Mathematics, 30.03.2020 18:39 rebecabosca

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.

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Answer from: tishaadams2160

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

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

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

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.

Answer from: Quest

Ido not know my friend

Answer from: Quest

first anwser choice.

step-by-step explanation:

the greater/less or equal to symbol (> or < with the line at the bottom) is a closed/filled in circle.

when multiplying or dividing by a negative on both sides you have to flip the inequality sign. this rule only applies to multiplying or dividing by a negative number in inequality equations. this rule is really important.

-5c + 2 _< _27

subtract from both sides by 2

-5c _< _ 27

divide both sides by -5

c _< _ 5.4

flip the inequality sign

c _> _5.4

the only possible answer that would match the simplified inequality is the first answer choice

5c+2< 27 i really do not understand this.

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