Mathematics, 26.02.2020 18:59 vrentadrienneoqug1a

A random sample of 1,700 cars is checked for a serious defect. The chance that a car has the defect is 0.6%. Approximate the probability that at least 10 cars in the sample have the defect. State and check the appropriate conditions.

Answers: 2

Show answers

Answers

Answer from: tburlew01

52.39% probability that at least 10 cars in the sample have the defect.

Conditions:

Each car has only two possible outcomes. Either it is defective, or it is not. The cars are chosen at random, which means that the probability of a car being defective is independent from other cars. So we use the binomial probability distribution to solve this questions.

Step-by-step explanation:

Conditions:

We are working with a large sample, which means that we can approximate the binomial distribution to the normal.

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 = 1700, p = 0.006

E(X) = np = 1700*0.006 = 10.2

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{1700*0.006*0.994} = 3.18$

Approximate the probability that at least 10 cars in the sample have the defect.

This is 1 subtracted by the pvalue of Z when X = 10. So

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

$Z = \frac{9 - 10.2}{3.18}$

Z = -0.06

Z = -0.06 has a pvalue of 0.4761.

1 - 0.4761 = 0.5239

52.39% probability that at least 10 cars in the sample have the defect.

Answer from: Quest

step-by-step explanation:

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