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A humane society reports that 19% of all pet dogs were adopted from an animal shelter.Assuming the truth of this assertion, find the probability that in a random sample of80 per dogs, between 15% and 20% were adopted from a shelter. You may assume thatthe normal distribution applies.

Respuesta :

Answer:

The probability of pet dogs adopted between 15% and 20% is 0.4096.

Explanation:

Let X = number of pet dogs adopted from an animal shelter.

The proportion of pet dogs adopted from an animal shelter is, p = 0.19.

The sample of pet dogs selected is of size, n = 80.

A Normal approximation to Binomial can be applied in this case since,

  • np = 80 × 0.19 = 15.2 > 10
  • n(1 - p) = 80 × (1 - 0.19) = 64.8 > 10

So the sample proportion ([tex]\hat p[/tex]) of pet dogs adopted from an animal shelter follows a normal distribution.

Mean of [tex]\hat p[/tex] is:

[tex]\mu_{\hat p}=p=0.19[/tex]

Standard deviation of [tex]\hat p[/tex] is:

[tex]\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n} }[/tex]

Compute the probability of [tex]\hat p[/tex] between 15% and 20% as follows:

[tex]P(0.15\leq \hat p\leq 0.20)=P(\frac{0.15-0.19}{0.044}\leq \frac{\hat p-\mu_{\hat p}}{\sigma_{\hat p}}\leq \frac{0.20-0.19}{0.044}} )\\=P(-0.91<Z<0.23)\\=P(Z<0.23)-P(Z<-0.91)\\=0.591-0.1814\\=0.4096[/tex]

Thus, the probability of pet dogs adopted between 15% and 20% is 0.4096.

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