(a) With n=12 and p =0.4 find the binomial probability that p(9) by using a binomial probability table. (b) np ¡Ý5, nq¡Ü5, also estimate the indicated probability by using the normal distribution as an approximation to the binomial, is np < 5 or nq < 5 than state the normal approximation cannot be used

(a) To find the binomial probability P(9), we can use a binomial probability table. The table provides the probabilities for different values of n and p.

Given n = 12 and p = 0.4, we want to find P(9).

Looking at the binomial probability table, we locate the row corresponding to n = 12. Then, in that row, we find the column corresponding to p = 0.4. The value at the intersection gives us the probability.

For P(9), the table value is 0.0960.

Therefore, the binomial probability of P(9) with n = 12 and p = 0.4 is 0.0960.

(b) To determine if the normal approximation to the binomial can be used, we need to check if np ≥ 5 and nq ≥ 5.

Here, n = 12 and p = 0.4.

Calculating np and nq:

np = 12 * 0.4 = 4.8
nq = 12 * (1 - 0.4) = 7.2

Since np < 5 and nq < 5 (4.8 < 5 and 7.2 < 5), the normal approximation cannot be used.

Therefore, the normal distribution cannot be used to approximate the indicated probability.

(a) To find the binomial probability P(9) with n = 12 and p = 0.4 using a binomial probability table, you need to refer to the table that provides the values for each n, p, and x combination.

1. Locate the row in the binomial probability table corresponding to n = 12.
2. Find the column in the table that corresponds to the given value of p = 0.4.
3. Look for the intersection point of the selected row and column; this will give you the value of P(x) for each x value.
4. Find the value of P(9) at the intersection point of the selected row and column.

(b) To determine if the normal approximation can be used when np ≥ 5 and nq ≤ 5, you need to check if both conditions are met.

1. Calculate np: Multiply the number of trials (n) by the probability of success (p).
np = n * p

2. Calculate nq: Multiply the number of trials (n) by the probability of failure (q), which is equal to 1 - p.
nq = n * (1 - p)

3. If np ≥ 5 and nq ≤ 5, then the normal approximation can be used for estimating the indicated probability.

However, if np < 5 or nq < 5, then the normal approximation cannot be used, and you would need to use the binomial distribution for accuracy.