Question

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.

Answer 1

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 ($\hat p$) of pet dogs adopted from an animal shelter follows a normal distribution.

Mean of $\hat p$ is:

$\mu_{\hat p}=p=0.19$

Standard deviation of $\hat p$ is:

$\sigma_{\hat p}=\sqrt{\frac{p(1-p)}{n} }$

Compute the probability of $\hat p$ between 15% and 20% as follows:

$P(0.15\leq \hat p\leq 0.20)=P(\frac{0.15-0.19}{0.044}\leq \frac{\hat p-\mu_{\hat p}}{\sigma_{\hat p}}\leq \frac{0.20-0.19}{0.044}} )\\=P(-0.91$

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