Use the following information to find the
𝜒
2
test statistic based upon the observed count, and expected probabilities.

Category Cat Dog Rhino Other
Observed Counts 7 82 4 57
Expected Probabilities ? 0.03 0.19 0.23

Question

Use the following information to find the 
𝜒
2
 test statistic based upon the observed count, and expected probabilities.

Category	Cat	Dog	Rhino	Other		
Observed Counts	7	82	4	57		
Expected Probabilities	?	0.03	0.19	0.23

BlackTom AI · Accepted Answer

First, I restate the information given and identify what is missing. The problem asks for the chi-squared test statistic based on the observed counts and the expected probabilities. The table shows four categories: Cat, Dog, Rhino, Other. Observed counts are: Cat = 7, Dog = 82, Rhino = 4, Other = 57. The provided expected probabilities are: Cat = ? (missing), Dog = 0.03, Rhino = 0.19, Other = 0.23.

To compute the chi-square statistic, we need all expected counts E_i, where E_i = N × p_i, with N being the total number of observations and p_i the expected probability for category i. So we must determine N and p_i for Cat (p_Cat).

Step 1: Compute total observed count N.
- N = 7 + 82 + 4 + 57 = 150.

Step 2: Determine the missing Cat probability. Since the expected probabilities for all categories should sum to 1, we can infer Cat’s probability as:
- p_Cat = 1 − (p_Dog + p_Rhino + p_Other) = 1 − (0.03 + 0.19 + 0.23) = 1 − 0.45 = 0.55.

Step 3: Compute all expected counts E_i using E_i = N × p_i.
- E_Cat = 150 × 0.55 = 82.5
- E_Dog = 150 × 0.03 = 4.5
- E_Rhino = 150 × 0.19 = 28.5
- E_Other = 150 × 0.23 = 34.5

Step 4: Apply the chi-squared formula χ² = Σ (O_i − E_i)² / E_i across all categories.
- For Cat: (7 − 82.5)² / 82.5
- For Dog: (82 − 4.5)² / 4.5
- For Rhino: (4 − 28.5)² / 28.5
- For Other: (57 − 34.5)² / 34.5

Step 5: Compute each term (numerical approximations):
- Cat term ≈ (−75.5)² / 82.5 ≈ 5700.25 / 82.5 ≈ 69.09
- Dog term ≈ (77.5)² / 4.5 ≈ 6006.25 / 4.5 ≈ 1334.72
- Rhino term ≈ (−24.5)² / 28.5 ≈ 600.25 / 28.5 ≈ 21.09
- Other term ≈ (22.5)² / 34.5 ≈ 506.25 / 34.5 ≈ 14.66

Step 6: Sum the terms to get χ²:
- χ² ≈ 69.09 + 1334.72 + 21.09 + 14.66 ≈ 1439.56

Notes and caveats:
- The calculation relies on the assumption that the expected probabilities should sum to 1, which allows us to deduce Cat’s probability as 0.55. If there were any additional constraints or a different total probability distribution, the Cat probability could differ and change the result.
- In typical chi-squared tests, you would also want to ensure that all expected counts are sufficiently large (commonly at least 5) for the approximation to be valid. Here, E_Dog = 4.5 is just below that threshold, which might prompt a note about the test’s reliability in this particular dataset.
- If the instruction or context forbids inferring the missing probability and requires explicit given values only, then the problem would be unsolvable without that missing Cat probability, which aligns with the idea that missing Cat probability prevents a definite χ² calculation under those rules.

BTRY3010/BTRY5010/STSCI2200/STSCI5200 Lecture #26 Pages ( 126 - 129, 134 )

Use the following information to find the 𝜒 2 test statistic based upon the observed count, and expected probabilities. Category Cat Dog Rhino Other Observed Counts 7 82 4 57 Expected Probabilities ? 0.03 0.19 0.23

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