Question

Fish Ventilation Most fish use gills for respiration in water and researchers can observe how fast a fish’s gill cover beats to study ventilation, much like we might observe breathing rate for a person. Professor Brad Baldwin is interested in how water chemistry might affect gill beat rates. In one experiment he randomly assigned fish to tanks with different levels of calcium. One tank was low in calcium (0.71 mg/L), the second tank had a medium amount (5.24 mg/L), and the third tank had water with a high calcium level (18.24 mg/L). His research team counted gill rates (beats per minute) for samples of 30 fish in each tank. The results9 are stored in FishGills3. (Note: You may also see a file called FishGills12 which is a more extensive experiment with 12 tanks.) Use technology to check that the conditions for an ANOVA model are reasonable for these data. Include a plot that compares the gill rates for the three calcium conditions.If the conditions are met, use technology to find the ANOVA table and complete the test. If the conditions are not reasonable, use a randomization test (scrambling the calcium levels) to complete the test.

Answer 1

Answer: The ‘FishGills3’ dataset fulfils the assumptions of analysis of variance (ANOVA). The data are normally distributed (Figure 1 and Figure 3) and the sample sizes are more than 10. One-way ANOVA was used as there was a single independent variable.

Figure 1 Histograms of gill rate data at a different level (low, medium, and high) of calcium concentration treatment.

Figure 2 Boxplot showing the difference in the fish gill rate in different calcium treatment.

The computed ANOVA results are shown in the below Table (1) of the ‘FishGills3’ dataset.

Table 1 Summary of ANOVA results for ‘FishGills3’ dataset.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
FishGills3$Calcium	2	2037	1018.6	4.648	0.0121
Residual	87	19064	219.1

As per the ANOVA results, there is a significant difference (p < 0.05) in the fish gill rates in different calcium concentrations.

Figure 3 Histogram showing the gill rate data distribution in different calcium concentration (mg/L).

Figure 4 Scatterplot showing the relationship between the calcium concentration and the fish gill rates.

As per the correlation analysis (Figure 4), the fish gill rate is significantly (p < 0.05) correlated to the calcium concentration. Increase in calcium concentration decreases the gill rate in fish.