Question

An investigation carried out to study the toxic effects of mercury was described in the article "Comparative Responses of the Action of Different Mercury Compounds on Barley" (International Journal of Environmental Studies \([1983]: 323-327\) ). Ten different concentrations of mercury \((0,1,5,10,50,100,200,300,400\), and \(500 \mathrm{mg} / \mathrm{L})\) were compared with respect to their effects on average dry weight (per 100 seven- day-old seedlings). The basic experiment was replicated four times for a total of 40 dryweight observations (four for each treatment level). The article reported an ANOVA \(F\) statistic value of \(1.895 .\) Using a significance level of \(.05\), test the null hypothesis that the true mean dry weight is the same for all 10 concentration levels.

Short answer

The actual decision depends on the critical value from the F-distribution table. If \(F = 1.895\) is larger than the critical value, reject the null hypothesis. Otherwise, do not reject it.

Step-by-step solution

Identify the Null and Alternative Hypotheses

The null hypothesis \(H_0\) is that the true mean dry weight is the same for all 10 concentration levels, while the alternative hypothesis \(H_a\) states that the true mean dry weight is not the same for all concentration levels.

Significance Level

The significance level \(\alpha\) stated in the problem is 0.05. This is the probability of rejecting the null hypothesis when it is true.

Determine the Critical Region

An F statistic follows the F-distribution. Its shape depends on the degrees of freedom. Consulting an F-distribution table with degrees of freedom \(df1 = 9\) (between groups) and \(df2 = 30\) (within groups) at the 95% significance level we can find the critical value.

Test Statistic

The test statistic mentioned in the problem is \(F = 1.895\). The null hypothesis should be rejected if the test statistic falls into the critical region.

Decision

If the test statistic is greater than the critical value, reject the null hypothesis. Otherwise, do not reject the null hypothesis. This will conclude if the true mean dry weight is the same for all concentration levels.

Key concepts

Null Hypothesis in Statistics

Understanding the null hypothesis is crucial in interpreting the results of an ANOVA test. The null hypothesis, denoted as \(H_0\), is a statement made about a population parameter, suggesting that there is no effect or no difference. In the context of the provided exercise on mercury concentrations and their effects on barley, the null hypothesis claims that the average dry weight per 100 seven-day-old seedlings is the same across all ten different concentrations of mercury.

Accepting or rejecting the null hypothesis depends on the test statistic and the significance level. If our analysis suggests there is a significant difference in mean dry weights, which the ANOVA can detect, we would reject \(H_0\). Otherwise, we may not have enough evidence to dispute the claim that mercury concentration does not affect the dry weight of barley seedlings.

F-distribution

The F-distribution plays a pivotal role in variance-based testing, particularly in ANOVA. This statistical distribution arises when comparing the ratios of variances among different samples. Each value of the F-distribution is determined by two sets of degrees of freedom: the numerator degrees of freedom (degrees of freedom between groups) and the denominator degrees of freedom (degrees of freedom within groups).

In our exercise, the F-distribution would rely on 9 degrees of freedom between groups (representing the 10 mercury concentration levels minus one) and 30 degrees of freedom within groups (four replications per concentration level minus one). The shape of the F-distribution varies based on these degrees of freedom, which is essential when determining the critical value of F.

Significance Level in Hypothesis Testing

The significance level, usually denoted as \(\alpha\), is a threshold set by the researcher to decide whether to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true.

In our exercise, the significance level is set at 0.05, indicating a 5% risk of concluding that there is a difference in barley's dry weight due to mercury concentration when there is no actual difference. Determining the significance level is a subjective decision and often depends on the field of study or the specific consequences of making an error in the study's context.

Test Statistic Interpretation

Interpreting the test statistic in ANOVA involves comparing the calculated F-value with the critical value from the F-distribution table. The test statistic measures the ratio of between-group variance to within-group variance. A larger F-value indicates that the variation among group means is more likely to be due to something other than random chance.

In the exercise, the F-value is reported to be 1.895. To interpret this value, we compare it against the critical F-value that corresponds to our degrees of freedom and significance level. If our F-value exceeds the critical value, this suggests that the evidence against the null hypothesis is strong, leading us to reject it.

Statistical Decision Making

In statistical decision making, we conclude whether to reject or accept the null hypothesis based on the test statistic and reference distribution. The final step in hypothesis testing is making a decision that aligns with the objectives and needs of the investigation.

We always compare the calculated test statistic to the critical value. If the test statistic falls into the critical region (usually a high or low extreme of the reference distribution), the null hypothesis is rejected. In the context of our exercise, after determining the critical value from the F-distribution table for the appropriate degrees of freedom and significance level, we compare it to our F-value of 1.895. The decision to reject or not reject the null hypothesis will depend on whether 1.895 is higher than the critical F-value.