Question

Consider the probability that no less than 96 out of 145 people will not get the flu this winter. Assume the probability that a given person will not get the flu this winter is 61%. Approximate the probability using the normal distribution. Round your answer to four decimal places.

Answer 1

Answer:

0.1324 = 13.24% probability that no less than 96 out of 145 people will not get the flu this winter.

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:

$n = 145, p = 0.61$

So

$\mu = E(X) = np = 145*0.61 = 88.45$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{145*0.61*0.39} = 5.87$

Consider the probability that no less than 96 out of 145 people will not get the flu this winter.

More than 95 people, which is the same as 1 subtracted by the pvalue of Z when X = 95. So

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

$Z = \frac{95 - 88.45}{5.87}$

$Z = 1.115$

$Z = 1.115$ has a pvalue of 0.8676

1 - 0.8676 = 0.1324

0.1324 = 13.24% probability that no less than 96 out of 145 people will not get the flu this winter.