Final answer:
The test-statistic for the observed frequencies of Mongolian gerbils' colors, calculated using the chi-square test for goodness of fit, is 4.63.
Step-by-step explanation:
The student has been asked to calculate the test-statistic for observed frequencies of different color gerbils. The assumption is that gerbils are equally likely to be brown, white, or black. To calculate the test-statistic, we will use the chi-square test for goodness of fit, which compares the observed frequencies with the expected frequencies assuming there is an equal chance for each color.
First, let's calculate the total number of gerbils observed:
- Black: 40
- Brown: 59
- White: 42
Total gerbils = 40 + 59 + 42 = 141
If each color is equally likely, then each color should make up about one-third of the total. So, the expected frequency for each color is:
Expected frequency for each color = Total gerbils / 3 = 141 / 3 = 47
Now, we calculate the chi-square test-statistic using the formula:
Chi-square χ² = ∑((Observed - Expected)² / Expected)
For each color:
- Black: ((40 - 47)²) / 47
- Brown: ((59 - 47)²) / 47
- White: ((42 - 47)²) / 47
Chi-square χ² = (49 / 47) + (144 / 47) + (25 / 47)
Chi-square χ² = 1.04 + 3.06 + 0.53
Chi-square χ² = 4.63
The calculated test-statistic is 4.63 for the observed frequencies of the Mongolian gerbils' colors.