Question

A horticulturalist devised a scale to measure the freshness of roses that were packaged and stored for varying periods of time before transplanting. The freshness measurement \(y\) and the length of time in days that the rose is pack-aged and stored before transplanting \(x\) are given below. $$ \begin{array}{l|lllll} x & 5 & 10 & 15 & 20 & 25 \\ \hline y & 15.3 & 13.6 & 9.8 & 5.5 & 1.8 \\ & 16.8 & 13.8 & 8.7 & 4.7 & 1.0 \end{array} $$ a. Fit a least-squares line to the data. b. Construct the ANOVA table. c. Is there sufficient evidence to indicate that freshness is linearly related to storage time? Use \(\alpha=.05 .\) d. Estimate the mean rate of change in freshness for a 1 -day increase in storage time usig a \(98 \%\) confidence interval. e. Estimate the expected freshness measurement for a storage time of 14 days with a \(95 \%\) confidence interval. f. Of what value is the linear model in reference to \(\bar{y}\) in predicting freshness?

Short answer

Short Answer: Given the dataset involving the freshness measurements of roses and their storage time, first organize and calculate summary statistics. Then fit a least-squares line and perform an ANOVA analysis to test for the significance of the linear relationship. Estimate the rate of change in the freshness of roses for a 1-day increase in storage time with a 98% confidence interval and estimate the expected freshness for 14-day storage time using a 95% confidence interval. Finally, evaluate the usefulness of the linear model by comparing its predictive ability using the coefficient of determination (\(R^2\)).

Step-by-step solution

Organize the data

Organize the given data in a two-column table with x as the independent variable (storage time) and y as the dependent variable (freshness measurement).

Calculate summary statistics

Calculate the necessary summary statistics, which include the mean of x (\(\bar{x}\)), the mean of y (\(\bar{y}\)), and the sums of squares and cross products (SSCP) for x and y: $$SSCP_{xx} = \sum(x^2) - \frac{(\sum{x})^2}{n}$$ $$SSCP_{yy} = \sum(y^2) - \frac{(\sum{y})^2}{n}$$ $$SSCP_{xy} = \sum(xy) - \frac{(\sum{x})(\sum{y})}{n}$$

Calculate the least-squares line coefficients

Calculate the slope, \(b_1\), and the intercept, \(b_0\), of the least-squares line using the following formulas: $$b_1 = \frac{SSCP_{xy}}{SSCP_{xx}}$$ $$b_0 = \bar{y} - b_1\bar{x}$$

Fit the least-squares line

Having calculated the coefficients \(b_0\) and \(b_1\), the fitted least-squares line is $$\hat{y} = b_0 + b_1x$$

Calculate error statistics

Calculate the residual sum of squares (RSS) and the total sum of squares (TSS) to estimate the goodness of fit of the line: $$RSS = SSCP_{yy} - b_1 SSCP_{xy}$$ $$TSS = SSCP_{yy}$$ Calculate the coefficient of determination, \(R^2\), as: $$R^2 = 1 - \frac{RSS}{TSS}$$

Construct the ANOVA table

The ANOVA table typically includes columns for Source of Variation, Sum of Squares, Degrees of Freedom, Mean Squares, and the \(F\)-Statistic. For this problem, the sources of variation are Regression (explained) and Residual (unexplained). Fill in the ANOVA table using the calculated statistics and the degrees of freedom (df): $$df_{Regression}=1$$ $$df_{Residual}=n-2$$ $$F = \frac{MSR}{MSE}$$

Significance Test for Linear Relationship

To test if there is sufficient evidence to indicate a linear relationship between freshness and storage time at a significance level of \(\alpha = 0.05\), compare the calculated F-statistic from the ANOVA table to the critical F-value by looking up the F-distribution table with \((df_{numerator}, df_{denominator}) = (1, n-2)\). If the calculated F-statistic is greater than the critical F-value, reject the null hypothesis.

