Question

1) The weights of newborn children in the U.S. vary according to the normal distribution with mean 7.5 pounds and standard deviation 1.25 pounds. The government classifies newborns as having low birth weight if the weight is less than 5.5 pounds. a) What is the probability that a baby chosen at random weighs less than 5.5 pounds at birth? b) Suppose we select 10 babies at random. What is the probability that their average birth weight is less than 5.5 pounds? 2) The amount of snowfall falling in a certain mountain range is normally distributed with a mean of 105 inches and a standard deviation of 10 inches. What is the probability that the mean annual snowfall during 25 randomly picked years will exceed 107.8 inches? 0.5808 0.0808 0.0026 0.4192

Answer 1

Solution :

1. Given that ,

mean = $\mu$ = 7.5

standard deviation = $\sigma$ = 1.25

(a) P(x < 5.5)

= P[(x - $\mu$ ) / $\sigma$ < (5.5 - 7.5) / 1.25]

= P(z < -1.6)

Using z table,

= 0.0548

Probability = 0.0548

(b) n = 10

$\mu$_{$\bar x$} = 7.5

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 1.25 / $\sqrt$ 10 = 0.3953

P($\bar x$ < 5.5)

= P(($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (5.5 - 7.5) / 0.3953)

= P(z < -5.06)

Using z table

= 0.00

Probability = 0.00

2. Given that,

mean = $\mu$ = 105

standard deviation = $\sigma$ = 10

n = 25

$\mu$_{$\bar x$} = 105

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 10 / $\sqrt$ 25 = 2

P($\bar x$ > 107.8)

= 1 - P($\bar x$ < 107.8)

= 1 - P[($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (107.8 - 105) / 2]

= 1 - P( z < 1.4)

Using z table,    

= 1 - 0.9192

= 0.0808

Probability = 0.0808

Correct option :- 0.0808