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The mean head circumference of U

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The mean head circumference of U.S. newborns is 35.0 centimeters (cm) with a standard deviation of 1.3 cm. Assuming a normally distributed population of head circumferences:

A. What percent of newborns have a head circumference greater than 36.0 cm or less than 33.5 cm?

B. Microencephaly in newborns is a condition where an infant’s head is significantly smaller than normal. Some medical researchers consider a newborn to be microcephalic if its head circumference is in the bottom 3% of all newborn head circumferences. Given this definition, what would be the head circumference cutoff (in cm) for classifying a newborn as microcephalic?

C. If a random sample of 26 newborns is selected and their head circumferences are measured, what is the probability that the mean head circumference for this sample is less than 35.1 cm?

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Solution :

Given that,

mean = \mu = 35.0

standard deviation = \sigma = 15

a ) P (x > 36.0 )

= 1 - P (x < 36.0 )

= 1 - P ( x -  \mu/ \sigma ) < ( 36.0 - 35.0 / 1.3)

= 1 - P ( z < 1 / 1.3 )

= 1 - P ( z < 0.77)

Using z table

= 1 - 0.7794

= 0.2206

P( x < 33.5 )

P ( x - \mu / \sigma ) < ( 33.5 - 35.0 / 1.3)

P ( z < -1.5 / 1.3 )

P ( z < -1.15)

= 0.1251

= 0.1251 + 0.2206 = 0.3457

Probability = 34.57%

b ) P(Z < z) = 3%

= P(Z < z) = 0.03

= P(Z < -1.88 ) = 0.03

z = -1.88

Using z-score formula,

x = z * \sigma + \mu

x = -1.88 * 1.3 + 35

x = 32.56

c ) n = 26

\mu\bar x = 35

\sigma\bar x = \sigma / √n = 1.3 √26 = 0.2550

P( x < 35.1)

P ( x - \mu / \sigma ) < ( 35.1 - 35.0 / 0.2550)

P ( z < 0.1 / 0.2550 )

P ( z < 0.39)

Probability = 0.6517

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