Answer:
A. P(X<20,000) = 0.0392
B. P(20,000 < x < 30,000) = 0.488
C. Amount = $39,070
Step-by-step explanation:
From XO group website
Cost of a wedding = $29,858
Mean, μ = $29,858
Standard Deviation, σ =$5,600
a. Calculating the probability that a wedding costs less than $20,000
P(X<20,000)?
First, the z value needs to be calculated.
z = (x - μ)/σ
x = 20,000
z = (20,000 - 29,858)/5600
z = -1.76
So, P(X<20,000) = P(Z<-1.76)
From the z table,
P(Z<-1.76) = 0.0392
P(X<20,000) = 0.0392
b. Calculating the probability that a wedding costs between $20,000 and $30,000
P(20,000 < x < 30,000)
First, the z value needs to be calculated.
z = (x - μ)/σ
When x = 20,000
z = (20,000 - 29,858)/5600
z = -1.76
When x = 30,000
z = (30,000 - 29,858)/5600
z = 0.02
P(20,000 < x < 30,000) = P(-1.76 < z <0.02) --- using the z table
P(-1.76 < z <0.02) = 0.508 - 0.0392
P(-1.76 < z <0.02) = 0.4688
P(20,000 < x < 30,000) = 0.488
c. Using the following formula, we'll get the amount it'll a wedding to be among the 5% most expensive
z = (x - μ)/σ where x = amount
Make x the subject of formula
x = σz + μ
Fist we need to get the z value of 5%
z0.05 = 1.645
x = σz + μ becomes
x = 5600 * 1.645 + 29,858
x = $39,070
Amount = $39,070
and differentiating both sides with respect to 
yields![\dfrac{\mathrm d}{\mathrm dx}\bigg[3x^2+y^2\bigg]=\dfrac{\mathrm d}{\mathrm dx}\bigg[7\bigg]\implies 6x+2y\dfrac{\mathrm dy}{\mathrm dx}=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cbigg%5B3x%5E2%2By%5E2%5Cbigg%5D%3D%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cbigg%5B7%5Cbigg%5D%5Cimplies%206x%2B2y%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%3D0)
![\dfrac{\mathrm d}{\mathrm dx}\bigg[6x+2y\dfrac{\mathrm dy}{\mathrm dx}\bigg]=\dfrac{\mathrm d}{\mathrm dx}\bigg[0\bigg]\implies 6+2\left(\dfrac{\mathrm dy}{\mathrm dx}\right)^2+2y\dfrac{\mathrm d^2y}{\mathrm dx^2}=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cbigg%5B6x%2B2y%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%5Cbigg%5D%3D%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cbigg%5B0%5Cbigg%5D%5Cimplies%206%2B2%5Cleft%28%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%5Cright%29%5E2%2B2y%5Cdfrac%7B%5Cmathrm%20d%5E2y%7D%7B%5Cmathrm%20dx%5E2%7D%3D0)
and 
, we have