Answer:
Step-by-step explanation:
The formula for binomial distribution is expressed as
P(x = r) = nCr × p^r × q^(n - r)
Where
x represent the number of successes.
p represents the probability of success.
q = (1 - p) represents the probability of failure.
n represents the number of trials or sample.
From the information given,
p = 36.5% = 36.5/100 = 0.365
q = 1 - p = 1 - 0.365
q = 0.635
n = 15
a) P(x = 0) = 15C0 × 0.365^0 × 0.635^(15 - 0) = 0.0011
P(x = 1) = 15C1 × 0.365^1 × 0.635^(15 - 1) = 0.0095
P(x = 2) = 15C2 × 0.365^2 × 0.635^(15 - 2) = 0.038
P(x = 3) = 15C3 × 0.365^3 × 0.635^(15 - 3) = 0.095
P(x = 4) = 15C4 × 0.365^4 × 0.635^(15 - 4) = 0.16
P(x = 5) = 15C5 × 0.365^5 × 0.635^(15 - 5) = 0.21
k P(X = k)
0 0.0011
1 0.0095
2 0.038
3 0.095
4 0.16
5 0.21
b) mean = np = 15 × 0.365 = 5.475
c) standard deviation = √npq
= √15 × 0.365 × 0.635
= 1.86
d) z = (x - mean)/standard deviation
x = 2
z = (2 - 5.475)/1.86 = - 1.87
5 + 4/5t