10C Coreq

In the next preview assignment and in the next class, you will need to be comfortable working with fractions, proportions, and percentages.
Parts of the Whole
In many types of research, we work with data that can be expressed with simple “yes” or “no” answers. Do you support the new law proposed in Congress? Does that patient have diabetes? Do cars come equipped with automatic braking? When we report the number of “yes” responses as a portion of the population or sample, we use a proportion.

Question 1

A small dairy farm is trying to decide whether or not to expand their operations. To  gauge demand, they ask 100 different people a single question: “Do you buy more  than one gallon of milk per week?” 20 people answer “yes.”

  1. One way to write this information would be as a fraction: [latex]\frac{successes}{attempts}[/latex]. Write the result from the survey in this format.
  2. Convert your fraction to a decimal. We usually call this value the sample proportion, or [latex]\hat{p}[/latex].
  3. Now, convert your decimal to a percentage.

Question 2

When working with proportions in research, the results may be written in any of these three formats. Complete the following table by converting between fractions, decimals, and percentages.

Question 3

In research, when using proportions, we can never have a result higher than [latex]\hat{p} = 1[/latex].  Consider our dairy farm example. If 100 people are surveyed, there is no way we  can get more than 100 “yes” answers. This allows us to determine the proportion of  failures for any given value of [latex]\hat{p}[/latex] (the proportion of successes in the sample) using the formula [latex]1 - \hat{p}[/latex].

  1. Our sample proportion of “yes” answers in the dairy farm example is 0.2.  What is the proportion of “no” answers?
  2. The values of [latex]\hat{p}[/latex] and [latex]1 - \hat{p}[/latex] are important parts of the formula we use to  determine how much error there might be in our sample. Complete the  following table to practice finding [latex]\hat{p}[/latex] and [latex]\hat{p}[/latex].
    [latex]\hat{p}[/latex] [latex]1 - \hat{p}[/latex]
    0.1 0.9
    0.23
    0.5
    0.78
    0.91
    0.85
    0.48

Question 4

It is also important to understand how changing the numerator or denominator of a  fraction or proportion changes the overall value. Consider several different possible scenarios for the dairy farm example.

  1. Complete the following table by finding the missing values. The first row has been completed for you.
    Number

    Surveyed

    		 Number of

    “Yes”

    Responses

    		 Fraction [latex]\hat{p}[/latex]
    100 20 20/100 0.2
    100 40
    100 80
    50 20
    500 20
    1000 20
  2. As the numerator increases and the denominator is held constant, what  happens to the value of the fraction?
  3. As the denominator increases and the numerator is held constant, what  happens to the value of the fraction?
  4. Consider the fraction [latex]\frac{x}{y}[/latex]. As [latex]x[/latex] increases and [latex]y[/latex] is held constant, what happens to the overall value of the fraction?