Question

Ferulic acid is a compound that may play a role in disease resistance in corn. A botanist measured the con- \(-\) centration of soluble ferulic acid in corn seedlings grown in the dark or in a light/dark photoperiod. The results (nmol acid per gm tissue) were as shown in the table. \({ }^{41}\) $$ \begin{array}{|ccc|} \hline & \text { Dark } & \text { Photoperiod } \\ \hline n & 4 & 4 \\ \bar{y} & 92 & 115 \\ s & 13 & 13 \\ \hline \end{array} $$ (a) Construct a \(95 \%\) confidence interval for the difference in ferulic acid concentration under the two lighting conditions. (Assume that the two populations from which the data came are normally distributed.) [Note: Formula (6.7.1) yields 6 degrees of freedom for these data. (b) Repeat part (a) for a \(90 \%\) level of confidence.

Short answer

The 95% confidence interval for the difference in ferulic acid concentrations between dark and photoperiod-grown corn seedlings is approximately \([-45.497, -0.503]\). The 90% confidence interval is approximately \([-40.865, -5.135]\).

Step-by-step solution

State the Formula for the Confidence Interval

For estimating the confidence interval for the difference between two means with the assumption of the populations being normally distributed, we use the following formula: \[ CI = (\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2, u} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(s_1\) and \(s_2\) are the sample standard deviations, n_1 and n_2 are the sample sizes, \(t_{\alpha/2, u}\) is the t-value for \(\alpha/2\) significance level with \(u\) degrees of freedom.

Calculate the Standard Error of the Difference

The standard error (SE) of the difference between the two means is given by the following formula: \[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]Substitute the given values: \[ SE = \sqrt{\frac{13^2}{4} + \frac{13^2}{4}} = \sqrt{\frac{169}{4} + \frac{169}{4}} = \sqrt{\frac{338}{4}} = \sqrt{84.5} = 9.1924 \]

Determine the t-Value for 95% Confidence Level

For a 95% confidence interval with 6 degrees of freedom, we use a t-distribution table or calculator to find the t-value for \(\alpha/2 = 0.025\). The t-value \(t_{0.025, 6}\) is approximately 2.447.

Calculate the 95% Confidence Interval for the Difference

Now we can apply the values to the confidence interval formula: \[ CI = (92 - 115) \pm 2.447 \cdot 9.1924 \]\[ CI = (-23) \pm 22.497 \]This results in the following confidence interval: \[ CI = [-45.497, -0.503] \]

Determine the t-Value for 90% Confidence Level

For a 90% confidence interval with 6 degrees of freedom, the t-value for \(\alpha/2 = 0.05\) is approximately 1.943.

Calculate the 90% Confidence Interval for the Difference

Using the 90% confidence level t-value: \[ CI = (92 - 115) \pm 1.943 \cdot 9.1924 \]\[ CI = (-23) \pm 17.865 \]This results in the following confidence interval: \[ CI = [-40.865, -5.135] \]

Key concepts

Difference Between Two Means

Understanding the difference between two means is essential in statistics, especially when comparing the results of two groups. In our exercise, we're interested in the difference in ferulic acid concentrations in corn seedlings grown under dark conditions compared to those grown in light/dark photoperiod. The mean difference is calculated simply by subtracting one sample mean from the other, as demonstrated by \( \bar{x}_1 - \bar{x}_2 \).

This calculation tells us if there's a significant variation between the two groups. However, knowing just the mean difference isn't enough; we need to determine whether this difference is statistically significant, which is where the confidence interval comes into play. It's a way to quantify the uncertainty around the mean difference, helping us understand the range wherein the true mean difference between the two populations lies with a certain level of confidence.

Standard Error

When we talk about the standard error (SE), we refer to the precision of our sample mean estimate. It's a measure of how far we can expect the sample mean to be from the true population mean. The SE is especially important when comparing the means from two different groups because it takes both the sample variability and the size into account.

In our exercise, the SE for the difference between two means is calculated with the formula \( SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \), where \(s_1\) and \(s_2\) are the sample standard deviations and \(n_1\) and \(n_2\) are the sample sizes for the dark and photoperiod groups, respectively. A smaller SE suggests a more precise estimate of the mean difference which enhances the reliability of the confidence interval we are trying to estimate.

T-Distribution

The t-distribution plays a pivotal role when we're dealing with small sample sizes, which is the case in our corn seedlings exercise. It's similar to the normal distribution but accounts for the extra uncertainty that comes with smaller samples.

When constructing a confidence interval for the difference between two means, we use a t-distribution to find the critical value known as the t-value. This reflects the level of confidence we want for our interval. For the 95% and 90% confidence intervals in our exercise, we looked up the respective t-values for 6 degrees of freedom. These t-values are crucial in stretching our interval to capture the true mean difference with a specified level of confidence.

Degrees of Freedom

Degrees of freedom (df) are a concept linked to the t-distribution and refer to the number of independent pieces of information in our data set that can be used to estimate a parameter, such as variance. In the formula for finding a confidence interval for the difference between two means, we used 6 degrees of freedom — a number derived from the sample sizes of our two groups.

The df often equals the sample size minus one for each group, but since we're comparing two means here, we use a combined approach to calculate the degrees of freedom, which in this case, yielded 6. This number then guides which t-value we should use from the t-distribution table to construct our confidence intervals. The concept of degrees of freedom is crucial for ensuring the validity of our t-distribution application in confidence interval estimation.