Answered

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.


Have SnowboardedNever SnowboardedTotal

Skateboard		351348

No Skateboard		64147211

Total		99160256






B.


Have SnowboardedNever SnowboardedTotal

Skateboard		3599134

No Skateboard		13147160

Total		48246294






C.


Have SnowboardedNever SnowboardedTotal

Skateboard		356499

No Skateboard		13147160

Verified Answer :

  • Identify the given values: \(|S| = 99\), \(|S \cap H| = 35\), \(|NS \cap H| = 13\), and \(|NS \cap N| = 147\).
  • Calculate the remaining cells: \(|S \cap N| = 99 - 35 = 64\), \(|NS| = 13 + 147 = 160\), \(|H| = 35 + 13 = 48\), and \(|N| = 64 + 147 = 211\).
  • Verify the grand total: \(99 + 160 = 259\) or \(48 + 211 = 259\).
  • Select the third option: \(\boxed{\text{Option 3}}\).

Explanation

  1. Analyze the provided data

To organize the survey data effectively, we will construct a \(2 \times 2\) contingency table. A contingency table helps us display the frequency distribution of variables. In this case, our two variables are:

  1. Owning a skateboard (Skateboard / No Skateboard)
  2. Experience with snowboarding (Have Snowboarded / Never Snowboarded)

We are given the following information:

  • The total number of students who own a skateboard is \(|S| = 99\).
  • Of these \(99\) students, the number who have snowboarded is \(|S \cap H| = 35\).
  • The number of students who have snowboarded but do not own a skateboard is \(|NS \cap H| = 13\).
  • The number of students who have never snowboarded and do not own a skateboard is \(|NS \cap N| = 147\).
  1. Calculate the missing values

Now, let us calculate the missing values in our contingency table:

  1. Skateboard and Never Snowboarded (\(S \cap N\)): Since we know there are \(99\) skateboard owners in total and \(35\) of them have snowboarded, the number of skateboard owners who have never snowboarded is:

\[|S \cap N| = |S| - |S \cap H| = 99 - 35 = 64\]

  1. Total Have Snowboarded (\(H\)): This is the sum of skateboard owners who have snowboarded and non-skateboard owners who have snowboarded:

\[|H| = |S \cap H| + |NS \cap H| = 35 + 13 = 48\]

  1. Total Never Snowboarded (\(N\)): This is the sum of skateboard owners who have never snowboarded and non-skateboard owners who have never snowboarded:

\[|N| = |S \cap N| + |NS \cap N| = 64 + 147 = 211\]

  1. Total No Skateboard (\(NS\)): This is the sum of non-skateboard owners who have snowboarded and those who have never snowboarded:

\[|NS| = |NS \cap H| + |NS \cap N| = 13 + 147 = 160\]

  1. Grand Total (\(T\)): This is the sum of the row totals (or column totals):

\[T = |S| + |NS| = 99 + 160 = 259\] (or \(T = |H| + |N| = 48 + 211 = 259\))

  1. Construct the final table and select the correct option

Now we organize our findings into the final contingency table:

\[ \begin{array}{|l|l|l|l|} \ & \text{Have Snowboarded} & \text{Never Snowboarded} & \text{Total} \\ \text{Skateboard} & 35 & 64 & 99 \\ \text{No Skateboard} & 13 & 147 & 160 \\ \text{Total} & 48 & 211 & 259 \\ \end{array} \]

Comparing this result to the provided options, we see that it matches the third option perfectly.

Examples

Contingency tables are a fundamental tool in statistics used to study the relationship between two categorical variables. For example, a doctor might use a \(2 \times 2\) table to compare two groups of patients (e.g., treated vs. placebo) against two outcomes (e.g., recovered vs. not recovered) to evaluate the effectiveness of a new medicine. By organizing data into these clear rows and columns, we can easily perform probability calculations and identify patterns in survey responses.

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