Forty-five percent of all customers of a dairy company prefer large curd cottage cheese to small curd cottage cheese. What is the approximate probability that a simple random sample of 100 of these customers would contain at least 57 persons who preferred small curd cottage cheese?

1.04% probability that a simple random sample of 100 of these customers would contain at least 57 persons who preferred small curd cottage cheese

Step-by-step explanation:

I am going to use the normal approximation to the binomial 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:

What is the approximate probability that a simple random sample of 100 of these customers would contain at least 57 persons who preferred small curd cottage cheese?

Using continuity correction, this is , which is 1 subtracted by the pvalue of Z when X = 56.5. So

has a pvalue of 0.9896

1 - 0.9896 = 0.0104

1.04% probability that a simple random sample of 100 of these customers would contain at least 57 persons who preferred small curd cottage cheese

i got 86 for this

hope this !

if you need an explanation just ask