Question Video: Estimating Population Percentages from Sample Data Mathematics

A teacher wants to estimate the percentage of the school students who consider science to be their favorite subject. He asked a random sample of 80 students entering the school and found that 52 students consider science to be their favorite subject, while 28 students do not. Which of the following is true? [A] We can conclude that about 65% of the school students consider science to be their favorite subject. [B] We can conclude that about 65% of the school students consider mathematics to be their favorite subject. [C] We can conclude that about 35% of the school students consider science to be their favorite subject. [D] We can conclude that about 65% of the school students do not consider science to be their favorite subject.

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Video Transcript

A teacher wants to estimate the percentage of the school students who consider science to be their favorite subject. He asked a random sample of 80 students entering the school and found that 52 students consider science to be their favorite subject, while 28 students do not. Which of the following is true? (A) We can conclude that about 65 percent of the school students consider science to be their favorite subject. (B) We can conclude that about 65 percent of the school students consider mathematics to be their favorite subject. (C) We can conclude that about 35 percent of the school students consider science to be their favorite subject. Or (D) We can conclude that about 65 percent of the school students do not consider science to be their favorite subject.

Let’s begin by noting that each of the options (A), (B), (C), and (D) state a conclusion about the population of all the students in the school. Our population, therefore, is all students in the school. We note also that each of these conclusions is an estimate. So the teacher is taking a sample of 80 students and trying to estimate a percentage for the population. That is, the percentage of students who consider science to be their favorite subject.

Now, using the information in the question, we know that 52 out of 80 students consider science to be their favorite subject. And to convert this to a percentage, we multiplied by 100. This gives us 65 percent. So we know that 65 percent of the students consider science to be their favorite subject. We know also from the question that 28 students do not consider science to be their favorite subject. And as a percentage, 28 out of 80 is 35 percent. So 35 percent do not consider science to be their favorite subject.

We’ve worked out these percentage from the exact data. And given that the sample was randomly selected, we can infer these percentages about the population. We can therefore infer or conclude that about 65 percent of the school students consider science to be their favorite subject. This corresponds to option (A). Therefore, option (A) is true.

If we consider the other options. Option (B) refers to mathematics. And since we have no information about mathematics preferences of students, we cannot claim option (B) to be true. Option (C) claims that about 35 percent of students consider science to be their favorite subject, whereas in fact we found this to be 65 percent. So option (C) is not true. Option (D) concludes that about 65 percent of the school students do not consider science to be their favorite subject. But we found this to be 35 percent. So option (D) is not true. Our answer then is that option (A) is true.

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