Given a binomial distribution with n = 21 and p = 0.76, would the normal distribution provide a reasonable approximation? Why or why not?

To determine whether the normal distribution provides a reasonable approximation for a given binomial distribution, we need to check if the conditions for applying the normal approximation are met.

The conditions for a binomial distribution to be approximated by a normal distribution are:
1. The number of trials, n, is large (typically larger than 30).
2. The probability of success, p, is not too close to 0 or 1 (generally p > 0.10 and 1 - p > 0.10).

In the given problem, we have n = 21 and p = 0.76.

First, we check if n is large enough. While 21 is smaller than the commonly suggested minimum of 30, it is still possible to use the normal approximation if the other condition is met.

Next, we need to check if p is not too close to 0 or 1. In this case, p = 0.76, which is quite far from the extremes of 0 and 1. Therefore, this condition is satisfied.

Based on these conditions, we can conclude that the normal distribution can provide a reasonable approximation for the given binomial distribution with n = 21 and p = 0.76. However, keep in mind that the approximation will be better as n approaches a larger value.