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According to a Gallup poll, 60% of American adults prefer saving over spending. You randomly select 4 adults and ask if they prefer spending over saving. Construct a probability distribution for the random variable x. Then graph the distribution. (Histogram)



Expected Value ______________________________

Standard Deviation _____________________________

B –

I-

N-

S-

Interpretation

Sagot :

Using the binomial distribution, it is found that:

  • The expected value is of 2.4.
  • The standard deviation is of 0.76.
  • The distribution is given by the histogram.

For each person, there are only two possible outcomes, either they prefer saving, or they prefer spending. The preferences of each person are independent of any other person, hence, the binomial distribution is used to solve this question.

Binomial probability distribution

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are:

  • x is the number of successes.
  • n is the number of trials.
  • p is the probability of a success on a single trial.

The expected value is:

[tex]E(X) = np[/tex]

The standard deviation is:

[tex]\sqrt{V(X)} = \sqrt{np(1 - p)}[/tex]

In this problem:

  • 60% prefer saving over spending, hence [tex]p = 0.6[/tex]
  • 4 adults are selected, hence [tex]n = 4[/tex].

Then:

[tex]E(X) = np = 4(0.6) = 2.4[/tex]

[tex]\sqrt{V(X)} = \sqrt{np(1 - p)} = \sqrt{2.4(0.6)(0.4)} = 0.76[/tex]

The histogram is sketched at the end of the answer.

You can learn more about the binomial distribution at https://brainly.com/question/24863377

View image joaobezerra
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