The two-way table shows the results from a survey of dog and cat owners about whether their pet prefers dry food or wet food. begin{array}{c|c|c} &text { Dry } & text { Wet } hline text{ Cats } & 10&30 hline text{ Dogs} & 20&20 end{array} Does the two-way table show any difference in preferences between dogs and cats? Explain.

Two-way tables
asked 2020-12-03
The two-way table shows the results from a survey of dog and cat owners about whether their pet prefers dry food or wet food.
\(\begin{array}{c|c|c} &\text { Dry } & \text { Wet }\\ \hline \text{ Cats } & 10&30 \\ \hline \text{ Dogs} & 20&20\\ \end{array}\\)
Does the two-way table show any difference in preferences between dogs and cats? Explain.

Answers (1)

\(\begin{array}{c|cc|c} &\text { Dry } & \text { Wet }& \text { Total }\\ \hline \text{ Cats } & 10&30 \\ \text{ Dogs} & 20&20\\ \hline \text{Total} \end{array}\\)
Let us first determine the cumulative frequencies which are obtained by adding the frequencies in the corresponding column or the corresponding row.
\(\begin{array}{c|cc|c} &\text { Dry } & \text { Wet }& \text { Total }\\ \hline \text{ Cats } & 10&30&10+30=40 \\ \text{ Dogs} & 20&20&20+20=40\\ \hline \text{Total}&10+20=30&30+20=50&40+40=80 \end{array}\\)
Next, we determine the percentages by dividing the frequency by the row total for the cells not in the last column, while we divide the frequency by the table total for the cells in the last column.
\(\begin{array}{c|cc|c} &\text { Dry } & \text { Wet }& \text { Total }\\ \hline \text{ Cats } & 10(\frac{10}{40}=0.25=25\% )&30(\frac{30}{40}=0.75=75\% )&40(\frac{40}{80}=0.5=50\% ) \\ \text{ Dogs} & 20(\frac{20}{40}=0.5=50\% )&20(\frac{20}{40}=0.5=50\% )&40(\frac{40}{80}=0.5=50\% )\\ \hline \text{Total}&30(\frac{30}{80}=0.375=35.5\% )&50(\frac{50}{80}=0.625=62.5\% )&80(\frac{80}{80}=1=100\% ) \end{array}\\)
We note that 75% of the cats prefer the wet food, while 50% of the dogs prefer the wet food. This then implies that there appears to be a different, in the preferences of dogs and cats as the percentages 75% and 50% differ a lot.
Answer: YES

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