Binomial experiment is given. Determine whether you can use...

Binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why. A survey of full-time workers found that $72 \%$ go to work when they are sick. You randomly select 25 full-time workers and ask them whether they go to work when they are sick. (Source: CareerBuilder)

Binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.
A survey of full-time workers found that $72 \%$ go to work when they are sick. You randomly select 25 full-time workers and ask them whether they go to work when they are sick. (Source: CareerBuilder)

Key Concepts

Standard Deviation of a Binomial Distribution

The standard deviation measures the variability or dispersion of the binomial distribution and is determined by the formula ? = ?(n * p * (1-p)). This value indicates how much the number of successes is likely to deviate from the mean.

Mean of a Binomial Distribution

The mean (or expected value) of a binomial distribution is calculated using the formula ? = n * p, where n is the number of trials and p is the probability of success in each trial. This value represents the average number of expected successes over many repetitions of the experiment.

Criteria for Normal Approximation

To justify using the normal approximation to a binomial distribution, certain criteria should be met—typically, both the expected number of successes (np) and the expected number of failures (n(1-p)) should be at least 10. These conditions ensure the binomial distribution is not too skewed and is sufficiently symmetric for the approximation to be accurate.

Normal Approximation to the Binomial Distribution

The normal approximation is a method used to approximate the binomial distribution with a normal distribution, particularly when calculations with the binomial formula are cumbersome. This approximation simplifies probability calculations by substituting a continuous distribution for a discrete one.

Binomial Distribution

The binomial distribution is a discrete probability model that describes 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 throughout the trials.

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