Answer :
The best description of the area under the normal curve that would be used to approximate binomial probability is 'Area to the left of 46.5'.
To approximate the binomial probability, we can use the normal distribution as an approximation for the binomial distribution when the sample size is large enough, and the probability of success is not too close to 0 or 1. The mean of the normal distribution is equal to the mean of the binomial distribution, which is np, and the standard deviation of the normal distribution is equal to the square root of npq, where q is the probability of failure.
In this case, the number of DVDs that may malfunction follows a binomial distribution with parameters n = 319 and p = 1/100, where 1/100 is the probability of a DVD malfunctioning. Therefore, the mean of the binomial distribution is np = 3.19, and the standard deviation is [tex]\sqrt{npq}[/tex] = [tex]\sqrt{3.19*0.99}[/tex] ≈ 1.77.
To find the probability that at most 46 out of 319 DVDs will malfunction, we need to find the area under the normal curve to the left of 46.5 (since we are interested in the probability of at most 46 malfunctions, which includes 46.5 as well). Therefore, the answer is 'Area to the left of 46.5'.
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