Respuesta :
Answer:
96.42% probability that the number of tails is between 40 and 60.
Step-by-step explanation:
I am going to use the binomial approximation to the normal to solve this question.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
In this problem, we have that:
100 tosses, so [tex]n = 100[/tex]
Two outcomes, both equally as likely. So [tex]p = \frac{1}{2} = 0.5[/tex]
So
[tex]E(X) = np = 100*0.5 = 50[/tex]
[tex]\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{100*0.5*0.5} = 5[/tex]
Estimate the approximate probability that the number of tails is between 40 and 60.
Using continuity correction.
[tex]P(40 - 0.5 \leq X \leq 60 + 0.5) = P(39.5 \leq X \leq 60.5)[/tex]
This is the pvalue of Z when X = 60.5 subtracted by the pvalue of Z when X = 39.5. So
X = 60.5
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{60.5 - 50}{5}[/tex]
[tex]Z = 2.1[/tex]
[tex]Z = 2.1[/tex] has a pvalue of 0.9821
X = 39.5
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{39.5 - 50}{5}[/tex]
[tex]Z = -2.1[/tex]
[tex]Z = -2.1[/tex] has a pvalue of 0.0179
0.9821 - 0.0179 = 0.9642
96.42% probability that the number of tails is between 40 and 60.
