16B Coreq

In the next preview assignment, you will need to understand complex fractions to calculate the value of an F-statistic and use the DCMP F Distribution tool to calculate the correspondingP-value.

Complex Fractions

A complex fraction is a fraction in which the numerator and/or the denominator include fractions. An example of a complex fraction is shown below:

[latex]\frac{\frac{1}{2}}{\frac{3}{4}}[/latex]

In keeping with the order of operations, you should evaluate the numerator and the denominator separately before dividing the numerator by the denominator. When using a calculator, use parentheses to indicate that the operations in the numerator and the denominator should be completed first, as shown:

[latex](1/2)/(3/4)≈0.667[/latex]

Question 1

1) Use a calculator to evaluate the following complex fractions. Express your answers using decimals. Round to three decimal places.

Part A: 149⁄2139/12

Part B: 4⁄13/12

F-statistics always involve complex fractions. For example, In-Class Activity 14.B introduced the one-way ANOVA F-statistic. The formula for the one-way ANOVA F-statistic is:

[latex]𝐹=\frac{𝑀𝑆𝐺𝑟𝑜𝑢𝑝}{𝑀𝑆𝐸𝑟𝑟𝑜𝑟}=\frac{𝑆𝑆𝐺𝑟𝑜𝑢𝑝/𝑑𝑓𝐺𝑟𝑜𝑢𝑝}{𝑆𝑆𝐸𝑟𝑟𝑜𝑟/𝑑𝑓𝐸𝑟𝑟𝑜𝑟}[/latex]

Question 2

2) Use the values given in the following partially filled-in ANOVA table to calculate the F-statistic. Round your answer to three decimal places. Source DfSum sqMean sqF valueGroup2614.6307.3Error69391156.681Total714525.6During In-Class Activity16.B, you will consider how various factors affect the value of the F-statistic.

Question 3

3) First let’s try changing some of the values in the complex fractions you saw in Question 1. Fill in the blanks.

Part A: 𝑥⁄2139/12As 𝑥increases, the size of the overall fraction ______.

Part B: 4⁄13/𝑥As 𝑥increases, the size of the overall fraction ______.

Question 4

4) Now let’s try changing some of the values in the formula for the F-statistic.You may assume that we are only changing one value at a time—all other values in the formula are being held constant.

Part A: As SSGroupincreases, the size of the F-statistic ______.

Part B: As the degrees of freedom for Groupincreases, the size of the F-statistic ______.

Part C: As SSErrorincreases, the size of the F-statistic ______.

Part D: As the degrees of freedom for Errorincreases, the size of the F-statistic ______.

Using the F Distribution Tool

The F Distribution models the values of the F-statistic that would occur if the null hypothesis was true. The F Distribution tool (i.e., data analysis tool) can be used to calculate a P-value—the probability of getting an F-statistic as large or larger than the one from the sample if the null hypothesis was true. Note: This is an upper tail test. There are different F Distributions, depending on the numerator degrees of freedom (𝑑𝑓1) and the denominator degrees of freedom (𝑑𝑓2). These degrees of freedom come from the numerator and the denominator of the F-statistic, so for a one-way ANOVA,𝑑𝑓1is the degrees of freedom for groups and 𝑑𝑓2is the degrees of freedom for error. Follow these steps to calculate a P-value using the DCMP F Distribution tool at https://dcmathpathways.shinyapps.io/FDist/.

•Click the Find Probability tab at the top of the page.

•Change the numerator and denominator degrees of freedom to match the given scenario.

•Switch to “Upper Tail” probability.

•Type the F-statistic into the “Value of x” box.

•The P-value is the percentage shown in orange above the graph. This value can also be converted to a proportion by moving the decimal two places to the left.

Question 5

5) Suppose that F = 2.89, 𝑑𝑓1= 3, and 𝑑𝑓2= 76.

Part A: Use the DCMP F Distribution tool to find the P-value. Express your answer as a percentage. Do not round.

Part B: Convert the P-value to a decimal. Do not round.

Question 6

6) Use the values given in the following partially filled-in ANOVA table to carry out a one-way ANOVA F-test. SourceDfSum sqMean sqF valueGroup236.018.0Error21134.66.41Total23170.6

Part A: Calculate the F-statistic. Round your answer to three decimal places.