Final answer:
For a uniformly distributed length of yarn from 40 to 50 inches, the density curve height must be 0.1 inches^-1, 20% of guests cut yarn longer than 48 inches, and the 25th percentile is at 42.5 inches.
Step-by-step explanation:
At a baby shower, if the length of yarn cut by guests for guessing the distance around the mom's baby bump is uniformly distributed between 40 and 50 inches, we can calculate the necessary elements of this distribution.
a. Height of the Density Curve
For a uniform distribution, the area under the density curve must equal 1 since it represents the total probability. The density curve is a rectangle with a base from 40 to 50 inches, so the width is 50 - 40 = 10 inches. Therefore, the height must be 1/10 or 0.1 inches-1 to give the rectangle an area of 1.
b. Percent of Guests Who Cut Yarn Longer than 48 Inches
The length from 48 to 50 inches is 2 inches long. The area of this section of the rectangle, which represents the probability, is 2 inches * 0.1 inches-1 = 0.2 or 20%. Thus, about 20% of guests cut their yarn longer than 48 inches.
c. 25th Percentile of the Distribution
To find the 25th percentile (Q1), we look for the length such that 25% of the distribution lies below it. The area from 40 inches to Q1 should be 25% of the total area. Since the distribution is uniform, Q1 is 40 + (10 * 0.25) = 42.5 inches. Therefore, the 25th percentile is at 42.5 inches, meaning 25% of the yarn lengths cut by guests are less than or equal to 42.5 inches.