In Francis Galton's time some political candidates included in their campaign material the "total marks" (score) they had received in a grueling (44 hours over 8 days) but prestigious mathematics examination. Galton felt many politicians claimed higher scores than they received. He obtained marks actually given on two successive examinations and found them to compare favorably to a N(m, s2 ) distribution. His data consisted of the scores received by 800 men, and only 6.7% of them were greater than 1500 marks, which was minimally sufficient to be awarded the title of "wrangler of mathematics."
a. If the data are from a normal distribution with m = 900, show how to find s2 = 1600.
b. The one of the approximately 400 students who receives the greatest number of marks is called "senior wrangler." If scores are normally distributed, what score is likely to qualify for that distinction.
c. To address the concern Galton was investigating, suppose 140 candidates have reported scores they claim they received on the examination.
i. What assumptions must be made in order to use the normal distribution for inference?
ii. If the assumptions can be made, what is the expected number with scores greater than 1500 marks
iii. Suppose 24 of the 140 claim they received scores greater than 1500 marks, what would you conclude about the truthfulness of the scores claimed?