Respuesta :
Answer:
a) 0.4761
b) 0.2118
Step-by-step explanation:
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:
[tex]n = 1000, p = 0.5[/tex]
So
[tex]\mu = E(X) = np = 1000*0.5 = 5000[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{1000*0.5*0.5} = 15.81[/tex]
a)P(x>500)
This is 1 subtracted by the pvalue of Z when X = 501. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{501 - 500}{15.81}[/tex]
[tex]Z = 0.06[/tex]
[tex]Z = 0.06[/tex] has a pvalue of 0.5239
1 - 0.5239 = 0.4761
b)P(490≤x<500)
This is the pvalue of Z when X = 499 subtracted by the pvalue of Z when X = 490. So
X = 499
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{499 - 500}{15.81}[/tex]
[tex]Z = -0.06[/tex]
[tex]Z = -0.06[/tex] has a pvalue of 0.4761
X = 490
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{490 - 500}{15.81}[/tex]
[tex]Z = -0.63[/tex]
[tex]Z = -0.63[/tex] has a pvalue of 0.2643
0.4761 - 0.2643 = 0.2118
