Question

1. the density curve shown models the distribution of a quantitative variable that is equally likely to take any value in the interval from 0 to 2. what percent of the observations lie between 0.5 and 1.2? a. 25% b. 35% c. 50% d. 70%

Explanation:

Step1: Recall the formula for probability of a uniform - distribution

For a uniform distribution on the interval $[a,b]$, the probability density function is $f(x)=\frac{1}{b - a}$ for $a\leq x\leq b$ and $0$ otherwise. Here, $a = 0$ and $b = 2$, so $f(x)=\frac{1}{2-0}=0.5$ for $0\leq x\leq 2$. The probability $P(c\leq X\leq d)$ is given by $P(c\leq X\leq d)=f(x)\times(d - c)$.

Step2: Calculate the probability for the given interval

We want to find $P(0.5\leq X\leq 1.2)$. Since $f(x) = 0.5$, and $d=1.2$, $c = 0.5$, then $P(0.5\leq X\leq 1.2)=0.5\times(1.2 - 0.5)$.
$P(0.5\leq X\leq 1.2)=0.5\times0.7 = 0.35$.

Step3: Convert probability to percentage

To convert the probability to a percentage, we multiply by 100. So $0.35\times100 = 35\%$.

Answer:

b. 35%