Question

at larson middle school, mr. hernandez asked 90 randomly selected students from each grade level for their favorite subject, and 28 chose social studies as their favorite. he used this data to draw the inference that out of the 250 middle school students, about 20% prefer social studies. did he make a reasonable inference?
yes, mr. hernandezs sample is random, not biased, large enough compared to the population, and 28 is 20% of 250.
yes, mr. hernandezs sample is random, not biased, large enough compared to the population, and 28 is 20% of 90.
no, mr. hernandezs sample is random, not biased, large enough compared to the population, but 28 is not 20% of 90.
no, mr. hernandezs sample is not large enough compared to the population.

Explanation:

Step1: Calculate the percentage in the sample

To find the percentage of students who prefer social - studies in the sample, we use the formula $\text{Percentage}=\frac{\text{Number of students who prefer social - studies}}{\text{Total number of students in the sample}}\times100\%$. Here, the number of students in the sample is $n = 90$, and the number of students who prefer social - studies is $x = 28$. So the percentage in the sample is $\frac{28}{90}\times100\%\approx31.1\%$.

Step2: Analyze the inference

Mr. Hernandez inferred that about 20% of the 250 middle - school students prefer social studies. But $28$ is not 20% of $90$ (since $\frac{28}{90}\times100\%\approx31.1\%$) and the sample size of 90 (from each grade level) is relatively large compared to the population of 250, and the sample is random and not biased.

Answer:

No, Mr. Hernandez's sample is random, not biased, large enough compared to the population, but 28 is not 20% of 90.