Final answer:
The value of d where 80% of travelers purchase tickets fewer than d days in advance and the average time is 28 days, the exponential distribution CDF is used. Solving the equation 0.80 = 1 - e^(-(1/28)d), we find that d is approximately 39 days.
Step-by-step explanation:
The student's question deals with determining the value of d when the time to purchase an airline ticket is exponentially distributed, with an average amount of time equal to 28 days, and there is an 80% chance of a traveler purchasing the ticket fewer than d days in advance. To find the value of d, we use the cumulative distribution function (CDF) for the exponential distribution, which is 1 - e-λx, where λ is the rate parameter, λ = 1/mean, and x is the number of days.
Since the mean is 28, λ equals 1/28. We set the CDF to 0.80 and solve for d: 0.80 = 1 - e-(1/28)d. Solving for d yields that d ≈ 39 days. Therefore, travelers have an 80% chance of purchasing a ticket fewer than 39 days in advance.