Consider the probability that more than 91 out of 149 DVDs will work correctly. Assume the probability that a given DVD will work correctly is 66%. Approximate the probability using the normal distribution. Round your answer to four decimal places.

0.8810 = 88.10% probability that more than 91 out of 149 DVDs will work correctly.

Step-by-step explanation:

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

Consider the probability that more than 91 out of 149 DVDs will work correctly.

Using continuity correction, this is P(X > 91 + 0.5 = 91.5), which is one subtracted by the pvalue of Z when X = 91.5. So

has a pvalue of 0.1190

1 - 0.1190 = 0.8810

0.8810 = 88.10% probability that more than 91 out of 149 DVDs will work correctly.

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