The length of human pregnancies from conception to birth varies according to a d

Question

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 267 days and standard deviation 18 days.

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 267 days and standard deviation 18 days. a) What proportion of pregnancies last less than 270 days (about 9 months)? 0.5662x (Use 4 decimal places) (b) What proportion of pregnancies last between 240 and 270 days (roughly between 8 months and 9 months)? (No Response) (Use 4 decimal places) (c)How long do the longest 20% of pregnancies last? (No Response) (Use 2 decimal places) (d) What are the quartiles of the distribution of lengths of human pregnancies? (Use 2 decimal places) QI =(N°Response) Q3 (No Response)

Explanation / Answer

NORMAL DISTRIBUTION
the PDF of normal distribution is = 1/ * 2 * e ^ -(x-u)^2/ 2^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
equation of the normal curve is ( Z )= x - u / sd ~ N(0,1)
mean ( u ) = 267
standard Deviation ( sd )= 18
a.
P(X < 270) = (270-267)/18
= 3/18= 0.1667
= P ( Z <0.1667) From Standard Normal Table
= 0.5662
b.
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 240) = (240-267)/18
= -27/18 = -1.5
= P ( Z <-1.5) From Standard Normal Table
= 0.0668
P(X < 270) = (270-267)/18
= 3/18 = 0.1667
= P ( Z <0.1667) From Standard Normal Table
= 0.5662
P(240 < X < 270) = 0.5662-0.0668 = 0.4994
c.
P ( Z > x ) = 0.2
Value of z to the cumulative probability of 0.2 from normal table is 0.8416
P( x-u / (s.d) > x - 267/18) = 0.2
That is, ( x - 267/18) = 0.8416
--> x = 0.8416 * 18+267 = 282.1492
d.
Q1 = P ( Z = x ) = 0.25
Value of z to the cumulative probability of 0.25 from normal table is -0.6745
P( x-u/s.d < x - 267/18 ) = 0.25
That is, ( x - 267/18 ) = -0.6745
--> x = -0.6745 * 18 + 267 = 254.8592
Q1 = 254.8592

Q3 = P ( Z < x ) = 0.75
Value of z to the cumulative probability of 0.75 from normal table is 0.6745
P( x-u/s.d < x - 267/18 ) = 0.75
That is, ( x - 267/18 ) = 0.6745
--> x = 0.6745 * 18 + 267 = 279.1408
Q3 = 279.1408

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