A corporation receives 120 applications for positions from...

A corporation receives 120 applications for positions from recent college graduates in business. Assuming that these applicants can be viewed as a random sample of all such graduates, what is the probability that between $35 \%$ and $45 \%$ of them are women if $40 \%$ of all recent college graduates in business are women?

Key Concepts

Normal Approximation

For large sample sizes, the binomial distribution can be approximated by a normal distribution using the mean np and variance np(1-p). This approximation simplifies calculations, especially when computing probabilities over a range, such as determining the likelihood that the proportion of women applicants falls between 35% and 45%.

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 and a constant probability of success. In this scenario, each application is a trial with the outcome being whether the applicant is a woman, making the number of women applicants follow a binomial distribution.

Sampling Distribution of the Proportion

This concept refers to the distribution of the sample proportion that arises when taking repeated random samples from a larger population. The sample proportion, which is the number of women applicants divided by the total number of applicants, has its own distribution centered around the true proportion (0.4 in this case) with variability depending on the sample size.

Continuity Correction

When using the normal approximation to estimate probabilities for a discrete distribution like the binomial, a continuity correction is applied. This involves adjusting the discrete variable’s bounds by 0.5 to account for the fact that the normal distribution is continuous, thereby improving the accuracy of the approximation.

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