Question

Stereo speakers are manufactured with a probability of 0.100.10 being defective. TwentyTwenty speakers are randomly selected. Let the random variable X be defined as the number of defective speakers. Find the expected value and the standard deviation.

Answer 1

Answer:

The expected value of X is 2 with a standard deviation of 1.34.

Step-by-step explanation:

For each speaker, there are only two possible outcomes. Either it is defective, or it is not. The probability of a speaker being defective is independent of other speakers. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The expected value of the binomial distribution is:

$E(X) = np$

The standard deviation of the binomial distribution is:

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

Stereo speakers are manufactured with a probability of 0.1 of being defective

This means that $p = 0.1$

Twenty speakers are randomly selected.

This means that $n = 20$

Let the random variable X be defined as the number of defective speakers. Find the expected value and the standard deviation.

$E(X) = np = 20*0.1 = 2$

$\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{20*0.1*0.9} = 1.34$

The expected value of X is 2 with a standard deviation of 1.34.