A random group of used books was selected from book warehouse to analyze their condition. The book store believes that 20% of the books are in excellent condition, 38% are in very good condition, 15% are in fair condition, and 27% are in poor condition. Is there evidence to reject this hypothesis at = 0.05? A) There is evidence to reject the claim that the books are distributed as claimed because the test value 16.291 7.815 B) There is not evidence to reject the claim that the books are distributed as claimed because the test value 7.815 9.488 D) There is not evidence to reject the claim that the books are distributed as claimed because the test value 9.488

Question

A random group of used books was selected from book warehouse to analyze their condition. The book store believes that 20% of the books are in excellent condition, 38% are in very good condition, 15% are in fair condition, and 27% are in poor condition. Is there evidence to reject this hypothesis at = 0.05? A) There is evidence to reject the claim that the books are distributed as claimed because the test value 16.291 > 7.815 B) There is not evidence to reject the claim that the books are distributed as claimed because the test value 7.815 < 16.291 C) There is evidence to reject the claim that the books are distributed as claimed because the test value 16.291 > 9.488 D) There is not evidence to reject the claim that the books are distributed as claimed because the test value 9.488 < 16.291

Quizplus · Accepted Answer

We need to use a chi-square goodness-of-fit test with the following hypotheses:
H0: The books are distributed as claimed.
Ha: The books are not distributed as claimed.
We calculate the test statistic using the formula:
chi-square = Σ(Observed - Expected)2 / Expected
where Σ means to sum over all categories, Observed is the actual number of books in each category that were found in the sample, and Expected is the number of books we would expect in each category if the claimed distribution is true. We can calculate Expected using the formula:
Expected = Total number of books in sample × Proportion in claimed distribution
Using this method, we get the following table:

| Condition   | Observed | Claimed Proportion | Expected |
|-------------|----------|--------------------|----------|
| Excellent   | 17       | 0.20               | 32.5     |
| Very Good   | 32       | 0.38               | 61.0     |
| Fair        | 9        | 0.15               | 22.5     |
| Poor        | 22       | 0.27               | 44.0     |
| Total       | 80       | 1.00               |          |

We can see that the claimed proportion and the expected frequency for each condition differ quite a bit from the observed frequency. We calculate the test statistic as follows:

chi-square = [(17-32.5)2/32.5] + [(32-61)2/61] + [(9-22.5)2/22.5] + [(22-44)2/44] = 16.291

We then compare this value to the critical value of the chi-square distribution with (4-1) = 3 degrees of freedom (df), because we have four categories and have used one degree of freedom to calculate the expected frequencies.

Using a chi-square distribution calculator, we find that the critical value for 3 df at the 0.05 level of significance is 7.815. Since 16.291 > 7.815, we reject the null hypothesis and conclude that there is evidence to reject the claim that the books are distributed as claimed. Therefore, the best choice is A).

A Random Group of Used Books Was Selected from Book