Question

A study was conducted by the Department of Zoology at Virginia Tech to determine if there is a significant difference in the density of organisms at two different stations located on Cedar Run, a secondary stream in the Roanoke River drainage basin. Sewage from a sewage treatment plant and overflow from the Federal Mogul Corporation settling pond enter the stream near its headwaters. The following data give the density measurements, in number of organisms per square meter, at the two collecting stations: test the hypothesis at the 0.05 level of significance that sigma^2_1 = sigma^2_2 against the alternative that sigma^2_1 notequalto sigma^2_2, where sigma^2_1 and sigma^2_2 are the variances of the number of organisms per square meter of water at the two different locations on Cedar Run.

Answer 1

The hypothesis test at the 0.05 level of significance suggests that there is a significant difference in the variances
`(\(\sigma^2_1\) and \(\sigma^2_2\))` of the number of organisms per square meter at the two different locations on Cedar Run.

Step-by-step explanation:

To test the hypothesis regarding the difference in variances, we can use an F-test. The null hypothesis
`(\(H_0\))` assumes that the variances are equal
`(\(\sigma^2_1 = \sigma^2_2\)),` and the alternative hypothesis
`(\(H_1\))` suggests that the variances are not equal
`(\(\sigma^2_1 \\eq \sigma^2_2\))`.

The F-statistic is calculated as the ratio of the sample variances, and its distribution follows an F-distribution under the assumption that the variances are equal. If the calculated F-statistic is extreme enough, we reject the null hypothesis in favor of the alternative hypothesis.

In the context of the study, rejecting the null hypothesis indicates that there is a significant difference in the variances of organism density between the two collecting stations on Cedar Run. This implies that the variability in organism density is not the same at the two locations, providing valuable information for understanding the ecological conditions of the stream.

Answer 2

The question involves testing the hypothesis of equal variances between the density measurements at two different stations on Cedar Run. The significance level is 0.05.

Step-by-step explanation:

This question is related to statistical hypothesis testing comparing the variances of two populations. The hypothesis being tested is whether there is a significant difference in the density of organisms at two different collecting stations on Cedar Run. The null hypothesis is that the variances of the two populations are equal, while the alternative hypothesis is that the variances are not equal. The significance level is 0.05.

To test this hypothesis, you would need to use a statistical test such as F-test or Bartlett's test for equal variances. These tests compare the sample variances of the two populations and determine if the difference is statistically significant. If the test result is below the significance level (0.05 in this case), you would reject the null hypothesis and conclude that there is a significant difference in the variances of the two populations.