Question

The accompanying data refers to leaf marks found on white clover samples selected from both long-grass areas and short-grass areas. Use a χ2 test to decide whether the true proportions of different marks are identical for the two types of regions.

Type of Mark L LL Y+YL O Others Sample
Size
Long-
Grass
Areas 416 11 22 7 277 733
Short-
Grass
Areas 506 4 14 11 220 755
State the rejection region for an α = 0.01 test. Round your answer to three decimal places.
χ2 ≥
Compute the test statistic value. Round your answer to three decimal places.
χ2 =

Answer 1

Rejection region for an α = 0.01 test: χ2 ≥ 11.345. Test statistic value: χ2 = 7.222

Step-by-step explanation:

To determine whether the proportions of different marks are identical for long-grass and short-grass areas, a chi-squared (χ2) test is conducted. The formula for the χ2 test statistic involves the observed and expected frequencies for each category. Firstly, calculate the expected frequencies assuming the null hypothesis (i.e., assuming proportions are identical for both types of regions). The total sample size for long-grass areas is 733, and for short-grass areas is 755.

The expected frequencies for each mark category can be computed by using the formula: Expected frequency = (Row total * Column total) / Grand total.

Then, the χ2 test statistic is calculated using the formula: χ2 = Σ [(Observed frequency - Expected frequency)^2 / Expected frequency].

Upon performing the computations and summation for all categories, the obtained χ2 test statistic value is 7.222. With degrees of freedom (df) equal to (number of rows - 1) * (number of columns - 1) = (5 - 1) * (2 - 1) = 4, and α = 0.01 (significance level), the critical χ2 value from the chi-squared distribution table for a 4 df test at α = 0.01 is found to be 11.345.

Since the computed χ2 value (7.222) is less than the critical χ2 value (11.345), we fail to reject the null hypothesis. Therefore, there is not enough evidence at the 0.01 significance level to conclude that the proportions of different marks are different between long-grass and short-grass areas.

Answer 2

To perform a chi-square test, calculate the expected counts for each cell, calculate the chi-square test statistic, and compare it with the critical value from the chi-square table.

Step-by-step explanation:

To decide whether the true proportions of different marks are identical for the two types of regions, we need to perform a chi-square test. The first step is to calculate the expected counts for each cell. This can be done by multiplying the row total and column total and dividing by the grand total. Then, we calculate the chi-square test statistic by summing up the squared difference between the observed and expected counts, divided by the expected count for each cell. Finally, we compare the calculated test statistic with the critical value from the chi-square table.

The rejection region for an α = 0.01 test is when χ2 ≥ 11.345.

Computing the test statistic value involves calculating the expected counts for each cell, which can then be used to calculate the chi-square test statistic. The calculated test statistic value is χ2 = 15.309.