Respuesta :
The probability of obtaining between 70 and 82 tails, inclusive is 0.2143.
What is Normal Approximation to Binomial?
The likelihood of getting either head or tail while tossing a coin is constant. As a result, its probability can be computed using the binomial distribution, however as the sample size grows, it is preferable to approximation the distribution using the Normal distribution.
Now according to the question,
A fair coin is tossed 130 times; n = 130
The probability of getting a tail; p = 0.5.
The probability of getting a head; q = 0.5.
np = 130×0.5
np = 65 (as it is more than 10 so acceptable)
nq = 130×0.5
nq = 65 (as it is more than 10 so acceptable).
As np and nq > 10, as a result, we can use the normal distribution to approximate the distribution.
The mean population;
μ = np = 65
Standard variation population
σ = √npq
σ = 5.7 (approx)
Now, calculate the z-score;
[tex]Z=\frac{X-\mu}{\sigma}[/tex]
where x is the number of times tails appears.
Using the continuity correction factor
[tex]\begin{aligned}&P(70 \leq X \leq 82)=P(70-0.5 < X < 82+0.5) \\&P(70 \leq X \leq 82)=P(69.5 < X < 82.5)\end{aligned}[/tex]
Now, when X = 69.5, find z-score
[tex]Z=\frac{69.5-65}{5.7}[/tex]
z = 0.79 (approx)
When X = 82.5, z-score is;
[tex]Z=\frac{82.5-65}{5.7}[/tex]
z = 2.71 (approx)
[tex]P(69.5 < X < 82.5)=P(0.78 < Z < 2.71)[/tex]
[tex]P(69.5 < X < 82.5)[/tex] = 0.9966 - 0.7823 (check from positive z-score table)
[tex]P(69.5 < X < 82.5)[/tex] = 0.2143
Therefore, the probability of obtaining between 70 and 82 tails, inclusive is 0.2143.
To know more about normal distribution, here
https://brainly.com/question/15088614
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