A simple random sample of 10 pages from a dictionary is obtained. The numbers of words defined on those pages are found, with the results $n=10$, $\bar{x}=66.9$ words, $s=16.3$ words. Given that this dictionary has 1428 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 49.0 words. Use a 0.10 significance level to test the claim that the mean number of words per page is greater than 49.0 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. What are the null and alternative hypotheses? A. \[ \begin{array}{l} H_{0}: \mu=49.0 \text { words } \\ H_{1}: \mu>49.0 \text { words } \end{array} \] c. \[ \begin{array}{l} H_{0}: \mu=49.0 \text { words } \\ H_{1}: \mu \neq 49.0 \text { words } \end{array} \] Determine the test statistic. $\square$ (Round to two decimal places as needed.) Determine the P-value. $\square$ (Round to three decimal places as needed.) State the final conclusion that addresses the original claim. B. $\mathrm{H}_{0}: \mu=49.0$ words $H_{1}: \mu<49.0$ words D. \[ \begin{array}{l} H_{0}: \mu>49.0 \text { words } \\ H_{1}: \mu<49.0 \text { words } \end{array} \] $\mathrm{H}_{0}$. There is evidence to support the claim that there 70,000 defined words in the dictionary.
Solution
Step 1 ：State the null and alternative hypotheses: \n\[\begin{array}{l}H_{0}: \mu=49.0 \text { words } \H_{1}: \mu>49.0 \text { words }\end{array}\]
Step 2 ：Calculate the test statistic using the formula for the z-score: \((x̄ - μ) / (s / √n)\), where x̄ is the sample mean, μ is the population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size. The test statistic is approximately \boxed{3.47}.
Step 3 ：Calculate the P-value, which is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. The P-value is approximately \boxed{0.00026}.
Step 4 ：Compare the P-value to the significance level. If the P-value is less than the significance level, reject the null hypothesis. In this case, the P-value is less than the significance level of 0.10, so we reject the null hypothesis.
Step 5 ：Interpret the results in the context of the original claim. Rejecting the null hypothesis suggests that the mean number of words per page is greater than 49.0. In the context of the original claim, this suggests that there are more than 70,000 defined words in the dictionary.
