Answer to Question #177181 in Math for Kim Galvez

Question

Find the area in each of the following. 

a. 𝑧 = −1.47 and 𝑧 = 1.64

b. 𝑧 = −0.44 and 𝑧 = 1.82

c. The region where 𝑧 > 1.92

d. The region where 𝑧 < 2.22

2. A biologist found the wingspans of a group of monary butterflies to be normally 

distributed with a mean of 52.2 mm and a standard deviation of 2.3 mm. What percent of 

the butterflies had a wingspan

a. less than 48.5 mm? 

b. between 50 and 55 mm?

3. A highway study of 8000 vehicles that passed by a checkpoint found that their speeds 

were normally distributed, with a mean of 61 mph and a standard deviation of 7 mph. 

a. How many of the vehicles had a speed of more than 68 mph? 

b. How many of the vehicles had a speed of less than 40 mph?

Answer 1

1.

a.

$$P(-1.47<Z<1.64)$$

$$=P(Z<1.64)-P(Z\leq-1.47)$$

$$\approx0.949497-0.070781=0.878716$$

b.

$$P(-0.44<Z<1.82)$$

$$=P(Z<1.82)-P(Z\leq-0.44)$$

$$\approx0.965620-0.329969=0.635661$$

c.

$$P(Z>1.92)=1-P(Z\leq1.92)$$

$$\approx1-0.972571=0.027429$$

d.

$$P(Z<2.22)\approx0.986791$$

$$\approx1-0.972571=0.027429$$

2.

$\mu=52.2\ mm, \sigma=2.3\ mm$

a.

$$P(X<48.5)=P(Z<\dfrac{48.5-52.2}{2.3})$$

$$\approx P(Z<-1.6087)\approx0.0538$$

$5.38\%$

b.

$$P(50<X<55)=P(X<55)-P(X\leq 50)$$

$$=P(Z<\dfrac{55-52.2}{2.3})-(Z<\dfrac{50-52.2}{2.3})$$

$$\approx P(Z<1.2174)-(Z<-0.9565)$$

$$\approx 0.888274-0.169410\approx0.7189$$

$71.89\%$

3.

$\mu=61\ mph, \sigma=7\ mph$

a.

$$P(X>68)=1-P(X\leq 68)$$

$$=1-P(Z\leq \dfrac{68-61}{7})=1-P(Z\leq 1)$$

$$\approx 1-0.841345=0.158655$$

$0.158655(8000)=1269$

1269 vehicles

b.

$$P(X<40)=P(Z<\dfrac{40-61}{7})=P(Z<-3)$$

$$\approx 0.001350$$

$0.001350(8000)=11$

11 vehicles