Question

0.3 notetaking with vocabulary (continued)
. in a survey, 214 ninth graders played video games every day of the week and 22 nin did not play video games every day of the week. of those that played every day of tl 36 had trouble sleeping at night. of those that did not play every day of the week, 7 sleeping at night. make a two - way table that shows the joint and marginal relative fi

Explanation:

Step1: Identify Categories

We have two categories: "Play Video Games Daily" (with two sub - categories: Yes/No) and "Trouble Sleeping" (with two sub - categories: Yes/No). Let's define:

• Let \( A \): Play video games daily (\( n(A)=214 \))
• Let \(

eg A \): Do not play video games daily (\( n(
eg A) = 22\))

• Let \( B \): Have trouble sleeping. For \( A \), \( n(A\cap B)=36 \); for \(

eg A \), \( n(
eg A\cap B)=7 \)

Step2: Calculate Totals

First, calculate the total number of ninth - graders surveyed: \( N=n(A)+n(
eg A)=214 + 22=236\)
For the "Play daily" group:

• Number of students who play daily and do not have trouble sleeping: \( n(A\cap

eg B)=n(A)-n(A\cap B)=214 - 36 = 178\)
For the "Do not play daily" group:

• Number of students who do not play daily and do not have trouble sleeping: \( n(

eg A\cap
eg B)=n(
eg A)-n(
eg A\cap B)=22 - 7=15\)

Step3: Create the Two - Way Table (Frequency Table)

	Trouble Sleeping (Yes)	Trouble Sleeping (No)	Total
Do Not Play Daily	7	15	22
Total	\( 36 + 7=43\)	\( 178+15 = 193\)	236

Step4: Calculate Joint and Marginal Relative Frequencies

• Joint relative frequency: For a cell with frequency \( f \), joint relative frequency \(=\frac{f}{N}\)
• \( P(A\cap B)=\frac{36}{236}\approx0.1525\)
• \( P(A\cap

eg B)=\frac{178}{236}\approx0.7542\)

• \( P(

eg A\cap B)=\frac{7}{236}\approx0.0297\)

• \( P(

eg A\cap
eg B)=\frac{15}{236}\approx0.0636\)

• Marginal relative frequency: For a row or column total \( T \), marginal relative frequency \(=\frac{T}{N}\)
• Row marginal for "Play Daily": \( P(A)=\frac{214}{236}\approx0.9068\)
• Row marginal for "Do Not Play Daily": \( P(

eg A)=\frac{22}{236}\approx0.0932\)

• Column marginal for "Trouble Sleeping (Yes)": \( P(B)=\frac{43}{236}\approx0.1822\)
• Column marginal for "Trouble Sleeping (No)": \( P(

eg B)=\frac{193}{236}\approx0.8178\)

Answer:

The two - way table (with joint and marginal relative frequencies can be presented as above. The frequency two - way table is:

	Trouble Sleeping (Yes)	Trouble Sleeping (No)	Total
Do Not Play Daily	7	15	22
Total	43	193	236

And the joint relative frequencies are \( \frac{36}{236},\frac{178}{236},\frac{7}{236},\frac{15}{236}\) and marginal relative frequencies are \( \frac{214}{236},\frac{22}{236},\frac{43}{236},\frac{193}{236}\) (or their decimal approximations).