggests there is a significant difference or effect.
If the evidence strongly contradicts the null hypothesis, we "reject" it. However, if the data do not provide sufficient grounds to refute the null hypothesis, we "fail to reject" it. It’s crucial to note that we never "accept" the null hypothesis; rather, we gain or lack evidence against it.

Sample size

Sample size refers to the number of individual observations or elements in a sample. It's a critical factor in statistical tests as it impacts the accuracy and reliability of the results. In the context of Z-tests, a larger sample size generally provides more precise estimates, reducing the standard error and allowing for a more accurate determination of statistical significance.
In our problem, the sample size is 317. This number is used to calculate the standard error in the Z-test formula, affecting how we determine the significance of the results. Larger samples tend to give more reliable results because they better approximate the population.
It’s important to ensure that the sample size is not arbitrarily chosen but rather large enough to capture the variability within the population. Proper sample sizing ensures that the results are both reliable and valid.

Critical value

The critical value is a key concept in hypothesis testing, representing the point beyond which we consider the test statistic to be "extreme." It helps in deciding whether to reject the null hypothesis. In hypothesis tests, we compare the calculated test statistic with critical values derived from the chosen significance level (\( \alpha \)).
In our example, with a significance level of \( \alpha = 0.01 \) and for a two-tailed test, the critical values are approximately \( \pm 2.576 \).

• If the test statistic falls beyond these values, we reject the null hypothesis.
• If the test statistic is within these limits, we do not have sufficient evidence to reject the null hypothesis.

Hence, critical values act as a threshold separating results that are plausible under the null hypothesis from those that suggest a significant deviation.