A survey on gender status reveals that 56 % of the families...

A survey on gender status reveals that $56 \%$ of the families have more males than females. Assuming that this percentage is still valid, what is the probability that among 1000 randomly selected families, the male members exceed the female in between 570 and 650 (both included) of the families?

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Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant. In this problem, each family is a trial with a success defined as having more males than females, and the probability of success is 0.56.

Normal Approximation

When dealing with a large number of trials, the binomial distribution can be approximated by the normal distribution, which simplifies probability calculations. This approximation is particularly useful for calculating the probability that the number of successes falls within a specific interval, as is the case here with 570 to 650 families.

Continuity Correction

Because the normal distribution is continuous and the binomial distribution is discrete, a continuity correction is used when applying the normal approximation. This involves adjusting the interval bounds by 0.5 units to improve the approximation accuracy when converting a discrete probability problem into a continuous one.

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