Question

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. In one county, the conviction rate for speeding is 85%. Estimate the probability that of the next 100 speeding summonses issued, there will be at least 90 convictions.

Answer 1

Answer:

13.14% probability that of the next 100 speeding summonses issued, there will be at least 90 convictions.

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:

$n = 100, p = 0.85$

So

$\mu = E(X) = np = 100*0.85 = 85$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{100*0.85*0.15} = 3.57$

Estimate the probability that of the next 100 speeding summonses issued, there will be at least 90 convictions.

At least 90 is more than 89, which is 1 subtracted by the pvalue of Z when X = 89. So

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

$Z = \frac{89 - 85}{3.57}$

$Z = 1.12$

$Z = 1.12$ has a pvalue of 0.8686

1 - 0.8686 = 0.1314

13.14% probability that of the next 100 speeding summonses issued, there will be at least 90 convictions.