Name: Date: Period: Samples and Populations Investigation 1.1: Comparing Wait Times In your lifetime, you spend a lot of time waiting. Sometimes it feels like you could stand in line forever. For example, you may wait a long time for your favorite ride at an amusement park. During the summer, one estimate of average wait time at an amusement park is 60 minutes. The most popular rides can accommodate 1,500 people per hour. Lines form when more people arrive than the rides can fit. Amusement parks are designed to minimize wait times, but variability in the number of people who choose a particular ride can result in lines. Sally and her family spent the day at an amusement park. At the end of the day, Sally noticed the sign below. People often want to know what is typical in a given situation. For example, you might want to know the typical wait time at the amusement park. You can gather information as well as collect and gather data to determine how long people typically wait. All data sets include some variability. Not all waiting times are the same. Not all amusement parks are the same. The wait times provided are samples of the wait times at the amusement park. You can use data from samples to make general statements about overall performance. In problem 1.1, you will use measures of center and measures of variability, or spread, to determine which type of ride has the more consistent wait time. A.) Since Sally waited in line longer than the average wait time, she wondered how much wait times vary. The dot plot below shows a distribution of ten wait times for the Scenic Trolley ride. 1.) What is the median of the Scenic Trolley wait times?
2.) Sally says that the mean wait time is 25 minutes, just like the sign claimed. Do you agree? Explain. 3.) What do you notice about the mean and median? 4.) What is the range of this distribution? 5.) Sally wonders how typical a wait time of 25 minutes is. She says, I can find how much, on average, the data values vary from the mean time of 25 minutes. She uses the graph below to find the distance each data value is from the mean. Fred says, That s a good idea, but I used an ordered-value bar graph to show the same idea. a.) Describe how you can use each graph to find how much on average, the data values vary from mean time of 25 minutes. b.) What does this information tell you about how long you might have to wait in line to ride the Scenic Trolley?
c.) Sally noticed that the sum of the distances to the mean for the data values less than the mean equaled the sum of the distances to the mean for the data values greater than the mean. Does this make sense? Explain. Sally and Fred calculated a statistic called the mean absolute deviation (MAD) of the distribution. It is the average distance (or mean distance) from the mean of all data values. B.) Below is a sample of ten wait times for the Carousel, which also has a mean wait time of 25 minutes (indicated by the triangle). 1.) What is the mean absolute deviation (MAD) for this distribution? 2.) What is the range of this distribution? 3.) What is the median for this distribution? 4.) Compare the medians, means, ranges, and MADs for the Scenic Trolley and the Carousel. Why might you choose the Carousel over the Scenic Trolley? Explain.
C.) The Bumper Cars have a mean wait time of 10 minutes. Like other rides, the wait times are variable. Below is a sample of ten wait times for the Bumper Cars. 5.) What is the mean absolute deviation (MAD) for this distribution? 6.) What is the range of this distribution? 7.) What is the median for this distribution? 8.) Compare the medians, means, ranges, and MADs for the Scenic Trolley and the Bumper Cars. Why might you choose the Bumper Cars over the Scenic Trolley? Explain.
D.) Use these two signs for amusement park rides. Suppose you have to leave the park in 30 minutes. You want one last ride. Each ride lasts 3 minutes. Which ride would you choose? Explain. s E.) The following data are samples of wait times for the Merry-go-Round and for Superman. Wait Time for Merry-go-Round Wait Time for Superman 10 8 11 8 12 9 15 10 15 15 17 21 17 21 18 27 20 29 25 32 1.) Make a dot plot for each of the ride s wait time. Merry-go-Round
Superman 2.) Write three sentences that compare the distributions. 3.) The following three students came up with strategies to determine which ride has the best wait times. For each strategy, explain whether or not the strategy helps determine the best wait times. If the strategy does help, use it to determine the best wait time. a.) Bianca: For each ride, just add up all the wait times. Then compare the results of the two distributions. b.) Gianna: Find the mean wait time for each of the wait times. Then compare the results of the two distributions. c.) Jonah: Compare each amount of wait time to the mean of the wait times. On average, how far does each wait time differ from the ride s mean wait time? For each ride, find the MAD. Then compare the MAD s of the two rides.
F.) In question E1 and E2, you made dot plots of two sets of data. In question B3c, you found the mean absolute deviation (MAD) of each of the two distributions. The dot-plot below shows the Merry-go-Rounds wait times. The lines indicate the distances of one MAD and two MADs from the mean on either side. Count the data points located closer than, but not including, the distance of one MAD from the mean. (The triangle indicates the mean.) 1.) How many of Merry-go-Rounds data is located within one MAD (both less than and greater than the mean)? Write this number as a percent. 2.) How many of Merry-go-Rounds data is located within two MADs of the mean? Write this number as a percent. 3.) How many of Merry-go-Rounds data is located more than two MADs of the mean? Write this number as a percent. 4.) Repeat parts 1-3 for the wait times of Superman. 5.) Use the MAD locations from parts 1-4 to describe how the samples of wait times for the two rides are spread around the mean of the data.