Question

1. one hundred fifty random children were asked whether or not they own a bike and whether they had played outside after school this week.

• 104 children owned a bike and 46 children did not own a bike.
• of the children who owned a bike, 35 said they had played outside after school this week.
• of the children who did not own a bike, 27 said they had played outside after school this week.

a. construct a two-way frequency table of this information.
table with columns “own a bike (yes, no, total)” and rows “play outside after school (yes, no, total)”

b. construct a two-way relative frequency table of this information. round to the nearest whole number.
table with columns “own a bike (yes, no)” and rows “play outside after school (yes, no)”

c. if a child has a bike, are they more likely to play outside after school? explain your reasoning.

Answer:

Part a: Two - way Frequency Table

Step 1: Determine the number of children who own a bike and did not play outside

We know that 104 children own a bike and 35 of them played outside. So the number of children who own a bike and did not play outside is \(104 - 35=69\).

Step 2: Determine the number of children who do not own a bike and did not play outside

We know that 46 children do not own a bike and 27 of them played outside. So the number of children who do not own a bike and did not play outside is \(46 - 27 = 19\).

Step 3: Calculate the totals for "Play Outside After School"

• For "Yes": The number of children who played outside is \(35+27 = 62\).
• For "No": The number of children who did not play outside is \(69 + 19=88\).
• The total number of children is \(104 + 46=150\) (there was a miscalculation in the original table, the correct total is 150).

The correct two - way frequency table is:

	Own a Bike - Yes	Own a Bike - No	Total
Play Outside - No	69	19	88
Total	104	46	150

Part b: Two - way Relative Frequency Table

Step 1: Calculate relative frequencies for each cell

The relative frequency of a cell is calculated as \(\frac{\text{Frequency of the cell}}{\text{Total number of observations}}\times100\) (to get a percentage) or \(\frac{\text{Frequency of the cell}}{\text{Total number of observations}}\) (to get a proportion). We will use the total number of children \(n = 150\).

• For (Play Outside - Yes, Own a Bike - Yes): \(\frac{35}{150}\approx0.233\approx23\%\)
• For (Play Outside - Yes, Own a Bike - No): \(\frac{27}{150} = 0.18=18\%\)
• For (Play Outside - No, Own a Bike - Yes): \(\frac{69}{150}=0.46 = 46\%\)
• For (Play Outside - No, Own a Bike - No): \(\frac{19}{150}\approx0.127\approx13\%\)

The two - way relative frequency table (rounded to the nearest whole number as a percentage) is:

	Own a Bike - Yes	Own a Bike - No
Play Outside - No	46%	13%

Part c: Likelihood of playing outside with a bike

Step 1: Calculate the probability of playing outside given that a child owns a bike

The probability \(P(\text{Play Outside}|\text{Own a Bike})\) is calculated using the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In terms of frequencies, \(P(\text{Play Outside}|\text{Own a Bike})=\frac{\text{Number of children who own a bike and play outside}}{\text{Number of children who own a bike}}\)

We have the number of children who own a bike and play outside \(= 35\) and the number of children who own a bike \(= 104\). So \(P(\text{Play Outside}|\text{Own a Bike})=\frac{35}{104}\approx0.337\)

Step 2: Calculate the probability of playing outside given that a child does not own a bike

\(P(\text{Play Outside}|\text{Do not Own a Bike})=\frac{\text{Number of children who do not own a bike and play outside}}{\text{Number of children who do not own a bike}}\)

The number of children who do not own a bike and play outside \(= 27\) and the number of children who do not own a bike \(= 46\). So \(P(\text{Play Outside}|\text{Do not Own a Bike})=\frac{27}{46}\approx0.587\)

Since \(0.337<0.587\), a child who has a bike is less likely to play outside after school. Because the probability of playing outside given that a child owns a bike (\(\approx0.34\)) is less than the probability of playing outside given that a child does not own a bike (\(\approx0.59\)).

Final Answers

Part a

	Own a Bike - Yes	Own a Bike - No	Total
Play Outside - No	69	19	88
Total	104	46	150

Part b

	Own a Bike - Yes	Own a Bike - No
Play Outside - No	46%	13%

Part c

A child who has a bike is less likely to play outside after school. The probability of playing outside given a bike is \(\frac{35}{104}\approx0.34\) and given no bike is \(\frac{27}{46}\approx0.59\), and \(0.34<0.59\).