Question

. Suppose that a random sample of n = 102 children is selected from the population of newborn infants in Mexico. The probability that a child in this population weighs at most 2500 grams is presumed to be π = 0.15. Calculate the probability that thirteen or fewer of the infants weigh at m

Answer 1

30.85% probability that thirteen or fewer of the infants weigh at most 2500 grams

Explanation:

I am going to use the binomial approximation to the normal to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Normal probability distribution

Problems of normally distributed samples can be 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.

When we are approximating a binomial distribution to a normal one, we have that
`\mu = E(X)`,
`\sigma = √(V(X))`.

In this problem, we have that:

`n = 102, p = 0.15`

So

`\mu = E(X) = 102*0.15 = 15.3`

`\sigma = √(V(X)) = √(np(1-p)) = √(102*0.15*0.85) = 3.61`

Calculate the probability that thirteen or fewer of the infants weigh at most 2500 grams

This is
`P(X \leq 13)`, which using continuity correction is
`P(X \leq 13 + 0.5) = P(X \leq 13.5)`, which is the pvalue of Z when X = 13.5.

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

`Z = (13.5 - 15.3)/(3.61)`

`Z = -0.5`

`Z = -0.5` has a pvalue of 0.3085

30.85% probability that thirteen or fewer of the infants weigh at most 2500 grams