Estimate Rate of Change

To estimate the mean rate of change in freshness for a 1-day increase in storage time using a \(98\%\) confidence interval, calculate the standard error for the slope, \(SE_{b_1}\), and the degrees of freedom \((n-2)\) using the t-distribution table. Then, calculate the \(98\%\) confidence interval as follows: $$b_1 \pm t_{\alpha/2}(n-2) \times SE_{b_1}$$

Estimate Freshness for 14-day Storage Time

To estimate the expected freshness measurement for a storage time of 14 days with a \(95\%\) confidence interval, calculate the standard error for prediction and use the t-distribution table. Then, calculate the confidence interval by applying the linear model to the given storage time (14 days) and adding the margin of error: $$\hat{y}(14) \pm t_{\alpha/2}(n-2) \times SE_{prediction}$$

Evaluate the linear model

To determine the value of the linear model in predicting freshness, compare the fitted line (\(\hat{y}\)) to the mean of y (\(\bar{y}\)) and evaluate the coefficient of determination (\(R^2\)). If the \(R^2\) is close to 1, the linear model can reasonably predict freshness. If it is close to 0, it might not be a suitable predictor.

Key concepts

Least-Squares Method

The least-squares method is a popular statistical technique used to determine the best fit line through a set of data points. This involves finding the line where the sum of the squares of the vertical distances of each data point from the line is minimized. This line is often called the regression line, and in the context of this exercise, it is used to explore the relationship between the storage time of roses and their freshness.

To fit a least-squares line, we first organize our data into two columns: one for the independent variable, storage time (\(x\)), and the other for the dependent variable, freshness (\(y\)). Summary statistics, such as the mean of each variable and sums of squares, are then calculated.

• Slope (\(b_1\)) is calculated by dividing the sum of cross-products by the sum of squares of the independent variable.
• Intercept (\(b_0\)) is found by subtracting the product of the slope and mean of \(x\) from the mean of \(y\).

Once you have these coefficients, the equation of the regression line can be written as \(\hat{y} = b_0 + b_1x\). This line serves as a model to make predictions about the freshness of roses for any given storage duration.

ANOVA Table

ANOVA, or Analysis of Variance, is a method used in statistics to break down the variability in a dataset. An ANOVA table shows how much of the variance in the dependent variable is explained by the independent variable and how much is due to random error. In the context of regression analysis, we want to know how well the linear regression line explains the variation in the data.

Our ANOVA table includes the following details:

• Sum of Squares for Regression (explained variation).
• Sum of Squares for Residuals (unexplained variation).
• Total Sum of Squares (aggregate variability in the data).

From these, we can calculate Mean Squares for both regression and residuals. The F-statistic, an integral part of the ANOVA table, is used to test the model's significance. A significant F-statistic indicates that the linear model provides a better fit than a model with no independent variables. This calculation is based on the degrees of freedom and helps us understand if storage time significantly impacts rose freshness.

Coefficient of Determination

The coefficient of determination, commonly referred to as \(R^2\), is a key metric in regression analysis. It quantifies how much of the variability in the dependent variable is explained by the independent variable in a regression model. In simple terms, it's a measure of how well the regression line fits the data.

An \(R^2\) value ranges from 0 to 1. An \(R^2\) value near 1 implies that a large proportion of variability in the response variable can be accounted for through the linear relationship. Conversely, an \(R^2\) value near 0 suggests that the linear model does not adequately explain the variability observed.

• High \(R^2\) = Better fit and predictive power.
• Low \(R^2\) = Poor fit.

For the horticulturalist's study, \(R^2\) tells us how well the linear model (relationship between storage time and freshness) explains the changes in the freshness measurements. Assessing \(R^2\) is essential to deciding whether the linear model is valuable for prediction purposes.

Confidence Interval

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated parameter being the rate of change in freshness for this exercise. It's derived from the standard error of the estimate and the desired confidence level. In regression, confidence intervals can be calculated for both predictions of the dependent variable and estimates of the slope of the regression line.

In the context of the horticulturist's data, we might use:

• A \(98\%\) confidence interval to estimate the slope or the mean rate of change in freshness with storage time.
• A \(95\%\) confidence interval to estimate the expected freshness for a given number of days.

Creating a confidence interval involves using the t-distribution to determine the margin of error. The interval indicates that there is a certain level of confidence (such as \(95\%\) or \(98\%\)) the true parameter lies within this calculated range. This statistical tool aids in assessing the precision and reliability of the estimated parameters. By examining the confidence interval's width, one can judge the estimate's accuracy; narrower intervals signify more precise estimates.