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Question
Math
Posted 2 months ago

A zoo has separate habitats for its three types of bears: brown, black, and grizzly bears. Each habitat offers three areas for the bears to spend their time: an outdoor area, an indoor area, and a pool.

Gisele is the zookeeper who cares for the brown bear. She takes a random sample of 10 moments each day for a week and records which area the brown bear is at each of those moments. Bryan and Adele use a similar strategy and obtain separate samples of 70 observations for the black and grizzly bears, respectively. Here are their results:
\begin{tabular}{lrrrr} 
Bear & Outdoor & Indoor & Pool & Total \\
\hline Brown & 23 & 27 & 20 & 70 \\
Black & 15 & 18 & 37 & 70 \\
Grizzly & 32 & 12 & 26 & 70 \\
Total & 70 & 57 & 83 & 210
\end{tabular}

They want to perform a χ2\chi^{2} test for homogeneity to judge if the bears are spending their time differently in their habitats.

What is the expected count for the cell corresponding to the grizzly bear being by the pool?
You may round your answer to the nearest hundredth.
Expected:
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 1
Calculate the expected count for the grizzly bears by the pool using the formula for expected frequency in a chi-square test: Eij=(row total)×(column total)grand totalE_{ij} = \frac{(row\ total) \times (column\ total)}{grand\ total}
step 2
Identify the row total for grizzly bears and the column total for the pool: row total=70row\ total = 70, column total=83column\ total = 83, grand total=210grand\ total = 210
step 3
Substitute the identified totals into the formula: Egrizzly,pool=(70)×(83)210E_{grizzly,pool} = \frac{(70) \times (83)}{210}
step 4
Perform the calculation: Egrizzly,pool=(70)×(83)210=5810210E_{grizzly,pool} = \frac{(70) \times (83)}{210} = \frac{5810}{210}
step 5
Simplify the fraction to find the expected count: Egrizzly,pool=27.666666666666666666666666666666666666666666666666666666666666666...E_{grizzly,pool} = 27.666666666666666666666666666666666666666666666666666666666666666...
step 6
Round the expected count to the nearest hundredth: Egrizzly,pool27.67E_{grizzly,pool} \approx 27.67
Answer
The expected count for the cell corresponding to grizzly bears being by the pool is approximately 27.67.
Key Concept
Expected Frequency in a Chi-Square Test
Explanation
The expected frequency for a cell in a chi-square test is calculated by multiplying the row total by the column total and dividing by the grand total. This provides the expected count if there is no difference in the distribution across the categories.

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