The North American Mathematics Challenge for high school students was held last month. The challenge consists of two rounds. The difficulty of round 1 is at the same level as last year's, and the average round 1 score of contestants who took this challenge last year was 85. Given that the contestants this year are generally younger and from lower grades, the organizer believes that the contestants will perform worse than last year. If the average round 1 score of the contestants who took this challenge this year is below 75, the organizer wants to adjust the difficulty level of round 2 to make it slightly easier. The following are the scores of selected 30 contestants who took the challenge this year. The scores with an asterisk correspond to those of contestants from Canada:

83, 87*, 68, 72, 64, 75*, 65, 95, 94*, 68, 63*, 85*, 64, 59, 98,

67, 70, 78*, 89, 62, 69*, 83, 76, 83*, 67, 64, 78*, 55, 72, 61*

The organizer is now more concerned about the true proportion of all contestants from Canada whose score is lower than 80.

(a) Construct a 95% confidence interval for the true proportion of all contestants from Canada whose score is lower than 80. Make sure to state all your assumptions in the context of the question.

(b) Give an interpretation to the confidence interval in the context of the question.

(c) The confidence interval and its interpretation are valid only when the assumptions you stated in (a) are met. Provide

an example in the context of the question where the assumptions can be violated. Also explain the potential

consequence. Please limit your answer to no more than two sentences.