Final answer:
To test the claim that the mean number of words per page is greater than 48.1, we set up null and alternative hypotheses. Using the t-statistic and the t-distribution, we calculate the p-value. The p-value is very small, suggesting that the mean number of words per page is indeed greater than 48.1 words, supporting the claim of more than 70,000 defined words.
Step-by-step explanation:
To test the claim that the mean number of words per page is greater than 48.1 words, we need to set up the null and alternative hypotheses. The null hypothesis, denoted as H0, assumes that the mean number of words per page is equal to or less than 48.1 words. The alternative hypothesis, denoted as Ha, assumes that the mean number of words per page is greater than 48.1 words.
The test statistic for this hypothesis test is the t-statistic, which is calculated by taking the difference between the sample mean (x) and the hypothesized mean (μ), and dividing it by the standard error of the mean (s/√n). This test statistic follows a t-distribution with n-1 degrees of freedom, where n is the sample size. The p-value can be calculated using the t-distribution.
Comparing the p-value to the significance level of 0.01, if the p-value is less than 0.01, we reject the null hypothesis. If the p-value is greater than or equal to 0.01, we fail to reject the null hypothesis.
In this case, the calculated t-statistic is (53.5 - 48.1) / (16.5 / √20) = 13.06. Using the t-table or a t-distribution calculator, we find that the p-value is extremely small, much less than 0.01. Therefore, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the mean number of words per page is greater than 48.1 words. This suggests that there are more than 70,000 defined words in the dictionary.