A simple random sample of 20 pages from a dictionary is obtained. The numbers of words defined on those pages are found, with the results $n=20, \bar{x}=58.3$ words, $s=16.3$ words. Given that this dictionary has 1488 pages with defined words, the claim that there are more than 70,000 defined words is equivalent to the claim that the mean number of words per page is greater than 47.0 words. Use a 0.10 significance level to test the claim that the mean number of words per page is greater than 47.0 words. What does the result suggest about the claim that there are more than 70,000 . defined words? Identify the null and alternative hypotheses, test statistic, P. value, and state the final conclusion that addresses the original claim. Assume that the population is normally distributed. Determine the test statistic. 3.10 (Round to two decimal places as needed.) Determine the P-value. (Round to three decimal places as needed.)

Step 1 ：We are given a simple random sample of 20 pages from a dictionary. The numbers of words defined on those pages are found, with the results $n=20$, $\bar{x}=58.3$ words, $s=16.3$ words. We are asked to test the claim that the mean number of words per page is greater than 47.0 words at a 0.10 significance level.

Step 2 ：The null hypothesis is that the mean number of words per page is equal to 47.0 words, and the alternative hypothesis is that the mean number of words per page is greater than 47.0 words.

Step 3 ：The test statistic for a hypothesis test for a population mean is given by the formula: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$ where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesized population mean, $s$ is the sample standard deviation, and $n$ is the sample size.

Step 4 ：Substituting the given values into the formula, we get $t = \frac{58.3 - 47.0}{16.3 / \sqrt{20}} \approx 3.10$.

Step 5 ：The P-value is the probability of observing a test statistic as extreme as the one we calculated, assuming the null hypothesis is true. We can find the P-value using a t-distribution table or a statistical software. Since we are performing a one-tailed test (because the alternative hypothesis is that the mean is greater than 47.0 words), the P-value is the area to the right of the test statistic on the t-distribution.

Step 6 ：The P-value is approximately 0.003. Since the P-value is less than the significance level of 0.10, we reject the null hypothesis.

Step 7 ：This suggests that the mean number of words per page is greater than 47.0 words. Therefore, it is likely that there are more than 70,000 defined words in the dictionary.

Step 8 ：Final Answer: The test statistic is \(\boxed{3.10}\) and the P-value is \(\boxed{0.003}\). We reject the null hypothesis that the mean number of words per page is equal to 47.0 words. This suggests that the mean number of words per page is greater than 47.0 words, and therefore, it is likely that there are more than 70,000 defined words in the dictionary.