Question

2.Assume that x is a binomial random variable with n=1000 and p=0.50. Use a normal approximation to find each of the following probabilities:a)P(x>500)0.4880b)P(490≤x<500)0.2334

Answer 1

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:

$E(X) = np$

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 $\mu = E(X)$, $\sigma = \sqrt{V(X)}$.

In this problem, we have that:

$n = 1000, p = 0.5$

So

$\mu = E(X) = np = 1000*0.5 = 5000$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{1000*0.5*0.5} = 15.81$

a)P(x>500)

This is 1 subtracted by the pvalue of Z when X = 501. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{501 - 500}{15.81}$

$Z = 0.06$

$Z = 0.06$ 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

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{499 - 500}{15.81}$

$Z = -0.06$

$Z = -0.06$ has a pvalue of 0.4761

X = 490

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{490 - 500}{15.81}$

$Z = -0.63$

$Z = -0.63$ has a pvalue of 0.2643

0.4761 - 0.2643 = 0.2118