. 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

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

Step-by-step 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:

The standard deviation of the binomial distribution is:

Normal probability distribution

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

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

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 , .

In this problem, we have that:

So

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

This is , which using continuity correction is , which is the pvalue of Z when X = 13.5.

has a pvalue of 0.3085

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