Respuesta :
Answer:
Using a binomial model, the probability can be calculated using the binomial probability formula. For this scenario, where n (number of trials) is 20, p (probability of success on each trial) is 0.27, and x (number of successes) is 5, the probability is obtained as follows:
�
(
�
≤
5
)
=
∑
�
=
0
5
(
20
�
)
×
(
0.27
)
�
×
(
0.73
)
20
−
�
P(X≤5)=∑
k=0
5
(
k
20
)×(0.27)
k
×(0.73)
20−k
b) To use a Normal model, the conditions for applying the Normal approximation to the binomial distribution need to be satisfied. If np and n(1-p) are both greater than 5, then a Normal distribution can be used. Calculate the mean
and standard deviation
for the binomial distribution, and then use the Z-score formula to find the probability.
c) Apply the same approach as in part a using the binomial model, with n = 700, p = 0.27, and x = 175.
d) Utilize the Normal model as in part b, checking the conditions for approximation and calculating the probability using the Z-score formula.
Keep in mind that the Normal approximation is more accurate for larger sample sizes, and the answers from the binomial and Normal models should be reasonably close for sufficiently large sample sizes.
