Associations in Categorical Data

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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
Standards: HSS-ID.B.5
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17 questions

Associations in Categorical Data

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Cake or Pie

The table displays the dessert preference and dominant hand (left- or right-handed) for a sample of 300 people.

prefers cakeprefers pietotalleft-handed102030right-handed90180270total100200300\begin{array}{|c|c|c|c|} \hline \\[-1em] & \textbf{prefers cake} & \textbf{prefers pie} & \textbf{total} \\[-1em] \\ \hline \\[-1em] \textbf{left-handed} & 10 & 20 & 30 \\[-1em] \\ \hline \\[-1em] \textbf{right-handed} & 90 & 180 & 270 \\[-1em] \\ \hline \\[-1em] \textbf{total} & 100 & 200 & 300 \\[-1em] \\ \hline \end{array}

For each of the calculations, describe the interpretation of the percentage in terms of the situation.

1) 10% from 10100=0.1\frac{10}{100} = 0.1

2) 67% from 1802700.67\frac{180}{270} \approx 0.67

3) 30% from 90300=0.3\frac{90}{300} = 0.3

Associations in Categorical Data


A template for answering this question. Ask your instructor for an alternative.

The two-way table displays data about 55 different locations. Scientists have a list of possible chemicals that may influence the health of the coral. They first look at how nitrate concentration might be related to coral health. The table displays the health of the coral (healthy or unhealthy) and nitrate concentration (low or high).

low nitrate concentrationhigh nitrate concentrationtotalhealthy20525unhealthy82230total282755\begin{array}{|l|c|c|c|} \hline \\[-1em] & \textbf{low nitrate concentration} & \textbf{high nitrate concentration} & \textbf{total}\\[-1em] \\ \hline \\[-1em] \textbf{healthy} & 20 & 5 & 25\\[-1em] \\ \hline \\[-1em] \textbf{unhealthy} & 8 & 22 & 30\\[-1em] \\ \hline \\[-1em] \textbf{total} & 28 & 27 & 55\\[-1em] \\ \hline \end{array}

Complete the two-way relative frequency table for the data in the two-way table in which the relative frequencies are based on the total for each column. Find the values that belong in the labeled cells in the table, rounding to the nearest whole percent.

low nitrate concentrationhigh nitrate concentrationhealthyABunhealthyCDtotal100%100%\begin{array}{|c|c|c|} \hline \\[-1em] & \textbf{low nitrate concentration} & \textbf{high nitrate concentration} \\[-1em] \\ \hline \\[-1em] \textbf{healthy} & \text{A} & \text{B} \\[-1em] \\ \hline \\[-1em] \textbf{unhealthy} & \text{C} & \text{D} \\[-1em] \\ \hline \\[-1em] \textbf{total} & 100\% & 100\% \\[-1em] \\ \hline \end{array}

4) Cell A

5) Cell B

6) Cell C

7) Cell D

8) When there is a low nitrate concentration, which had a higher relative frequency, healthy or unhealthy coral?

a)

Healthy coral

b)

Unhealthy coral

9) When there is a high nitrate concentration, is there a higher relative frequency of healthy or unhealthy coral?

a)

Healthy coral

b)

Unhealthy coral

10) Based on this data, is there a possible association\textbf{association} between coral health and the level of nitrate concentration?

True or false? Write below.

11) Explain your reasoning.

The scientists next look at how silicon dioxide concentration might be related to coral health. The relative frequencies based on the total for each column are shown in the table.

low silicon dioxide concentrationhigh silicon dioxide concentrationhealthy44%46%unhealthy56%54%total100%100%\begin{array}{|l|c|c|} \hline \\[-1em] & \textbf{low silicon dioxide concentration} & \textbf{high silicon dioxide concentration} \\[-1em] \\ \hline \\[-1em] \textbf{healthy} & 44\% & 46\% \\[-1em] \\ \hline \\[-1em] \textbf{unhealthy} & 56\% & 54\% \\[-1em] \\ \hline \\[-1em] \textbf{total} & 100\% & 100\% \\[-1em] \\ \hline \end{array}

12) Based on this data, is there a possible association between coral health and the level of silicon dioxide concentration?

True or false? Write below.

13) Explain your reasoning.

Jada surveyed 300 people from various age groups about their shoe preference. The two-way table summarizes the results of the survey.

prefers sneakersprefers sneakersprefers shoes thattotalwithout laceswith lacesare not sneakers4-10 years old211233611-17 years old21483910818-24 years old155487156total57114129300\begin{array}{|l|c|c|c|c|} \hline \\[-1em] & \textbf{prefers sneakers} & \textbf{prefers sneakers} & \textbf{prefers shoes that} & \textbf{total} \\ & \textbf{without laces} & \textbf{with laces} & \textbf{are not sneakers} & \\[-1em] \\ \hline \\[-1em] \textbf{4-10 years old} & 21 & 12 & 3 & 36 \\[-1em] \\ \hline \\[-1em] \textbf{11-17 years old} & 21 & 48 & 39 & 108\\[-1em] \\ \hline \\[-1em] \textbf{18-24 years old} & 15 & 54 & 87 & 156\\[-1em] \\ \hline \\[-1em] \textbf{total} & 57 & 114 & 129 & 300\\[-1em] \\ \hline \end{array}

Jada concludes that there is a possible association between age and shoe preference.

14) Is Jada’s conclusion reasonable?

True or false? Write below.

15) Explain your reasoning.

The two-way table summarizes data on writing utensil preference and the dominant hand for a sample of 100 people.

left-handedright-handedtotalprefers pen78289prefers pencil6511total1387100\begin{array}{|l|c|c|c|} \hline \\[-1em] & \textbf{left-handed} & \textbf{right-handed} & \textbf{total}\\[-1em] \\ \hline \\[-1em] \textbf{prefers pen} & 7 & 82 & 89\\[-1em] \\ \hline \\[-1em] \textbf{prefers pencil} & 6 & 5 & 11\\[-1em] \\ \hline \\[-1em] \textbf{total} & 13 & 87 & 100\\[-1em] \\ \hline \end{array}

16) Is there a possible association between dominant hand and writing utensil preference?

True or false? Write below.

17) Explain your reasoning.