Question

An experiment published in The American Biology Teacher studied the efficacy of using \(95 \%\) ethanol or \(20 \%\) bleach as a disinfectant in removing bacterial and fungal contamination when culturing plant tissues. The experiment was repeated 15 times with each disinfectant, using eggplant as the plant tissue being cultured. \({ }^{8}\) Five cuttings per plant were placed on a petri dish for each disinfectant and stored at \(25^{\circ} \mathrm{C}\) for 4 weeks. The observation reported was the number of uncontaminated eggplant cuttings after the 4 -week storage. $$ \begin{array}{lcl} \text { Disinfectant } & 95 \% \text { Ethanol } & 20 \% \text { Bleach } \\ \hline \text { Mean } & 3.73 & 4.80 \\ \text { Variance } & 2.78095 & .17143 \\ n & 15 & 15 \end{array} $$ a. Are you willing to assume that the underlying variances are equal? b. Using the information from part a, are you willing to conclude that there is a significant difference in the mean numbers of uncontaminated eggplants for the two disinfectants tested?

Short answer

Answer: No, we cannot assume that the variances of the two disinfectants are equal, since the p-value for the F-test is less than 0.05. There is a significant difference between the mean numbers of uncontaminated eggplants for the two disinfectants, as the p-value for the two-sample t-test with unequal variances is also less than 0.05.

Step-by-step solution

State the hypotheses.

The null hypothesis \(H_0\) is that the two variances are equal, while the alternative hypothesis \(H_1\) is that the variances are not equal. Mathematically, we can express this as: $$ H_0: \sigma_1^2 = \sigma_2^2 \\ H_1: \sigma_1^2 \neq \sigma_2^2 $$

Calculate the F statistic.

We need to calculate the F statistic using the variances of the two samples. The F statistic is the ratio of the larger variance to the smaller variance: $$ F = \frac{s_1^2}{s_2^2} = \frac{2.78095}{0.17143} \approx 16.223 $$

Determine the degrees of freedom for both samples.

Since we have 15 observations in each sample, the degrees of freedom for each sample are \(n_1 - 1 = 14\) and \(n_2 - 1 = 14\).

Find the p-value.

Using an F-distribution table or calculator, we find the p-value associated with the F statistic of 16.223 and degrees of freedom (14, 14) to be less than 0.001.

Draw a conclusion.

Since the p-value is less than 0.05 (our chosen significance level), we reject the null hypothesis. Therefore, we conclude that the variances are not equal. #b. Test for Significance in Mean Differences#

State the hypotheses.

The null hypothesis, \(H_0\), is that there is no significant difference between the means. The alternative hypothesis, \(H_1\), is that there is a difference. Mathematically, we can express this as: $$ H_0: \mu_1 = \mu_2 \\ H_1: \mu_1 \neq \mu_2 $$

Calculate the t statistic.

Since the variances are unequal, we will use the formula for the t statistic with unequal variances: $$ t = \frac{(\bar{x}_1-\bar{x}_2)-0}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}} = \frac{(3.73-4.80)}{\sqrt{\frac{2.78095}{15}+\frac{0.17143}{15}}} \approx -5.407 $$

Calculate the degrees of freedom.

Use Welch's approximation to approximate the degrees of freedom: $$ v \approx \frac{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2})^2}{\frac{(s_1^2/n_1)^2}{n_1 - 1} + \frac{(s_2^2/n_2)^2}{n_2 - 1}} \approx 17.56 $$

Find the p-value.

Using a t-distribution table or calculator, find the p-value associated with the t statistic of -5.407 and the approximated degrees of freedom of 17.56. The p-value is less than 0.001.

Draw a conclusion.

Since the p-value is less than 0.05 (our chosen significance level), we reject the null hypothesis. This implies that there is a significant difference between the mean numbers of uncontaminated eggplants for the two disinfectants tested.

Key concepts

F-test

The F-test is a statistical test used to compare two variances and determine if they are significantly different from each other. This test is particularly useful when comparing datasets to understand variability among them, such as when assessing the effectiveness of two different disinfectants, like ethanol and bleach in the given experiment.

To perform an F-test, you calculate the F statistic, which is the ratio of the larger variance to the smaller variance. In this example, the variances for ethanol and bleach are 2.78095 and 0.17143, respectively. The F statistic is calculated as follows: \( F = \frac{2.78095}{0.17143} \approx 16.223 \). A high F statistic, like 16.223, indicates a large discrepancy between the variances.

Once the F statistic is obtained, it is compared against a critical value from the F-distribution table at a set level of significance, usually 0.05. If the F statistic exceeds this critical value, it suggests that the variances are significantly different. In this case, the p-value is found to be less than 0.001, leading to the conclusion that the variances are not equal.

t-test

A t-test is a statistical test used to evaluate if there is a significant difference between the means of two groups. In the context of the experiment discussed, the t-test helps assess whether ethanol or bleach is more effective as a disinfectant.

There are different types of t-tests, including those for samples with equal variances and those for samples with unequal variances. Since our F-test indicated that the variances are unequal, we use the formula for the t statistic for unequal variances, also known as Welch's t-test. The formula is given by:

\[ t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]

With substituting our means and variances, this becomes \( t \approx -5.407 \). A t statistic is then compared to a critical value from the t-distribution to infer significance. Here, a negative t-value indicates that the mean for ethanol is lower than that for bleach.

p-value

A p-value is a measure of the evidence against a null hypothesis. Lower p-values indicate stronger evidence against the null hypothesis. In hypothesis testing, a p-value is compared to a significance level (commonly 0.05) to decide whether to reject the null hypothesis.

In this experiment, both the F-test and t-test resulted in p-values less than 0.001, which is much lower than the typical significance level of 0.05. This strong evidence suggests rejecting the null hypothesis in both cases. For the F-test, it means the variances are not equal, and for the t-test, it indicates a significant difference in the means of uncontaminated eggplants between the two disinfectants.

Variance

Variance is a statistical measure that represents the dispersion or spread within a dataset. It quantifies how far each data point is from the mean. High variance means data points are very spread out, while low variance indicates they are close to the mean.

In the given experiment, ethanol and bleach have variances of 2.78095 and 0.17143, respectively. The large variance for ethanol indicates that the number of uncontaminated eggplant cuttings is more varied compared to bleach.

Understanding variance is crucial when performing tests like the F-test, which directly compares the variances of two samples. This helps in understanding the reliability and consistency of results between different treatments or conditions.

Degrees of freedom

Degrees of freedom are values that specify the number of independent values in a calculation. They are crucial in the context of statistical tests, like the F-test and t-test, as they influence the shape of the corresponding distribution and assist in determining critical values.

For the F-test in our problem, each sample has degrees of freedom of 14, calculated as \( n - 1 \) where \( n \) is the number of observations, equating to 15 for both ethanol and bleach. Hence, the degrees of freedom are (14, 14) for our F statistic.

When computing the t-test with unequal variances, degrees of freedom are approximated using Welch's approximation. This complex formula approximates our degrees of freedom to around 17.56 in this case, allowing us to reference the appropriate statistical table to find the critical t-value and p-value necessary for hypothesis testing.