Final answer:
a. Display the data using a line plot and stem-and-leaf plot. b. Find the mean, median, mode, and range. c. Determine whether a normal distribution is appropriate. d. Estimate percentages using a normal distribution.
Step-by-step explanation:
a. Line Plot and Stem-and-Leaf Plot:
A line plot displays the data as single marks above a number line. The stem-and-leaf plot organizes the data by separating each value into a stem and a leaf. Both plots show the distribution of hat sizes.
b. Mean, Median, Mode, and Range:
The mean is the sum of all values divided by the number of values. The median is the middle value when the data is ordered. The mode is the value that appears most often. The range is the difference between the largest and smallest values.
c. Normal Distribution:
It is appropriate to use a normal distribution to model this data if the distribution is symmetric and bell-shaped.
d. Percentages Using Normal Distribution:
d1. To find the percentage of recruits taller than 72 inches, we use the standard normal distribution table or a calculator to find the area to the right of 72. d2. To find the percentage of recruits shorter than 67.5 inches, we use the standard normal distribution table or a calculator to find the area to the left of 67.5. d3. To find the percentage of recruits between 69 and 72 inches, we subtract the area to the left of 69 from the area to the left of 72. d4. To find the heights that capture 95% of new recruits, we use the standard normal distribution table or a calculator to find the z-score corresponding to a cumulative area of 0.975 (1 - 0.95/2), then use the mean and standard deviation to find the corresponding heights.