~ 8.16 ~
Estimating Population Proportions
Learning Targets
I can estimate the proportion of population data that are in a certain category based on a sample.
Notes
Sometimes a data set consists of information that fits into specific categories. For example, we could survey students about whether they have a pet cat or dog. The categories for these data would be {neither, dog only, cat only, both}. Suppose we surveyed 10 students. Here is a table showing possible results:
In this sample, 3 of the students said they have both a dog and a cat. We can say that the proportion of these students who have a both a dog and a cat is 3/10 or 0.3. If this sample is representative of all 720 students at the school, we can predict that about 3/10 of 720, or about 216 students at the school have both a dog and a cat.
In general, a proportion is a number from 0 to 1 that represents the fraction of the data that belongs to a given category.
Vocabulary
proportion
A proportion of a data set is the fraction of the data in a given category.
For example, a class has 18 students. There are 2 left-handed students and 16 right-handed students in the class. The proportion of students who are left-handed is 2/20, or 0.1.
Activities
16.1 Getting to School
A teacher asked all the students in one class how many minutes it takes them to get to school. Here is a table of their responses.
What fraction of the students in this class say:
it takes them 5 minutes to get to school?
it takes them more than 10 minutes to get to school?
If the whole school has 720 students, can you use this data to estimate how many of them would say that it takes them more than 10 minutes to get to school?
16.3 A New Comic Book Hero
Here are the results of a survey of 20 people who read The Adventures of Super Sam regarding what special ability they think the new hero should have.
What proportion of this sample want the new hero to have the ability to fly?
If there are 2,024 dedicated readers of The Adventures of Super Sam, estimate the number of readers who want the new hero to fly.
Two other comic books did a similar survey of their readers.
In a survey of people who read Beyond Human, 42 out of 60 people want a new hero to be able to fly.
In a survey of people who read Mysterious Planets, 14 out of 40 people want a new hero to be able to fly.
Do you think the proportion of all readers who want a new hero that can fly are nearly the same for the three different comic books? Explain your reasoning.
If you were in charge of these three comics, would you give the ability to fly to any of the new heroes? Explain your reasoning using the proportions you calculated.
16.5 More than 48 Grams
A chemical engineer is trying to increase the amount of the useful product in a reaction. She performs the reaction with her new equipment 10 times and gets the following amounts of the useful product.
What proportion of the reactions were above the 48 grams threshold?
Other chemists typically get 65% of their reactions to produce more than 48 grams. Should the engineer say that she was able to increase the useful product when compared to the other chemists?
Add to Your Notes
When using data, what is a proportion? How is it calculated?
A proportion is the fraction of the data that are in a certain category.
It is calculated by counting the number of data values in the category and dividing by the total number of data values in the sample.
In order to say that more than half of the people in a sample responded with a certain answer, what would the proportion for that answer be?
any value greater than 0.5
A random sample indicates that a 0.45 proportion of people shopping at a certain store prefer wheat bread to white bread. The store has 3,000 customers. Estimate the number of people shopping at the store who prefer wheat bread.
about 1,350 0.45 • 3,000 = 1,350
Assignment
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