The place kicker on a team in the American National Football League (NFL) has an all-time success rate (total number ?eld goals "made" divided by total number of ?eld goals attempted) of 0.82 on ?eld goal attempts of 55 yards or shorter. An attempt to quantify his performance with a probability model resulted in a Beta B(4.5, 1) distribution.

(i) Is this model consistent with the computed all-time success rate?

(ii) To be considered an elite place kicker, the success rate from this distance (D ≤ 55) needs to improve to at least 0.9. Determine the probability that this particular place kicker achieves elite status in any season, assuming that he maintains his current performance level.

(iii) It is known that the computed probability of attaining elite status is sensitive to the model parameters, especially α. For the same ?xed value β = 1, compute the probability of attaining elite status for the values α = 3.5, 4.0, 4.5, 5.0, 5.5. Plot these probabilities as a function of α.