Question

test 3
score: 0/6 answered: 0/6
question 1
find probabilities for the standard and non - standard normal distribution.
you may round your answers to four decimal places
a. ( p(z>-0.25)=)
b. ( p(z<-2.06)=)
c. ( p(-0.77<z<0.57)=)
d. if ( mu = 84 ) and ( sigma = 13 ), find ( p(77<x<107)=)
question help: video

Explanation:

Step1: Calculate \(P(z > - 0.25)\)

Using the property \(P(Z>z)=1 - P(Z\leq z)\). For \(z=-0.25\), from the standard - normal table \(P(Z\leq - 0.25)=0.4013\). So \(P(Z > - 0.25)=1 - 0.4013=0.5987\)

Step2: Calculate \(P(z < - 2.06)\)

From the standard - normal table, for \(z =-2.06\), \(P(Z < - 2.06)=0.0197\)

Step3: Calculate \(P(-0.77 < z < 0.57)\)

Using the property \(P(a < Z < b)=P(Z < b)-P(Z < a)\). For \(a=-0.77\), \(P(Z < - 0.77)=0.2206\); for \(b = 0.57\), \(P(Z < 0.57)=0.7157\). Then \(P(-0.77 < Z < 0.57)=0.7157 - 0.2206=0.4951\)

Step4: Calculate \(P(77 < x < 107)\) when \(\mu = 84\) and \(\sigma=13\)

First, standardize the values. \(z_1=\frac{77 - 84}{13}=\frac{-7}{13}\approx - 0.54\), \(z_2=\frac{107 - 84}{13}=\frac{23}{13}\approx1.77\)
Then \(P(77 < X < 107)=P(-0.54 < Z < 1.77)\)
Using \(P(a < Z < b)=P(Z < b)-P(Z < a)\). For \(a=-0.54\), \(P(Z < - 0.54)=0.2946\); for \(b = 1.77\), \(P(Z < 1.77)=0.9616\)
So \(P(-0.54 < Z < 1.77)=0.9616 - 0.2946=0.6670\)

Answer:

a. \(0.5987\)
b. \(0.0197\)
c. \(0.4951\)
d. \(0.6670\)