Bicycle sharing exists in a variety of cities around the country. Los Angeles has the Metro Bike Share system. Users pick up a bike from one station, go for a ride, and return the bike to any station. Go to www.pearsonhighered.com/sullivanstats and download the file 8_1_34. The data represent the duration (in minutes) of all rides in the fourth quarter (October through December) of 2018.
(a) Draw a relative frequency histogram of the data with a first-class having a lower class limit of 0 and a class width of 50. Describe the shape of the distribution.
(b) Determine the population mean and standard deviation duration.
(c) Based on the shape of the distribution of the variable “Duration” from the histogram in part (a), what must be true about the sample size in order for the distribution of the sample mean to be approximately normal?
(d) Suppose a random sample of n = 40 bike rides is obtained from the population of all bike rides in the fourth quarter of 2018. Describe the sampling distribution of the sample mean of the duration of all bike rides.
(e) The column “Sample1” represents the duration (in minutes) of a simple random sample of 40 bike rides where the ride was a “Round Trip” (that is, return the bike to the same location it was rented from). What is the sample mean?
(f) Determine the probability of obtaining a sample mean greater than that obtained in part (e) from the population of all bike rides in the fourth quarter of 2018 using the model described in part (d).
(g) The column “Sample2” represents the duration (in minutes) of a second (independent) simple random sample of 40 bike rides where the ride was a “Round Trip” (that is, return the bike to the same location it was rented from). What is the sample mean?
(h) Determine the probability of obtaining a sample mean greater than that obtained in part (g) from the population of all bike rides in the fourth quarter of 2018 using the model described in part (d).
(i) Explain why the probabilities in parts (f) and (h) differ. What does this suggest?