Answer to Question #177575 in Math for Kim Galvez

Question #177575

1. 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?



1
Expert's answer
2021-04-15T06:34:59-0400

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



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