To find out how many sixth graders should be chosen to accurately represent the population based on the given survey of seventh graders, we need to look at the proportions of each grade in the population.
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Calculate the total population of middle school students: \[ \text{Total} = 160 \text{ (sixth graders)} + 180 \text{ (seventh graders)} + 140 \text{ (eighth graders)} = 480 \]
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Find the proportion of sixth graders in the population: \[ \text{Proportion of sixth graders} = \frac{160}{480} = \frac{1}{3} \]
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Knowing that 9 seventh graders are chosen in the sample, now we can use proportions to determine how many sixth graders should also be included.
The total ratio of grades should remain consistent. If 9 seventh graders represent their proportion, we can find the corresponding sixth grade representation: \[ \text{Let } x \text{ be the number of sixth graders.} \]
Since sixth graders make up \(\frac{1}{3}\) of the total population, for every 1 sixth grader, there should be approximately 3 seventh graders in the sample.
The calculation based on the sample size of seventh graders is: \[ \frac{x}{9} = \frac{160/480}{180/480} = \frac{160}{180} \]
Rearranging gives: \[ x = 9 \times \frac{160}{180} = 9 \times \frac{8}{9} = 8 \]
Therefore, about 8 sixth graders should be chosen to accurately represent the population. The answer is:
8