Question 1165132
This is a hypothesis test for a single population proportion ($p$).

### 1. State the Hypotheses

The goal is to determine if the proportion in the new manuscript is **different** (either way) from Plato's *Republic*.

* **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 ($p = 0.261$).
    $$H_0: p = 0.261$$
* **Alternative Hypothesis ($H_1$):** The population proportion is different from Plato's Republic ($p \neq 0.261$). (This is a two-tailed test.)
    $$H_1: p \neq 0.261$$

### 2. Identify the Given Data

* **Hypothesized Population Proportion ($p_0$):** $0.261$
* **Sample Size ($n$):** $317$
* **Number of Successes ($x$):** $61$
* **Significance Level ($\alpha$):** $0.01$

### 3. Calculate the Sample Proportion ($\hat{p}$)

$$\hat{p} = \frac{x}{n} = \frac{61}{317} \approx 0.1924$$

### 4. Calculate the Test Statistic ($Z_{\text{test}}$)

The formula for the $Z$-test statistic for a proportion is:
$$Z_{\text{test}} = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$$

1.  **Calculate the Standard Error (SE) of the Proportion:**
    $$\text{SE} = \sqrt{\frac{0.261(1 - 0.261)}{317}} = \sqrt{\frac{0.261 \times 0.739}{317}} \approx \sqrt{0.00060907}$$
    $$\text{SE} \approx 0.02468$$

2.  **Calculate the $Z$-score:**
    $$Z_{\text{test}} = \frac{0.1924 - 0.261}{0.02468} = \frac{-0.0686}{0.02468}$$
    $$Z_{\text{test}} \approx -2.7796$$

Rounding the answer to two decimal places:
$$\mathbf{Z_{\text{test}} \approx -2.78}$$

### 5. Find the $P$-value of the test statistic.

Since this is a two-tailed test, the $P$-value is twice the area in the tail defined by the test statistic.
$$P\text{-value} = 2 \times P(Z \le -|Z_{\text{test}}|) = 2 \times P(Z \le -2.7796)$$

Using a standard normal distribution table or calculator:
$$P(Z \le -2.7796) \approx 0.00273$$

$$P\text{-value} = 2 \times 0.00273 \approx 0.00546$$

Rounding the answer to four decimal places:
$$\mathbf{P\text{-value} \approx 0.0055}$$

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### Conclusion (Optional)
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*.