Question

The weights of newborn children in the United States vary according to the Normal distribution with mean 7.5 pounds and standard deviation 1.25 pounds. The government classifies a newborn 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) You choose three babies at random. What is the probability that their average birth weight is less than 5.5 pounds?

Answer 1

5.48%

0.28

Explanation:

Answer 2

a) 5.48% probability that a baby chosen at random weighs less than 5.5 pounds at birth

b) 0.28% probability that their average birth weight is less than 5.5 pounds

Explanation:

To solve this question, the normal probability distribution and the central limit theorem are used.

Normal probability distribution:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sample means n of at least 30 can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`

In this problem, we have that:

`\mu = 7.5, \sigma = 1.25`

(a) What is the probability that a baby chosen at random weighs less than 5.5 pounds at birth?

This is the pvalue of Z when X = 5.5. So

`Z = (X - \mu)/(\sigma)`

`Z = (5.5 - 7.5)/(1.25)`

`Z = -1.6`

`Z = -1.6` has a pvalue of 0.0548

5.48% probability that a baby chosen at random weighs less than 5.5 pounds at birth

(b) You choose three babies at random. What is the probability that their average birth weight is less than 5.5 pounds?

`n = 3, s = (1.25)/(√(3)) = 0.7217`

This is the pvalue of Z when X = 5.5. So

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

This is the pvalue of Z when X = 5.5. So

`Z = (X - \mu)/(s)`

`Z = (5.5 - 7.5)/(0.7217)`

`Z = -2.77`

`Z = -2.77` has a pvalue of 0.0028

0.28% probability that their average birth weight is less than 5.5 pounds