A market research organization has found that 40 % of all...

A market research organization has found that $40 \%$ of all supermarket shoppers refuse to cooperate when questioned by its pollsters. If 1,000 shoppers are approached, what is the probability that fewer than 500 will refuse to cooperate?

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Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. It applies to situations where each trial has two possible outcomes (commonly labeled as success and failure) and is defined by two parameters: the number of trials (n) and the probability of success (p) in each trial.

Normal Approximation to the Binomial

When the number of trials in a binomial distribution is large, the distribution can be approximated by a normal distribution. This approximation is particularly useful for calculating probabilities because it converts a discrete distribution into a continuous one, making the computation more tractable using the properties of the normal curve, provided that both np and n(1-p) are sufficiently large.

Continuity Correction

The continuity correction is used when a continuous distribution, such as the normal distribution, is employed to approximate a discrete distribution like the binomial. It involves adjusting the discrete variable by 0.5 (either adding or subtracting) in order to better capture the probability mass of the discrete outcome, thereby improving the accuracy of the approximation.

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