Question

"The density curve shown takes the value 0.5 on the interval 0 SX S2 and takes the value 0 everywhere else. What percent of the observations lie between 0.5 and 1.2.

A) 35%
B) 68%
C) 70%
D) 50%
E) 25%"

Answer 1

The percentage of observations that lie between 0.5 and 1.2 on the density curve is 35%. Therefore, the correct answer is:

A) 35%

To determine the percentage of observations that lie between 0.5 and 1.2 on a density curve, we need to calculate the area under the curve for this interval. The density curve takes the value of 0.5 for the interval
`\( 0 \leq x \leq 2 \)` and the value of 0 everywhere else.

The area under the density curve for the entire interval from 0 to 2 is equal to the base times the height since the curve is flat (rectangular shape). The base of this rectangle is the length of the interval from 0 to 2, which is 2 units, and the height is the value of the density, which is 0.5.

The total area under the curve from 0 to 2, which represents 100% of the observations, is:

`\[ \text{Total area} = \text{base} * \text{height} = 2 * 0.5 = 1. \]`

Now, let's calculate the area under the curve from 0.5 to 1.2:

`\[ \text{Area from 0.5 to 1.2} = \text{base from 0.5 to 1.2} * \text{height}`

`\[ \text{Area from 0.5 to 1.2} = (1.2 - 0.5) * 0.5. \]`

`\[ \text{Area from 0.5 to 1.2} = 0.7 * 0.5. \]`

`\[ \text{Area from 0.5 to 1.2} = 0.35`

The percentage of observations that lie between 0.5 and 1.2 on the density curve is 35%. Therefore, the correct answer is:

A) 35%