HiSET: Math : Generalizing information about a population from a sample of the population

Study concepts, example questions & explanations for HiSET: Math

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Example Questions

Example Question #1 : Generalizing Information About A Population From A Sample Of The Population

Researchers want to find the average age of indigenous people who inhabit a remote island. It is believed that the people of this island live abnormally long lives, so the average age of people is higher than in the rest of the world. The previously reported average age of people on the island is 47 years.

Researchers wanted to test the validity of this claim. They took a random sample of 50 people on the island and found that the average age of the sample was 28.5 years. 

Which of the following statements can best be made, given the findings?

Possible Answers:

The average age of the indigenous people on the island is 28.5 years

50 people on the island have ages between 28 and 29 years. 

It is very likely that the average age of people on the island is actually much lower than previously reported. 

The average age of indigenous people on the island is between 28 and 29 years. 

The average age of people on the island is the same as the rest of the world. 

Correct answer:

It is very likely that the average age of people on the island is actually much lower than previously reported. 

Explanation:

From the question, we know the average age from a random sample of 50 people was 28.5 years; however, because this data is only from a random sampling of people, it is difficult to say anything certain about the entire population.

While it seems that the previously reported age is wrong, and that the average age of people on the island is likely around 29, it is impossible to say for certain what the average age actually is without gathering data for every single person on the island. 

With findings from a random sample, you can draw conclusions about what is likely.

It is likely that the actual average age is lower than the previously reported average age; however, without data for every single person in the population, statements that are certain, such as, "The average age of the indigenous people on the island is 28.5 years," cannot be made.

Note that the statement, "The average age of indigenous people on the island is between 28 and 29 years," is a certain statement, even though it does not give an exact value for the average. Even though researchers found an average age of 28.5 years, it is still very possible that the actual average age is outside the range of 28 to 29. 

Therefore, the statement, "It is very likely that the average age of people on the island is actually much lower than previously reported," is the answer because it does not try to make any certain claims and the conclusion is supported by the findings.

Example Question #2 : Generalizing Information About A Population From A Sample Of The Population

An ad campaign was run at 1200 fast food franchise locations. Two weeks later, a random sample of revenue of 80 locations was taken to measure the effectiveness of the ad campaign. Of the 80 locations, 60 locations reported increased sales.

Which of the following statements can be made with the greatest certainty?

Possible Answers:

Exactly 75% of franchise locations experienced increased sales in the last two weeks.

It is very likely that less than half of all franchise locations experienced increased sales in the last two weeks.

It is very likely that more than half of all franchise locations experienced increased sales in the last two weeks.

At least 75% of all franchise locations did not experience increased sales in the last two weeks.

It is very likely that at least half of all franchise locations did not experience increased sales in the last two weeks.

Correct answer:

It is very likely that more than half of all franchise locations experienced increased sales in the last two weeks.

Explanation:

From the question, we know that 60 out of a random sample of 80 locations experienced increased sales in the last two weeks; however, because this data is only from a random sampling of locations, it is difficult to say anything absolute about the population (all of the locations).

While 75% of the random sample did experience increased sales, we cannot know for certain what portion of the entire population experienced increased sales. What we can conclude from this data is that it is likely that the percentage of restaurants that experienced increased sales is roughly around 75%. 

Therefore, the statement that can be made with the most certainty is that "It is very likely that more than half of all franchise locations experienced increased sales in the last two weeks."

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