Question

At a large company banquet for several thousand employees and their families, many of the attendees became ill the next day. The company doctor suspects that the illness may be related to the fish, one of three options for the main course. Because all the dinner guests had to preorder their meal, the doctor was able to randomly select and contact 80 people that ate the fish, of which 64 people got sick. The doctor also randomly selected (and contacted) 60 people that did not eat the fish, of which 39 people got sick. The doctor also knows that at least 1000 attendees ordered the fish.
Required:
a. Is this convincing evidence that the true proportion of all attendees who ate the fish that got sick is more than the true proportion of all attendees who did not eat the fish that got sick?
b. Which mistake (a Type I error or a Type II error) could you have made? Interpret the potential error in context.

Answer 1

Answer:

a. Yes.  This provides convincing evidence that the true proportion of all attendees who ate the fish that got sick (80%) is more than the true proportion of all attendees who did not eat the fish that got sick.

b. The mistake here would have been the rejection of the Doctor's theory or hypothesis to the effect that more attendees who ate the fish got sick than those who did not eat the fish. This is a Type 1 error.   A Type 1 error occurs when a null hypothesis is rejected when it is true. On the other hand, a Type II error occurs when the null hypothesis is accepted when it should be rejected.  While a Type I error is equivalent to a false positive, a Type II error is equivalent to a false negative.

Step-by-step explanation:

Total number of attendees who ordered fish = 1,000

Sample size of the attendees who ate fish = 80

Number of attendees who ate the fish and got sick = 64 (80% or 64/80)

Sample size of attendees who did not eat fish = 60

Number of attendees who did not eat fish and got sick = 39 (65% or 39/60)