Question

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​, x=57.8 ​words, s=16.4 words. Given that this dictionary has 1463 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 47.8 words. Use a 0.01 significance level to test the claim that the mean number of words per page is greater than 47.8 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. State the final conclusion that addresses the original claim.

Answer 1

Given the numbers of words defined on those pages are​ found, with the results n=10​, and the sample mean as =57.8 ​words, sample standard deviation s=16.4 words.

The claim is that the mean number of words per page is greater than = 47.8 words. thus based on the claim the hypotheses are:

based on the hypothesis it will be right-tailed test, assuming the distribution as normal but the population standard deviation is unknown hence t-distribution is applicable, hence degree of freedom is used which is calculated as df = n-1= 10-1= 9.

Rejection region:

Given the significance level as 0.01 hence reject the Ho if the P-value is less than 0.01.

Test statistic:

P-value:

The p-value is calculated using the excel formula for t-distribution which is =T.DIST.RT( 1.928, 9), thus the P-value is computed as 0.043.

Conclusion:

Since the p-value is greater than 0.01 hence we fail to reject the Null hypothesis(Ho) and hence conclude that at 0.01 level of significance we do not have enough evidence to support the claim that the mean number of words per page is greater than 47.8 words