A sample of 400 observations will be taken from an infinite population. The population proportion equals .8. The probability that the sample proportion will be greater than .83 is:

6.68% probability that the sample proportion will be greater than .83.

Step-by-step explanation:

We are going to approximate the binomial distribution to the normal.

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

The probability that the sample proportion will be greater than .83 is:

83% of 400 is 0.83*400 = 332.

So this probability is 1 subtracted by the pvalue of Z when X = 332.

has a pvalue of 0.9332.

So 1-0.9332 = 0.0668 = 6.68% probability that the sample proportion will be greater than .83.

answer: the answer is 110