The sample size n, probability of success p, and probability...

The sample size $n,$ probability of success $p,$ and probability of failure $q$ are given for a binomial experiment. Determine whether you can use a normal distribution to approximate the distribution of $x .$
$$n=15, p=0.70, q=0.30$$

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Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is defined by the parameters n (number of trials) and p (probability of success) and is used when outcomes can only be classified as success or failure.

Normal Approximation to the Binomial

The normal approximation to the binomial distribution simplifies computation when dealing with large sample sizes. Instead of using the binomial formula directly, the normal distribution is used as an approximation, particularly because it is continuous and easier to work with compared to the discrete binomial distribution.

Criteria for Normal Approximation

For the normal approximation to be considered valid, both np and nq (with q being 1-p) should generally be greater than or equal to 10. This condition ensures that the binomial distribution is sufficiently symmetric and spread out, which makes the approximation reasonably accurate. If these criteria are not met, the normal approximation might not provide reliable results.

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