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 6 65 2 38
Expected Probabilities ? 0.24 0.03 0.11

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	6	65	2	38		
Expected Probabilities	?	0.24	0.03	0.11

BlackTom AI · Accepted Answer

The question asks us to compute the chi-squared statistic using the observed counts and the expected probabilities for each category.

First, restate the data we have: the observed counts are Cat = 6, Dog = 65, Rhino = 2, and Other = 38. The given expected probabilities are Dog = 0.24, Rhino = 0.03, and Other = 0.11. Since the probabilities must sum to 1, we can determine the Cat probability as 1 − (0.24 + 0.03 + 0.11) = 0.62.

Next, compute the total sample size N by summing the observed counts: N = 6 + 65 + 2 + 38 = 111.

Then, calculate the expected counts E for each category by multiplying N by the respective probabilities:
- Cat: E_cat = 111 × 0.62 = 68.82
- Dog: E_dog = 111 × 0.24 = 26.64
- Rhino: E_rhino = 111 × 0.03 = 3.33
- Other: E_other = 111 × 0.11 = 12.21

Now evaluate the chi-squared components for each category using (O − E)² / E, where O is the observed count and E is the expected count:
- Cat: (6 − 68.82)² / 68.82 ≈ (−62.82)² / 68.82 ≈ 3946 / 68.82 ≈ 57.3
- Dog: (65 − 26.64)² / 26.64 ≈ (38.36)² / 26.64 ≈ 1471 / 26.64 ≈ 55.2
- Rhino: (2 − 3.33)² / 3.33 ≈ (−1.33)² / 3.33 ≈ 1.77 / 3.33 ≈ 0.53
- Other: (38 − 12.21)² / 12.21 ≈ (25.79)² / 12.21 ≈ 665 / 12.21 ≈ 54.6

Finally, sum the four components to obtain the chi-squared statistic:
Chi-square ≈ 57.3 + 55.2 + 0.53 + 54.6 ≈ 167.6.

If the provided answer is 77.82, note that small differences can arise from rounding intermediate results or from slightly different handling of category definitions or continuity corrections. However, with the data as given and the standard (O − E)² / E calculation, the computed chi-square statistic is approximately in the 167 range, not 77.82. If a teacher or software has produced 77.82, it would be worth double-checking the category probabilities, totals, or any applied adjustments during the calculation to align with that specific result.

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 6 65 2 38 Expected Probabilities ? 0.24 0.03 0.11

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