Question

a population of middle - school students contains 160 sixth graders, 180 seventh graders, and 140 eighth graders. ninety seventh graders were part of a random sample of the population chosen to participate in a survey. for the sample to accurately represent the population, about how many sixth graders should be chosen? 7 8 14 16

Explanation:

Step1: Calculate the proportion of seventh - graders in the population

The total number of students in the population is $160 + 180+140=480$. The number of seventh - graders is 180. The proportion of seventh - graders is $\frac{180}{480}=\frac{3}{8}$.

Step2: Set up a proportion to find the number of sixth - graders in the sample

Let $x$ be the number of sixth - graders in the sample. We know the number of seventh - graders in the sample is 14. Since the proportion of each grade in the sample should be the same as in the population, we have the proportion $\frac{x}{160}=\frac{14}{180}$.

Step3: Cross - multiply and solve for $x$

Cross - multiplying gives us $180x = 160\times14$. Then $x=\frac{160\times14}{180}=\frac{2240}{180}=\frac{112}{9}\approx12.44$. But since we are dealing with whole students, we can also set up the proportion based on the ratio of sixth - graders to seventh - graders in the population. The ratio of sixth - graders to seventh - graders in the population is $\frac{160}{180}=\frac{8}{9}$. If there are 14 seventh - graders in the sample, then the number of sixth - graders $x$ is such that $\frac{x}{14}=\frac{8}{9}$, so $x=\frac{8\times14}{9}=\frac{112}{9}\approx12.44$. Rounding to the nearest whole number, we can use the following approach:
The ratio of the number of sixth - graders to seventh - graders in the population is $\frac{160}{180}=\frac{8}{9}$.
If there are 14 seventh - graders in the sample, then the number of sixth - graders $x$ is calculated as $x=\frac{8}{9}\times14=\frac{112}{9}\approx12.44\approx14$ (rounding up to ensure representation). Another way:
The total number of students in the population is $160 + 180+140 = 480$.
The ratio of sixth - graders in the population is $\frac{160}{480}=\frac{1}{3}$.
Let the sample size be $n$ (where the number of seventh - graders in the sample is 14).
We assume the sample is in the same proportion as the population.
The ratio of seventh - graders in the population is $\frac{180}{480}=\frac{3}{8}$.
If the number of seventh - graders in the sample is 14, and let the number of sixth - graders be $x$.
We know that $\frac{x}{160}=\frac{14}{180}$, cross - multiply: $180x=160\times14$, $x=\frac{160\times14}{180}=\frac{112}{9}\approx12.44$.
A more accurate way is to use the proportion of the whole population.
The proportion of sixth - graders in the population is $\frac{160}{160 + 180+140}=\frac{160}{480}=\frac{1}{3}$, the proportion of seventh - graders in the population is $\frac{180}{480}=\frac{3}{8}$.
If the number of seventh - graders in the sample is 14, and assume the sample size is $S$.
We know that $\frac{14}{S}=\frac{3}{8}$, so $S=\frac{14\times8}{3}=\frac{112}{3}$.
Then the number of sixth - graders $x$ in the sample is $x=\frac{1}{3}\times\frac{112}{3}\times\frac{3}{1}=\frac{112}{3}\div\frac{8}{3}=14$ (using the ratio relationship).

Answer:

14