Question

the age of a first grader on september 1 at garden elementary school is uniformly distributed from 5.8 to 6.8 years. we randomly select one first grader from the class. find the probability that she is between four and six years. a. 0.2 b. 0.3 c. 0.29 d. p412

Explanation:

Step1: Identify the range of the uniform - distribution

The uniform distribution has a lower bound $a = 5.8$ and an upper bound $b = 6.8$.

Step2: Determine the interval of interest

The interval of interest is from $x_1=4$ to $x_2 = 6$. But since the distribution is from $5.8$ to $6.8$, the part of the interval of interest that lies within the distribution is from $5.8$ to $6$.

Step3: Use the formula for the probability of a uniform - distribution

The formula for the probability of a uniform distribution $P(c\leq X\leq d)$ where $X\sim U(a,b)$ is $P(c\leq X\leq d)=\frac{d - c}{b - a}$ when $a\leq c\leq d\leq b$. Here, $c = 5.8$, $d = 6$, $a = 5.8$, $b = 6.8$.
$P(5.8\leq X\leq 6)=\frac{6 - 5.8}{6.8 - 5.8}$
$=\frac{0.2}{1}=0.2$

Answer:

a. 0.2