Question

The diameter of the stem of a wheat plant is an important trait because of its relationship to breakage of the stem, which interferes with harvesting the crop. An agronomist measured stem diameter in eight plants of the Tetrastichon cultivar of soft red winter wheat. All obser-- \(-\) wheat vations were made 3 weeks after flowering of the plant. The stem diameters (mm) were as follows \(^{50}\). $$ \begin{array}{llllllll} 2.3 & 2.6 & 2.4 & 2.2 & 2.3 & 2.5 & 1.9 & 2.0 The mean of these data is 2.275 and the standard deviation is \(0.238 .\) (a) Calculate the standard error of the mean. (b) Construct a \(95 \%\) confidence interval for the population mean. (c) Define in words the population mean that you estimated in part (b). (See Example 6.1.1.) \end{array} $$

Short answer

Standard error is 0.0842 mm. The 95% confidence interval for the population mean is from 2.076 mm to 2.474 mm. The estimated population mean refers to the average stem diameter of the Tetrastichon cultivar of soft red winter wheat plants.

Step-by-step solution

Understanding Standard Error

The standard error of the mean (SEM) is a measure of how much discrepancy there is likely to be in a sample's mean compared to the true population mean. The SEM is calculated by dividing the standard deviation (SD) by the square root of the sample size (n). The formula for SEM is SEM = SD / sqrt(n).

Calculate the Standard Error of the Mean

Use the given standard deviation (0.238 mm) and the number of observations (8) to compute the standard error. SEM = 0.238 / sqrt(8) = 0.238 / 2.828 = 0.0842 mm.

Understand Confidence Interval

A confidence interval gives a range within which we can say with a certain level of confidence that the true population mean lies. To construct a 95% confidence interval, we use the standard error and a critical value that corresponds to the desired level of confidence. The critical value for a 95% confidence interval with a small sample size is typically found using the t-distribution table with degrees of freedom (n - 1).

Determine the t-Value for 95% Confidence

The degrees of freedom (df) is n - 1, which is 8 - 1 = 7. Using a t-distribution table or calculator for df = 7 at the 95% confidence level, we find the t-value. Assume for this example that the t-value is approximately 2.365.

Construct the 95% Confidence Interval

Multiply the standard error by the t-value to get the margin of error, then subtract and add this margin to the sample mean to get the confidence interval. Margin of error = t-value * SEM = 2.365 * 0.0842. Confidence interval = mean - margin of error to mean + margin of error = 2.275 - (2.365 * 0.0842) to 2.275 + (2.365 * 0.0842).

Calculate the Margin of Error and the Confidence Interval

Margin of error = 2.365 * 0.0842 = 0.199. Confidence interval: Lower limit = 2.275 - 0.199 = 2.076, Upper limit = 2.275 + 0.199 = 2.474. Therefore, the 95% confidence interval is from 2.076 mm to 2.474 mm.

Define the Population Mean Estimation

The population mean that we estimated is the average diameter of Tetrastichon cultivar stems in the entire population of soft red winter wheat plants. The confidence interval indicates that we are 95% confident that the true population mean diameter is between 2.076 mm and 2.474 mm.

Key concepts

Confidence Interval

A confidence interval is a statistical tool used to estimate the range within which a population parameter, like the mean, is likely to fall. In simple terms, it tells us how confident we can be about our sample data representing the larger population. When we construct a 95% confidence interval, we’re saying that if we were to take many samples and calculate the confidence interval for each of them, we would expect the true population mean to fall within these intervals 95% of the time.

In our wheat plant stem diameter example, we used the standard error and a critical value from the t-distribution (because of our small sample size) to calculate the interval. The calculation provided a range from 2.076 mm to 2.474 mm, which means we can be 95% confident that the population mean diameter falls within this range. It’s crucial to remember that the width of the interval depends on the standard error and the level of confidence we desire: the higher the confidence level, i.e., 99% over 95%, the wider the interval will be.

Population Mean

The population mean is a measure of central tendency that represents the average of a particular trait across the entire population of interest. In the context of our agronomy study, it's the average stem diameter of all the Tetrastichon cultivar of soft red winter wheat, not just the sample we measured.

Since it's impossible or impractical to measure every single wheat plant, we use sample data to estimate this population mean. We make predictions about the population mean based on sample statistics, which are informed by the law of large numbers—the larger our sample, the closer our sample mean is likely to be to the true population mean. In part (c) of the exercise, we defined this estimate saying that, based on our sample, the average stem diameter for the Tetrastichon cultivar is likely between our calculated confidence interval of 2.076 mm and 2.474 mm.

Sample Size

The sample size, denoted as (n), plays a critical role in statistical analysis. It refers to the number of observations or measurements taken from a population to form a sample. A key point to understand is that the accuracy of our estimates of the population parameters, like the mean, increases with a larger sample size. This is because a bigger sample is more likely to be representative of the population.

However, larger samples also require more resources and, beyond a certain point, may offer diminishing returns. Returning to our example, we had a sample size of eight wheat plants. This small size increased the standard error and, subsequently, the width of the confidence interval. Had we measured more plants, our standard error would be smaller, leading to a narrower confidence interval and a more precise estimate of the population mean. It's a balance between practicality and precision which guides researchers in choosing an appropriate sample size for their studies.