Learn Probability Distributions with Plainmath's In-Depth Explanations and Real-World Examples

Sean and Evan are college roommates who have part-time jobs as servers in restaurants. The distribution of Sean’s weekly income is approximately normal with mean $225 and standard deviation $25. The distribution of Evan’s weekly income is approximately normal with mean $240 and standard deviation $15. Assuming their weekly incomes are independent of each other, which of the following is closest to the probability that Sean will have a greater income than Evan in a randomly selected week?
1) 0.67
2) 0.7000
3) 0.227
4) 0.303
5) 0.354

What is the formula of ${\left(A+B\right)}^3$?

A(n) _______ distribution has a​ "bell" shape.
A) normal
B) outlier
C) relative frequency
D) polygon

Use the Standard Normal Distribution table to find the indicated area under the standard normal curve.
Q1: Between z = 0 and z = 2.24
$$\begin{gathered} A1:0=0.5000 \\ 2.24=0.9875 \\ 0.9875–0.5000=0.4875 \end{gathered}$$
$$\begin{gathered} Q2:\text{ To the left of }z=1.09 \\ A2:0.8621 \end{gathered}$$
Q3: Between z = -1.15 and z = -0.56
$$\begin{gathered} A3:-<!--\; -\; -->1.15=0.1251 \\ -<!--\; -\; -->0.56=0.2877 \\ 0.1251-<!--\; -\; -->0.2877=-<!--\; -\; -->0.1626 \end{gathered}$$
Q4: To the right of z = -1.93
$A4:1–0.0268=0.9732$
Section 5.2: Normal Distributions: Find Probabilities
Q5: The diameters of a wooden dowel produced by a new machine arenormally distributed with a mean of 0.55 inches and a standarddeviation of 0.01 inches. What percent of the dowels will have adiameter greater than 0.57?
$$\begin{gathered} A5:z=x-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}/\mathrm{\sigma <!--\; \sigma \; -->} \\ =0.57–0.55/0.01=2 \\ =P(x>0.57) \\ =P(z>2) \\ =1–P(z<2) \\ =1–0.9772 \\ =0.0228 \end{gathered}$$
Q6: A loan officer rates applicants for credit. Ratings arenormally distributed. The mean is 240 and the standard deviation is50. Find the probability that an applicant will have a ratinggreater than 260.
$$\begin{gathered} A6:z=x-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}/\mathrm{\sigma <!--\; \sigma \; -->}=260–240/50 \\ =0.4 \\ =P(x>260) \\ =P(z>0.4) \\ =1–P(z<0.4) \\ =1–0.6554 \\ =0.9772 \end{gathered}$$