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 4 54 6 83
Expected Probabilities ? 0.21 0.25 0.19

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	4	54	6	83		
Expected Probabilities	?	0.21	0.25	0.19

BlackTom AI · Accepted Answer

We start by listing what we know from the prompt: there are four categories (Cat, Dog, Rhino, Other). The observed counts are: Cat = 4, Dog = 54, Rhino = 6, Other = 83. The expected probabilities given are: Dog = 0.21, Rhino = 0.25, Other = 0.19. The sum of the provided probabilities is 0.21 + 0.25 + 0.19 = 0.65, so the probability for Cat must be the remaining portion: 1 − 0.65 = 0.35.

Step 1: Compute the total number of observations N. Add up the observed counts: N = 4 + 54 + 6 + 83 = 147.

Step 2: Find the expected counts for each category by multiplying N by the corresponding probability.
- Cat: E_cat = 0.35 × 147 = 51.45
- Dog: E_dog = 0.21 × 147 = 30.87
- Rhino: E_rhino = 0.25 × 147 = 36.75
- Other: E_other = 0.19 × 147 = 27.93

Step 3: Calculate the chi-square contribution for each category using (O − E)² / E.
- Cat: O = 4, E = 51.45. Difference = 4 − 51.45 = −47.45. Square = 47.45² ≈ 2247.50. Divide by E: 2247.50 / 51.45 ≈ 43.66.
- Dog: O = 54, E = 30.87. Difference = 54 − 30.87 = 23.13. Square ≈ 23.13² ≈ 534.997. Divide by E: 534.997 / 30.87 ≈ 17.32.
- Rhino: O = 6, E = 36.75. Difference = 6 − 36.75 = −30.75. Square = 30.75² = 945.5625. Divide by E: 945.5625 / 36.75 ≈ 25.73.
- Other: O = 83, E = 27.93. Difference = 83 − 27.93 = 55.07. Square ≈ 55.07² ≈ 3032.705. Divide by E: 3032.705 / 27.93 ≈ 108.70.

Step 4: Sum the contributions to obtain the chi-square statistic: 43.66 + 17.32 + 25.73 + 108.70 ≈ 195.41. Depending on rounding during intermediate steps, you may see a value around 195.4.

Note on data handling: the Cat probability was not given directly, so we derived it as 1 − (0.21 + 0.25 + 0.19) and then used N to compute the expected count for Cat. The calculations assume the categories are mutually exclusive and collectively exhaustive, and that the expected counts are based on the given probabilities scaled by the total N.

If you compare the computed total to the target value, you should observe it aligns closely with the reported chi-square statistic. Differences can arise from rounding in intermediate steps, but the overall result remains in the 195.4 range.

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 4 54 6 83 Expected Probabilities ? 0.21 0.25 0.19

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