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 261 days and standard deviation 18 days.

(a) What proportion of pregnancies last less than 270 days (about 9 months)? .6914 Correct: Your answer is correct. (Please use 4 decimal places.)

(b) What proportion of pregnancies last between 240 and 270 days (roughly between 8 months and 9 months)? . Changed: Your submitted answer was incorrect. Your current answer has not been submitted. (Please use 4 decimal places.)

(c) How long do the longest 20% of pregnancies last? (Please use 2 decimal places.) The quartiles of any distribution are the values with cumulative proportions 0.25 and 0.75.

(d) What are the quartiles of the standard Normal distribution? (Use 2 decimal places.) Q1 = Q3 = (e) What are the quartiles of the distribution of lengths of human pregnancies? Please use 2 decimal places.

Q1 =

Q3 =

Explanation / Answer

( a )

we are given = 261 , = 18 for a normal distribution

To find the proportion of this distribution less than 270 days

we standardize the 270 ( convert into z-score )

270 --> 270 - 261 / 18 ==> 9 / 18 ==> 1/2 ==> 0.5

STANDARD NORMAL DISTRIBUTION: Table Values Represent AREA to the LEFT of the Z score.

for 0.5 is .69146

( b)

240 --> 240 - 261 / 18 ==> - 21/ 18 ==> - 7 / 6 ==> -1.1666666   and for 270 is 0.5

from STANDARD NORMAL DISTRIBUTION Table

for < - 1.166666 is .12507 and for < 0.5 is .69146

Therefore the proportion b/w z - values are .69146 - .12507 is 0.56639

( c )

since cut-off for which 20% of pregnancies are longer is same as the 80 th percentile

we look first for the z - value , z which satisfies P ( Z < z ) = 80 %

here Z = Standard normal random variable and   z is constant

Finally we convert z - value of 0.84 back to original scale by multplying = 261 , = 18

==> 0.84 ( 18) + 261=> 276.12 days

( d)

The z scores corresponding to probabilities of 0.25 and 0.75 are -0.67449 and 0.67449, respectively. Your table may show fewer digits, but these are the values correct to five digits.

Use this formula,

z = (x - µ)/

To get Q1,

-0.67449 = (x - 261)/18

x = 18(-0.67449) + 261 = 261 -12.14082 = 248.85918 days = Q1

To get Q2,

0.67449 = (x - 261)/18

x = 18(0.67449) + 261 = 261 + 12.14082= 273.14082 days = Q2

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