The highest grade for florida oranges is u. s. fancy and is based principally on color. suppose that shannon owns a small organic orchard and that the proportion of oranges she grows which can be classified as u. s. fancy is 0.40 . one morning, shannon randomly picks 44 oranges. estimate the probability that the number of oranges that will be classified as u. s. fancy, , is fewer than 15. use a normal approximation to the binomial distribution with continuity correction to obtain the solution. give your answer precise to at least four decimal places. p(x < 16) =

There is a 21.19% probability that the number of oranges that will be classified as U.S. Fancy is fewer than 15.

Step-by-step explanation:

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.

The binomial probability distribution is the probability of x sucesses on n trials, with a probability of p.

It has an expected value of

And a standard deviation of:

The binomial distribution can be approximated to the normal with n that is sufficiently large, with .

In this problem, we have that:

The proportion of oranges she grows which can be classified as U.S. Fancy is 0.40. One morning, Shannon randomly picks 44 oranges. This means that .

Estimate the probability that the number of oranges that will be classified as U.S. Fancy is fewer than 15.

This is the pvalue of Z when .

We have that:

So

has a pvalue of 0.2119.

This means that there is a 21.19% probability that the number of oranges that will be classified as U.S. Fancy is fewer than 15.

probability for j is 1/4

while

a is 1/6