2 A hospital in a large city records the weight of every inf

2. A hospital in a large city records the weight of every infant born at the hospital. The distribution of weights is normally distributed, has a mean of 2.90 kilograms, and has a standar4 deviation of 0.45. a Infants who weigh less than 1.90 kilograms have an increased risk of a variety of health problems. What percentage of infants weighed less than 1.90 kilograms? b What is the percentile rank of an infant who weighs 3.90 kilograms? c What percentage of infants weigh between 2.20 and 3.70 kilograms? d. What percentage of infants weigh between 3.35 and 4.40 kilograms? e. How heavy would a baby have to be to be in the top 1% of babies born at this hospital? f. How small would a baby have to be to be in the bottom 5% of babies born at this hospital? g. If 20,000 infants were born at this hospital, how many weighed less than 3.30 kilograms?

Solution

population mean,u = 2.9
standard deviation,sigma = 0.45

a.
P(X< 1.9 ) = P(Z< ((1.9-2.9) / 0.45)
= P(Z< -2.22 )
= 1 - P(Z<2.22)
= 1 - 0.9868
= 0.0132
= 1.32%

b.
P(X< 3.9 ) = P(Z< ((3.9-2.9) / 0.45)
= P(Z< 2.22 )
= 0.9868
98.68 percentile

c.
P( 2.2 <X< 3.7 )
= P( ((2.2-2.9) / 0.45) <Z< ((3.7-2.9) / 0.45) )
= P( -1.56 <Z< 1.78 )
= P(Z<1.78) - P(Z<-1.56)
= P(Z<1.78) - (1 - P(Z<1.56))
= 0.9625 - ( 1 - 0.9406)
= 0.9625 - 0.0594
= 0.9031

d.
P( 3.35 <X< 4.4 )
= P( ((3.35-2.9) / 0.45) <Z< ((4.4-2.9) / 0.45) )
= P( 1.00 <Z< 3.33 )
= P(Z<3.33) - P(Z<1)
= 0.9996 - 0.8413
= 0.1583

e.
top 1% corresponds to z-score=2.33
2.9 + 2.33*0.45
= 3.95 kg

f.
bottom 5% correspond to z-score = -1.645
2.9 - 1.645*0.45
= 2.16 kg

g.
P(X< 3.3 ) = P(Z< ((3.3-2.9) / 0.45)
= P(Z< 0.89 )
= 0.8133

20000*0.8133
= 16266