Question

Will surveyed students at his school about whether they have ever gone snowboarding and whether they own a skateboard. He found that 35 of the 99 students who own a skateboard have snowboarded. Also, there were 13 students who have snowboarded but do not own a skateboard, and 147 students who have never gone snowboarding and do not own a skateboard. Which two-way table correctly displays this data? A 4-column table has 3 rows. The first column has entries skateboard, no skateboard, total. The second column is labeled have snowboarded with entries 35, 64, 99. The third column is labeled never snowboarded with entries 13, 147, 160. The fourth column is labeled Total with entries 48, 211, 256. A 4-column table has 3 rows. The first column has entries skateboard, no skateboard, total. The second column is labeled have snowboarded with entries 35, 13, 48. The third column is labeled never snowboarded with entries 99, 147, 246. The fourth column is labeled Total with entries 134, 160, 294. A 4-column table has 3 rows. The first column has entries skateboard, no skateboard, total. The second column is labeled have snowboarded with entries 35, 13, 48. The third column is labeled never snowboarded with entries 64, 147, 211. The fourth column is labeled Total with entries 99, 160, 259. A 4-column table has 3 rows. The first column has entries skateboard, no skateboard, total. The second column is labeled have snowboarded with entries 35, 99, 134. The third column is labeled never snowboarded with entries 13, 147, 160. The fourth column is labeled Total with entries 48, 246, 294.

Answer 1

Two way table is description of two dimensions' data and their intersections' data. The correct two-way table for the given data is:

How to form two-way table?

Suppose two dimensions are there, viz X and Y. Some values of X are there as $X_1, X_2, ... , X_n$ and some values of Y are there as $Y_1, Y_2, ..., Y_k$. List them in title of the rows and left to the columns. There will be $n \times k$ table of values will be formed(excluding titles and totals), such that:

Value(ith row, jth column) = Frequency for intersection of $X_i$ and $Y_j$ (assuming X values are going in rows, and Y values are listed in columns).

Then totals for rows, columns, and whole table are written on bottom and right margin of the final table.

For n = 2, and k = 2, the table would look like:

$\begin{array}{cccc}&Y_1&Y_2&\rm Total\\X_1&n(X_1 \cap Y_1)&n(X_1\cap Y_2)&n(X_1)\\X_2&n(X_2 \cap Y_1)&n(X_2 \cap Y_2)&n(X_2)\\\rm Total & n(Y_1) & n(Y_2) & S \end{array}$

where S denotes total of totals, also called total frequency.

n is showing the frequency of the bracketed quantity, and intersection sign in between is showing occurrence of both the categories together.

For the given case, let we suppose:
X = Ownership for skateboards

• $X_1$ = Student owns a skateboard
• $X_2$ = Student not owning skateboard

Y = Ownership for snowboards

• $Y_1$ = Student owns a skateboard
• $Y_2$ = Student not owning skateboard

Their frequencies are given in the problem as:

35 of the 99 students who own a skateboard have snowboarded.

That means  $n(X_1 \cap Y_1) = 33$, and $n(X_1)$ = 99 (total frequency(number of students) is 99)

There were 13 students who have snowboarded but do not own a skateboard, so $n(X_2 \cap Y_1) = 13$

147 students who have never gone snowboarding and do not own a skateboard.  Thus, $n(X_2 \cap Y_2) = 147$

We get the table as:

$\begin{array}{cccc}&Y_1&Y_2&\rm Total\\X_1&35&n(X_1\cap Y_2)&99\\X_2&13&147&n(X_2)=13 + 147=160\\\rm Total & n(Y_1)=35+13=48 & n(Y_2) & S=160+99=259 \end{array}$

Thus, we get number of students who doesn't own snowboard but own skateboard = 99 - 35 = 64

and total students not owning either snowboard or skateboard = 35 + 147 = 182

Thus, the completed table would look like:

$\begin{array}{cccc}&Y_1&Y_2&\rm Total\\X_1&35&64&99\\X_2&13&147&160\\\rm Total & 48 & 211 & 259 \end{array}$

Learn more about two way frequency table here:

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