Final answer:
The problems involve using exponential distribution to calculate time between events and interpreting quartiles in statistical data. The average times between successive cars is 12 seconds, and the time for another seven cars is 84 seconds. For the 100-meter dash, the third quartile indicates that 75% of finishes occur within 11.5 seconds or less.
Step-by-step explanation:
To solve the problems provided effectively, we would need to apply principles of probability theory and statistics, specifically utilizing the exponential distribution and interpretation of quartiles.
Exponential Distribution and Car Passing
a. Given that cars pass at an average rate of five cars per minute, the time between successive cars is exponential with a mean of 1/5 minute or 12 seconds.
b. To find the average time for another seven cars to pass by, we would calculate 7 times the mean duration between cars, resulting in 84 seconds.
c. The probability that the next car passes within 20 seconds can be found using the exponential cumulative distribution function (CDF), which requires calculating 1 - e^(-5/60 × 20).
d. Conversely, the probability that a car will not pass for at least another 15 seconds is given by the survival function, which is e^(-5/60 × 15).
Third Quartile Interpretation
The third quartile of 11.5 seconds in the 100-meter dash indicates that 75% of race finishes occurred in 11.5 seconds or less.