2) Eight runners (called “A” through “H”) line up for the start of the 100 meter
ID: 3023271 • Letter: 2
Question
2) Eight runners (called “A” through “H”) line up for the start of the 100 meter dash. The distribution of runners’ times are all normal, with the same mean of exactly 11 seconds, but the SD’s differ. The SD’s for runners A, B, C, D, E, F, G, H are: .10, .13, .16, .19, .21, .24, .27, and .30, respectively.
(A) What is the probability that A beats H?
(B) Which runner has the greatest chance of winning the race? Note: for this part, you do not need to produce a rigorous proof; an intuitive argument will suffice.
3) The distance between Sophie’s house and the parking lot of the Orlando airport is, according to Google maps, exactly 120 miles. Sophie resets the odometer of her car to zero, and heads from her house to the airport. She stops at a service station close to the airport, notes that the odometer reading is X miles, then heads for the airport. When she returns from her trip, as she leaves the airport to drive back home, she resets the odometer, and when she arrives home she notes that the odometer reading is Z miles. Assume that the odometer readings are normally distributed, with mean equal to the distance travelled, and variance equal to ? 2 , which does not depend on the distance travelled (? is unknown). Assume further that X and Z are independent. Suppose that X is observed to be 70 miles and Z is observed to be 120.1 miles. Find a 95% confidence interval for the distance, call it d, between Sophie’s’s house and the service station.
Advice: Initially ignore the fact that X is observed to be 70 and Z is observed to be 120.1 and think of them only as random variables with the normal distributions stipulated above. Derive a formula for the CI for d, and at the last second plug the observed values into that formula.
E1 In class we stated that if U ~ n and ½ 2 then We did not prove that fact. Prove that E(U) does not depend on v. (Obviously, you can't use equation (1).)Explanation / Answer
2) the mean of all is 11 seconds
a) probability A beats H
STD DEV OF A =0.10
STD DEV OF H = 0.30
NOW THE DIFFERENCE = 0.30 - 0.10 = 0.20
SO THE PROBILITY = 0.10/0.20 = 0.5
B) INTUITIVELY THE PERSON HAVING THE LEAST STANDARD DEV WILL MOSTLY WIN THE RACE HENCE ACCORDING TO THE DATA, A HAS THE LEAST STANDARD DEVIATION = 0.10
HENCE A WILL WIN.
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