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.

Question
Two-way tables
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.

2020-12-04
Given:
$$\begin{array}{c|cc|c} &\text { Dry } & \text { Wet }& \text { Total }\\ \hline \text{ Cats } & 10&30 \\ \text{ Dogs} & 20&20\\ \hline \text{Total} \end{array}\$$
Let us first determine the cumulative frequencies which are obtained by adding the frequencies in the corresponding column or the corresponding row.
$$\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.

Relevant Questions

The two-way table shows the eye color of 200 cats participating in a cat show.
$$\begin{array}{c|ccc|c} &\text { Green } & \text { Blue }& \text { Yellow }& \text { Total }\\ \hline \text{ Male } & 40&24&16&80\\ \text{ Female} &30&60&30&120\\ \hline \text{Total}&70&84&46&200 \end{array}\$$
Make a two-way relative frequency table to show the distribution of the data with respect to gender. Round to the nearest tenth of a percent, as needed.
The following is a two-way table showing preferences for an award (A, B, C) by gender for the students sampled in survey. Test whether the data indicate there is some association between gender and preferred award.
$$\begin{array}{|c|c|c|}\hline &\text{A}&\text{B}&\text{C}&\text{Total}\\\hline \text{Female} &20&76&73&169\\ \hline \text{Male}&11&73&109&193 \\ \hline \text{Total}&31&149&182&360 \\ \hline \end{array}\\$$
Chi-square statistic=?
p-value=?
Conclusion: (reject or do not reject $$H_0$$)
Does the test indicate an association between gender and preferred award? (yes/no)
This two-way table shows the results of asking students if they prefer to have gym class in the morning or the afternoon A.How many students participate in the survey B. How many students in grade 8 prefer to have gym in the morning C. How many grade 10 students participated in the survey D. How many students prefer To have gym in the afternoon
$$\begin{array}{|c|c|c|}\hline&\text{morning}&\text{afternoon}&\text{total}\\\hline\text{grade 6} &15 & 8 & 23\\ \hline\text{grade 8}& 18 & 21&39\\\hline\text{grade 10}& 12 & 26&38\\ \hline \text{total}&45&55&100 \\ \hline \end{array}$$
Statistics students at a state college compiled the following two-way table from a sample of randomly selected students at their college:
$$\begin{array}{|c|c|c|}\hline&\text{Play chess}&\text{Don`t play chess}\\\hline \text{Male students} &25&162\\ \hline \text{Female students}&19&148 \\ \hline \end{array}\\$$
Answer the following questions about the table. Be sure to show any calculations.
What question about the population of students at the state college would this table attempt to answer?
State H^0 and H^1 for the test related to this table.
The following two-way contingency table gives the breakdown of the population of adults in a town according to their highest level of education and whether or not they regularly take vitamins:
$$\begin{array}{|c|c|c|c|c|} \hline \text {Education}& \text {Use of vitamins takes} &\text{Does not take}\\ \hline \text {No High School Diploma} & 0.03 & 0.07 \\ \hline \text{High School Diploma} & 0.11 & 0.39 \\ \hline \text {Undergraduate Degree} & 0.09 & 0.27 \\ \hline \text {Graduate Degree} & 0.02 & 0.02 \\ \hline \end{array$$
You select a person at random. What is the probability the person does not take vitamins regularly?
Gastric freezing was once a recommended treatment for ulcers in the upper intestine. One experiment compared 82 subjects randomly assigned to gastric freezing and 78 subjects randomly assigned to receive a placebo. The two-way table shows the results of the experiment. Is there convincing evidence of an association between treatment and outcome for subjects like these?
$$\begin{array}{c|cc|c} & \text { Gastric freezing } &\text{Placebo}& \text { Total } \\ \hline \text { Improved } & 28 & 30 & 58 \\\text { Didn't improve} & 54 & 48 & 102 \\ \hline \text { Total } & 82 & 78 & 160\\ \end{array}\$$
The Pew Research Center asked a random sample of 2024 adult cellphone owners from the United States their age and which type of cell phone they own: iPhone, Android, or other (including non-smartphones). The two-way table summarizes the data.
$$\begin{array}{c|ccc|c} & 18-34 & 35-54 & 55+ & \text { Total } \\ \hline \text { iPhone } & 169 & 171 & 127 & 467 \\ \text { Androod } & 214 & 189 & 100 & 503 \\ \text { Other } & 134 & 277 & 643 & 1054 \\ \hline \text { Total } & 517 & 637 & 870 & 2024 \end{array}$$
Suppose we select one of the survey respondents at random. What's the probability that: The person is not age 18 to 34 and does not own an iPhone?
A survey of 120 students about which sport , baseball , basketball , football ,hockey , or other , they prefer to watch on TV yielded the following two-way frequency table . What is the conditional relative frequency that a student prefers to watch baseball , given that the student is a girl? Round the answer to two decimal places as needed
$$\begin{array}{|c|c|c|}\hline &\text{Baseball}&\text{Basketball}&\text{Football}&\text{Hockey}&\text{Other}&\text{Total}\\\hline \text{Boys} &18&14&20&6&2&60\\ \hline \text{Girls}&14&16&13&5&12&60\\ \hline \text{Total}&32&30&33&11&14&120\\ \hline \end{array}\\$$
a) 11.67%
b) 23.33%
c) 43.75%
d) 53.33%
$$\begin{array}{|c|c|c|}\hline &\text{Less Than High School}&\text{High School}&\text{More Than High School}\\\hline \text{Better off} &140&440&430\\ \hline \text{Same as}&60&230&110\\ \hline \text{Worse off}&180&280&80\\ \hline\end{array}\\$$
$$P(\text{more than high school or worse off})=?$$
$$\begin{array}{ccc}\text{Original Counts}&\text{Exciting}&\text{Routine}&\text{Dull}&\text{Total}\\\hline \text{Male} &213&200&12&425\\ \text{Female}&221&305&29&555\\ \text{Female}&434&505&41&980 \end{array}$$