Question

The speed at which you can log into a website through asmartphone is an important quality characteristic of that website.In a recent test, the mean time to log into the JetBlue Airwayswebsite through a smartphone was 4.237 seconds. (Data extractedfrom N. Trejos, "Travelers Have No Patience for Slow MobileSites," USA Today, April 4, 2012, p. 3B.) Suppose that the downloadtime is normally distributed, with a standard deviation of1.3 seconds. What is the probability that

a download time isa. less than 2 seconds?
b. between 1.5 and 2.5 seconds?
c. above 1.8 seconds?
d. Ninety-nine percent of the download times are slower (higher)than how many seconds?
e. Ninety-five percent of the download times are between whattwo values, symmetrically distributed around the mean?

Answer 1

To find the required probabilities for the website login times, we calculate z-scores for the given times and use the normal distribution to determine the probabilities and times that correspond to different percentiles.

Step-by-step explanation:

To determine the probability of different download times for the JetBlue Airways website when logged in through a smartphone, we use the properties of the normally distributed variable, given that the mean login time is 4.237 seconds and the standard deviation is 1.3 seconds. For each of the probabilities required, we convert the specified time to a z-score and then use the normal distribution table or a calculator with normal distribution capabilities to find the corresponding probabilities.

1. To find the probability that the download time is less than 2 seconds, we calculate the z-score for 2 seconds and then look up this z-score in the normal distribution table.
2. For the probability of a download time between 1.5 and 2.5 seconds, we find the z-scores for both times and then find the area under the normal curve between these two z-scores.
3. To determine the probability that the download time is above 1.8 seconds, we subtract the cumulative probability up to 1.8 seconds from 1.
4. To find the time such that 99 percent of the download times are slower, we look for the z-score that corresponds to the 1st percentile and translate it back to the time in seconds.
5. Finally, to find the range within which 95 percent of the download times fall, symmetrically around the mean, we look for the z-scores that correspond to the 2.5th and 97.5th percentiles and convert these z-scores to times in seconds.