Question

at lawson 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 chose social - studies in the sample, we use the formula $\text{Percentage}=\frac{\text{Number of students who chose social - studies}}{\text{Total number of students in the sample}}\times100\%$. Here, the number of students who chose social - studies is 28 and the total number of students in the sample is 90. So, $\text{Percentage}=\frac{28}{90}\times 100\%\approx 31.1\%$.

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 we should base our percentage on the sample data. His sample is random and not biased and is relatively large compared to the population of 250 students. However, his percentage calculation is incorrect.

Answer:

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