Question

The birthweight of newborn babies is Normally distributed with a mean of 3.39 kg and a standard deviation of

0.55 kg.

(a) Find the probability that a baby chosen at random will have a birthweight of over 3.5 kg.

(b) Find the probability that an SRS of 16 babies will have an average birthweight of over 3.5 kg.

(c) Find the probability that an SRS of 100 babies will have an average birthweight of over 3.5 kg.

(d) What range of average birthweights would the largest 15% of samples of size 100 babies have?

Answer 1

Solution :

Given that ,

mean = = 3.39

standard deviation = = 0.55

(a)

P(x > 3.5) = 1 - P(x < 3.5)

= 1 - P((x - ) / < (3.5 - 3.39) / 0.55)

= 1 - P(z < 0.2)

= 1 - 0.5793

= 0.4207

P(x > 3.5) = 0.4207

Probability = 0.4207

(b)

n = 16

= 3.39 and

= / n = 0.55 / 16 = 0.55 / 4 = 0.1375

P( > 3.5) = 1 - P( < 3.5)

= 1 - P(( - ) / < (3.5 - 3.39) / 0.1375)

= 1 - P(z < 0.8)

= 1 - 0.7881

= 0.2119

P( >) = 0.2119

Probability = 0.2119

(c)

n = 100

= 3.39 and

= / n = 0.55 / 100 = 0.55 / 10 = 0.055

P( > 3.5) = 1 - P( < 3.5)

= 1 - P(( - ) / < (3. - 3.39) / 0.055)

= 1 - P(z < 2)

= 1 - 0.9772

= 0.0228

P( > 3.5) = 0.0228

Probability = 0.0228

(d)

P(Z > z) = 1%

1 - P(Z < z) = 0.15

P(Z < z) = 1 - 0.15 = 0.85

P(Z < 1.04) = 0.85

z = 1.04

= z * + = 1.04 * 0.55 + 3.39 = 3.962

Range = 3.96