Question

2. Adapted from Exercise 3.19 in the textbook. The length of human pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 266 days and standard deviation 16 days.

(a) Between what range of days do about 95% of pregnancies last?

(b) What proportion of pregnancies last more than 170 days?

(c) What proportion of pregnancies last less than 245 days? 1

(d) What proportion of pregnancies last between 245 and 280 days?

(e) A pregnancy that lasts an amount of time 0.3 standard deviations less than the means lasted for how many days?

Answer 1

Given :

(a)

According to the empirical rule , 95% of the data values lies between 2 standard deviation of the mean .

That is

Therefore about 95% of pregnancies last between 234 and 298 days.

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(b) Here we need to find P( x > 170)

Using z-score formula

Requitred probability is P(z>-6) = P(z< 6)

Using excel function =NORMSDIST(z)

=NORMSDIST(6)

= 1

The proportion of pregnancies last more than 170 days is 1 or 100%

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(c) P(x < 245)

P(z<-1.31)

Using Excel function =NORMSDIST(z)

=NORMSDIST( -1.31 )

=0.0951

The proportion of pregnancies last less than 245 days is 0.0951 or 9.51%

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(d) P( 245< x< 280 )

For x=245

For x = 280

Required probability is P( -1.31 < z < 0.88)

= P( z < 0.88) - P( z < -1.31)

=0.8106 - 0.0951                        ( usinge excel function =NORMSDIST(0.88)   and =NORMSDIST(-1.31))

=0.7155

The proportion of pregnancies last between 245 and 280 days is 0.7155 or 71.55%