Question

An experiment was conducted to determine the effect of soil applications of various levels of phosphorus on the inorganic phosphorus levels in a particular plant. The data in the table represent the levels of inorganic phosphorus in micromoles ( \(\mu\) mol) per gram dry weight of Sudan grass roots grown in the greenhouse for 28 days, in the absence of zinc. Use the MINITAB output to answer the questions. $$ \begin{array}{l} \text { Phosphorus Applied, } x \quad \text { Phosphorus in Plant, } y \\ \hline .5 \mu \mathrm{mol} & 204 \\ & 195 \\ & 247 \\ & 245 \\ & \\ .25 \mu \mathrm{mol} & 159 \\ & 127 \\ & 95 \\ & 144 \\ .10 \mu \mathrm{mol} & 128 \\ & 192 \\ & 84 \\ & 71 \end{array} $$ a. Plot the data. Do the data appear to exhibit a linear relationship? b. Find the least-squares line relating the plant phosphorus levels \(y\) to the amount of phosphorus applied to the soil \(x\). Graph the least-squares line as a check on your answer. c. Do the data provide sufficient evidence to indicate that the amount of phosphorus present in the plant is linearly related to the amount of phosphorus applied to the soil? d. Estimate the mean amount of phosphorus in the plant if \(.20 \mu \mathrm{mol}\) of phosphorus is applied to the soil, in the absence of zinc. Use a \(90 \%\) confidence interval.

Short answer

Question: Given the amount of phosphorus applied to the soil, estimate the mean amount of phosphorus in the plant with a 90% confidence interval when 0.20 µmol of phosphorus is applied. Assume you have already plotted the data, calculated the least-squares line (y = mx + b) and assessed linearity.

Step-by-step solution

Plot the Data

Using a suitable software, plot the x, y pairs provided in the problem. Based on the plot, decide whether the phosphorus applied (x) and the phosphorus in the plant (y) follow a linear trend.

Find the Least-squares Line

To calculate the least-squares line for the data, use the provided data to find the slope and y-intercept of the line using the following formulas: Slope, \(m = \frac{n\sum xy-\sum x\sum y}{n\sum x^{2} - (\sum x)^{2}}\) Y-intercept, \(b = \frac{\sum y - m\sum x}{n}\) Once you have the slope and y-intercept, you can find the equation of the least-squares line. In the form \(y=mx+b\).

Assess Linearity of the Relationship

Observe the plotted data and the least-squares regression line. If it appears that the relationship between the data is linear, then there's a possibility that the amount of phosphorus present in the plant is linearly related to the amount of phosphorus applied to the soil. However, additional statistical tests and analyses are required to determine the strength of this relationship and the degree of certainty.

Calculate the Mean with a Confidence Interval

Using the least-squares line you found earlier, plug in the value of x = 0.20 \(\mu\)mol to estimate the mean phosphorus content of the plant, given that 0.20 \(\mu\)mol of phosphorus is applied to the soil. The resulting value will be the point estimate for the mean. Next, calculate a 90% confidence interval for the mean phosphorus content in the plants using the standard error of the residuals and the "t" critical values. Using a suitable software or statistical techniques, you can calculate the standard error (SE) as well as determine the "t" values for 90% confidence interval. Once you have the standard error and "t" values, calculate the margins of error: Margin of Error (ME) = SE * t_critical Then, construct the 90% confidence interval for the mean phosphorus content: Lower Limit: Point estimate - ME Upper Limit: Point estimate + ME Thus, you will have computed a 90% confidence interval for the mean amount of phosphorus in the plant when 0.20 \(\mu\)mol phosphorus is applied to the soil.

Key concepts

Least-Squares Line

The least-squares line is a foundational concept in linear regression analysis, used to model the relationship between two variable quantities. When we conduct experiments, like the one described for the Sudan grass roots' phosphorus content, we often want to understand how one variable affects another. The least-squares line provides a way to visually and mathematically determine this relationship.

Consider a scatter plot with data points, each representing an experimental observation of applied phosphorus (the independent variable, x) and the resulting phosphorus in the plant (the dependent variable, y). The goal is to find the line that best fits these points. The 'best fit' criterion is that the sum of the squares of the vertical distances of the data points from the line should be as small as possible. This is where the term 'least-squares' comes from.

To calculate this line, we use formulas to find the slope and y-intercept:

• The slope (\(m\)) indicates the change in the dependent variable (\(y\)) for every one-unit change in the independent variable (\(x\)).
• The y-intercept (\(b\)) shows the value of (\(y\)) when (\(x\)) is zero.

After calculating (\(m\)) and (\(b\)), we articulate the line equation as (\(y = mx + b\)). This equation allows us to make predictions and interpret the relationship between the variables. For example, if the slope is positive, as the amount of phosphorus applied increases, the level of phosphorus in the plant also increases, suggesting a direct relationship.

Confidence Interval

A confidence interval is an essential tool in statistics that quantifies the uncertainty around an estimated value. It provides a range of values within which we can expect the true value to fall, with a given level of confidence, typically expressed as a percentage like 90%, 95%, or 99%. In the context of our phosphorus experiment, a 90% confidence interval around the estimated mean phosphorus content in the plant reflects our certainty that the true mean, for all possible plants of this type under the same conditions, would fall within this range.

Calculating a confidence interval involves several steps:

• First, determine the point estimate of the mean, which is the estimated value based on our sample data.
• Second, calculate the standard error of the estimate (SE), which measures the variability of the estimate.
• Lastly, use the standard error together with a critical value from the t-distribution – which reflects the desired level of confidence and the sample size – to define the margin of error (ME).

The confidence interval is then constructed by subtracting and adding this margin of error to the point estimate:

• Lower Limit: Point estimate - ME
• Upper Limit: Point estimate + ME

So, a 90% confidence interval tells us that we can be 90% certain the true mean lies within this range. It does not, however, mean that there is a 90% chance the true value is within the interval for one specific sample—the interval either contains the true mean, or it doesn't.

Statistical Hypothesis Testing

Statistical hypothesis testing is a methodology to make decisions or inferences about populations based on sample data. It involves formulating a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)), conducting an experiment, and then using the observed data to make a decision about which hypothesis is supported. The null hypothesis usually suggests that there is no effect or no difference, while the alternative hypothesis represents what we suspect to be true.

In our example of evaluating the relationship between soil phosphorus levels and plant growth, we might frame the null hypothesis as there being no linear relationship between the amount of phosphorus applied and the phosphorus present in the plant. The alternative hypothesis would be that there is indeed a linear relationship.

To test these hypotheses, we could use the least-squares regression line that we calculated and look at a statistical measure like the R-squared value, which indicates the proportion of the variance in the dependent variable (\(y\)) that is predictable from the independent variable (\(x\)). We might also examine the p-value associated with the slope of the regression line, which tells us whether the observed relationship could likely have occurred by chance. If the p-value is below a chosen significance level (often 0.05), we reject the null hypothesis in favor of the alternative hypothesis, concluding that there is a statistically significant linear relationship.