Question

a sample was taken of dogs at a local dog park on two random days. the counts are displayed in the table below. if there are estimated to be 50 dogs at the park at any given time, which proportion could be used to find the average number of shepherd mixes at the park at any given time?

dog park sample
labrador retriever	4	labrador retriever	5
shepherd mix	7	shepherd mix	9
chihuahua	3	chihuahua	5
poodle	1	poodle	2
australian cattle dog	2	australian cattle dog	4

o $\frac{8}{21}=\frac{x}{50}$
o $\frac{16}{21}=\frac{x}{50}$
o $\frac{8}{50}=\frac{x}{42}$

Explanation:

Step1: Calculate total shepherd - mixes in samples

Add the number of shepherd - mixes in sample 1 and sample 2. $7 + 9=16$.

Step2: Calculate total dogs in samples

Add all the dog counts in sample 1: $4 + 7+3 + 1+2=17$. Add all the dog counts in sample 2: $5 + 9+5 + 2+4 = 25$. Total dogs in samples is $17 + 25 = 42$. But we can also calculate it in another way. The sum of the number of each breed in both samples: $(4 + 5)+(7 + 9)+(3 + 5)+(1 + 2)+(2 + 4)=9 + 16+8 + 3+6 = 42$. The average number of dogs per sample for each breed is considered. The proportion is set up based on the ratio of shepherd - mixes to the total number of dogs in the samples and the total number of dogs in the park. Let $x$ be the average number of shepherd - mixes in the park. The ratio of shepherd - mixes to total dogs in the samples is $\frac{7 + 9}{42}=\frac{16}{42}=\frac{8}{21}$, and the ratio of the unknown number of shepherd - mixes $x$ to the total number of dogs in the park (50) should be the same. So the proportion is $\frac{16}{42}=\frac{x}{50}$, or simplified as $\frac{8}{21}=\frac{x}{50}$.

Answer:

The proportion $\frac{16}{42}=\frac{x}{50}$ (or $\frac{8}{21}=\frac{x}{50}$) could be used. Among the given options, if we assume the intended proportion is $\frac{16}{42}=\frac{x}{50}$ (simplified from the correct ratio - setup), and since $\frac{16}{42}=\frac{8}{21}$, the closest correct - looking option based on the work is $\frac{16}{42}=\frac{x}{50}$ (or $\frac{8}{21}=\frac{x}{50}$). But if we consider the options as they are, we note that the sum of shepherd - mixes in both samples is $7+9 = 16$, and if we assume the proportion is set up based on the total number of dogs in one - sample (sum of all breeds in one sample is 21), the correct proportion is $\frac{16}{21}=\frac{x}{50}$. So the answer is $\frac{16}{21}=\frac{x}{50}$.