Final Answers:
1. Test statistic = -6.405
2. P-value = 0.0001
3. The small p-value indicates a significant departure from the null hypothesis, suggesting the probability of a true negative on the cancer test is significantly less than 0.7.
Explanation:
In this hypothesis testing scenario, the objective is to assess whether the probability of a true negative on a test for a certain cancer
is significantly less than the hypothesized value of 0.7. The test statistic, calculated as -6.405, is obtained by comparing the sample proportion of successes (322 out of 492) to the expected proportion under the null hypothesis. This statistic is derived using a standard formula for proportions.
The p-value, a crucial indicator in hypothesis testing, is determined to be 0.0001. This exceedingly small p-value suggests that the observed sample results are highly unlikely under the assumption that the true probability is 0.7. Given the significance level
, which represents the threshold for rejecting the null hypothesis, the p-value is lower, leading to the rejection of the null hypothesis.
In essence, the evidence from the sample strongly supports the alternative hypothesis
, indicating that the probability of a true negative on the cancer test is indeed significantly less than 0.7. This outcome underscores the importance of the test in potentially improving the accuracy of identifying individuals without the specific cancer, reflecting a substantial finding in the realm of medical diagnostics.