Question

Compute P(X) using the binomial probability formula. Then determine whether the normal distribution can be used to estimate this probability. If so, approximate P(X) using the normal distribution and compare the result with the exact probability. n = 64, p = 0.3, and X = 31 Click the icon to view a table of areas under the normal curve. For n = 64, p =0.3, and X = 31, use the binomial probability formula to find P(X). 0008 (Round to four decimal places as needed.) Why can the normal distribution be used to approximate this probability? Because squareroot np(1- p) greaterthanorequalto 10 Because np(1- p) lessthanorequalto 10 Because np(1- p) greaterthanorequalto 10 Because squareroot np(1- P) lessthanorequalto 10 Approximate P(X) using the normal distribution. Use a standard normal distribution table. P(X) = 0006 (Round to four decimal places as needed.) By how much do the exact and approximated probabilities differ? (Round to four decimal places as needed.) Enter your answer in the answer box and then click Check Answer.

Answer 1

To find P(X) for given values, we use the binomial probability formula and normal distribution because the rule of thumb (np and n(1-p) > 10) is met. The z-score helps find the approximate P(X) in a z-table, and the difference between the exact and approximated probabilities is calculated.

Step-by-step explanation:

To compute P(X) using the binomial probability formula for n = 64, p = 0.3, and X = 31, we use the formula:

P(X) = C(n, X) * p^X * (1-p)^(n-X)

Where C(n, X) is the number of combinations of n things taken X at a time. After calculating, we get P(X) (rounded to four decimal places).

For the normal approximation to be used, the rule of thumb is that both np and n(1-p) should be greater than or equal to 10. In our case, with np = 64*0.3 = 19.2 and n(1-p) = 64*0.7 = 44.8, both conditions are satisfied, and thus normal distribution can be used.

Using the normal distribution, we calculate the z-score and find the corresponding area under the normal curve using a z-table to approximate P(X). The probabilities found this way are compared, and the difference between the exact and approximated probabilities is computed (rounded to four decimal places).