Answer:
(1) The correct answer is option (a) 0.510
(2) The correct answer is option (b) = 0.9664
(3) The correct answer is option (a) 0.8665
Step-by-step explanation:
1. Find attached of question 1
2.
n= 850
p= 0.1400
here mean of distribution(μ) =n*p
=850* 0.1400
= 119
standard deviation σ = √(np(1-p))
= √(850*0.1400(1-0.1400))
= √102.34
=10.1163
for normal distribution z score =(X-μ)/σx
therefore from normal approximation of binomial distribution and continuity correction:
probability = P(X>100.5)
= P(Z>-1.83)
= 1-P(Z<-1.83)
= 1-0.0336
= 0.9664
3.
n = 1000
p = 0.08
mean of distribution(μ) =np
= 1000*0.08
= 80
and standard deviation σ=√(np(1-p))
= √(1000*0.08(1-0.08))
= √736
= 8.5790
probability = P(X>70.5)
= P(Z>-1.11)
= 1-P(Z<-1.11)
= 1-0.1335
= 0.8665
The three questions are illustrations of normal approximation of binomial distribution and continuity correction
(a) Coins
The given parameters are:
[tex]\mathbf{n =35}[/tex]
[tex]\mathbf{p =0.3}[/tex]
Start by calculating the mean and the standard deviation
[tex]\mathbf{\bar x = np = 35 \times 0.3 = 10.5}[/tex]
[tex]\mathbf{\sigma = \sqrt{\bar x(1 - p)} = \sqrt{10.5 \times (1 - 0.3 )}= 2.71}[/tex]
The probability is then represented as:
[tex]\mathbf{P(9 < x < 14) = P(8.5 < x < 14.5)}[/tex]
Calculate the z-scores for x = 8.5, and 14.5
[tex]\mathbf{z = \frac{x - \bar x}{\sigma}}[/tex]
So, we have:
[tex]\mathbf{z = \frac{8.5 - 10.5}{2.71} = -0.3688}[/tex]
[tex]\mathbf{z = \frac{14.5 - 10.5}{2.71} = 1.1066}[/tex]
So, we have:
[tex]\mathbf{P(9 < x < 14) = P(-0.3688<z<1.1066)}[/tex]
Rewrite as:
[tex]\mathbf{P(9 < x < 14) = P(z<1.1066) - P(z<-0.3688)}[/tex]
Using z-scores of probabilities, we have:
[tex]\mathbf{P(9 < x < 14) = 0.86577- 0.3561}[/tex]
[tex]\mathbf{P(9 < x < 14) = 0.50967}[/tex]
Approximate
[tex]\mathbf{P(9 < x < 14) = 0.510}[/tex]
Hence, the probability of obtaining between 9 and 14 heads (exclusive) is (a) 0.510
(b) Bald men
The given parameters are:
[tex]\mathbf{n =850}[/tex]
[tex]\mathbf{p =14\%}[/tex]
Start by calculating the mean and the standard deviation
[tex]\mathbf{\bar x = np = 850 \times 14\% = 119}[/tex]
[tex]\mathbf{\sigma = \sqrt{\bar x(1 - p)} = \sqrt{119 \times (1 - 14\% )}= 10.12}[/tex]
The probability is then represented as:
[tex]\mathbf{P(x > 100) = P(x > 100.5)}[/tex]
Calculate the z-scores for x = 100.5
[tex]\mathbf{z = \frac{x - \bar x}{\sigma}}[/tex]
So, we have:
[tex]\mathbf{z = \frac{100.5 - 119}{10.12} = -1.83}[/tex]
So, we have:
[tex]\mathbf{P(x > 100) = P(z > -1.83)}[/tex]
Using z-scores of probabilities, we have:
[tex]\mathbf{P(x > 100) = 0.96638}[/tex]
Approximate
[tex]\mathbf{P(x > 100) = 0.9664}[/tex]
Hence, the probability that more than 100 in 850 are bald is (b) 0.9664
(c) Accidents
The given parameters are:
[tex]\mathbf{n =1000}[/tex]
[tex]\mathbf{p =0.08}[/tex]
Start by calculating the mean and the standard deviation
[tex]\mathbf{\bar x = np = 1000 \times 0.08 = 80}[/tex]
[tex]\mathbf{\sigma = \sqrt{\bar x(1 - p)} = \sqrt{80 \times (1 - 0.08)}= 8.58}[/tex]
The probability is then represented as:
[tex]\mathbf{P(x > 70) = P(x > 70.5)}[/tex]
Calculate the z-scores for x = 70.5
[tex]\mathbf{z = \frac{x - \bar x}{\sigma}}[/tex]
So, we have:
[tex]\mathbf{z = \frac{70.5 - 80}{8.58} = -1.11}[/tex]
So, we have:
[tex]\mathbf{P(x > 70) = P(z > -1.11)}[/tex]
Using z-scores of probabilities, we have:
[tex]\mathbf{P(x > 70) =0.8665}[/tex]
Hence, the probability that more than 70 workers will be involved in an accident is (a) 0.8665
Read more about probabilities at:
https://brainly.com/question/25347055
