91Ó°ÊÓ

The shelf life of packaged food depends on many factors. Dry cereal is considered to be a moisture-sensitive product (no one likes soggy cereal!) with the shelf life determined primarily by moisture content. In a study of the shelf life of one particular brand of cereal, \(x=\) time on shelf (stored at \(73^{\circ} \mathrm{F}\) and \(50 \%\) relative humidity) and \(y=\) moisture content were recorded. The resulting data are from "Computer Simulation Speeds Shelf Life Assessments" (Package Engineering [1983]: 72-73). \(\begin{array}{rrrrrrrr}x & 0 & 3 & 6 & 8 & 10 & 13 & 16 \\ y & 2.8 & 3.0 & 3.1 & 3.2 & 3.4 & 3.4 & 3.5 \\ x & 20 & 24 & 27 & 30 & 34 & 37 & 41 \\ y & 3.1 & 3.8 & 4.0 & 4.1 & 4.3 & 4.4 & 4.9\end{array}\) a. Summary quantities are $$ \begin{array}{ll} \sum x=269 & \sum y=51 \quad \sum x y=1081.5 \\ \sum y^{2}=7745 & \sum x^{2}=190.78 \end{array} $$ Find the equation of the estimated regression line for predicting moisture content from time on the shelf. b. Does the simple linear regression model provide useful information for predicting moisture content from knowledge of shelf time? c. Find a \(95 \%\) interval for the moisture content of an individual box of cereal that has been on the shelf 30 days. d. According to the article, taste tests indicate that this brand of cereal is unacceptably soggy when the moisture content exceeds 4.1. Based on your interval in Part (c), do you think that a box of cereal that has been on the shelf 30 days will be acceptable? Explain.

Step by step solution

01

Find the Regression Line

The regression line equation is given by \( y = a + b*x \). Where 'a' is the y-intercept which can be calculated using the formula \(a = (\sum y - b * \sum x) / n\) and 'b' is the slope of the line which can be calculated using the formula \(b = (n * \sum xy - \sum x * \sum y) / (n * \sum x^{2} - (\sum x) ^{2})\). Here, n is number of the observations, which is 16 (8 for each row of data given). After substituting given values, we get the regression line equation.

02

Check Model Information Usefulness

The simple linear regression model's usefulness is judged by its coefficient of determination, denoted by 'R' or 'R squared'. Closer the coefficient of determination is to 1, the more useful the model is. However, this value is not given in the question, so we can't evaluate how useful the model is for predicting moisture content from knowledge of shelf time.

03

Find the 95% Confidence Interval

The formula for confidence interval for prediction at given x is \(E(Y|X) = a + b * X \pm ts * \sqrt{(n+1)/n) + (x- \overline{x})^{2}/(n * \sigma_{x}^{2})} \), where t is the t-value from t-distribution table for (n-2) degree of freedom and for desired confidence level, 's' is the standard deviation given by \(\sqrt{(\sum y^{2} - a * \sum y - b * \sum xy)/n-2}\) , \(\overline{x}\) is mean of x-values and \(\sigma_{x}^{2}\) is variance of x values given by \(\sum x^{2}/n - (\overline{x})^{2}\). On using given values and computing, we get the confidence interval.

04

Judging the taste acceptance based on moisture content

From the confidence interval calculated in the previous step, if the upper limit is less than 4.1, we can conclude that the box of cereal that has been on the shelf for 30 days will be acceptable. However, if the upper limit is more than 4.1, then it can't be guaranteed that the cereal will be acceptable, since it might exceed the acceptable moisture level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Linear Regression

Regression analysis is an essential statistical tool for modeling the relationship between a dependent variable and one or more independent variables. In our case, we consider simple linear regression, which involves just one independent variable. The basic form is expressed as \( y = a + bx \), where \(y\) represents the dependent variable (moisture content), \(x\) is the independent variable (time on shelf), \(a\) is the y-intercept, and \(b\) is the slope of the line.

To predict the moisture content of cereal through regression, we first need to determine the values of \(a\) and \(b\) based on the provided summary quantities. By evaluating these values appropriately, we get the line that best fits the data. The slope, \(b\), shows how much the moisture content is expected to increase for each additional day the cereal remains on the shelf. In essence, if we're to forecast the moisture content for any given day, we would simply plug in the shelf time into our equation.

Predicting Moisture Content

In the context of packaged food like cereal, understanding and predicting moisture content is critical for product quality. By applying simple linear regression analysis, we can establish a model that predicts moisture content based on the time cereal has been on the shelf. Once we have found our regression line, predicting is straightforward: for any given time on the shelf, we can insert that value into our equation to get the corresponding predicted moisture content.

Using the regression line allows manufacturers and distributors to estimate whether a product will remain below a moisture threshold considered to be the point where the product turns soggy. This kind of prediction is vital for optimizing shelf life and ensuring customer satisfaction.

Confidence Interval Calculation

A confidence interval offers a range of values, derived from the regression model, within which we can be confident that the actual moisture content lies, to a certain degree of probability. When calculating a 95% confidence interval for the moisture content, we're saying that we are 95% certain that the true moisture content of a box of cereal will fall within this range after being on the shelf for a certain number of days.

The confidence interval takes into account the variability of the sample data and uses the t-distribution to adjust for small sample sizes. This interval is not just a single prediction but rather a range that indicates the reliability and precision of the estimated moisture content. It's important to note that this range widens for predictions made for values of \(x\) that are far from the sample's mean.

Residual Analysis

Residual analysis is crucial to validate the assumptions of a linear regression model. Residuals, the differences between observed and predicted values, should be randomly dispersed around the zero line if the model fits well. Analyzing these residuals enables us to check for non-random patterns that reveal potential problems such as non-linearity, outliers, or heteroscedasticity (non-constant variance).

Ideally, if we plotted the residuals against the predicted values, we would expect to see no clear pattern. The spread of residuals should be approximately the same across all levels of the independent variable. If this is not the case, or if there are outliers present, it may indicate that our model isn't capturing all the systematic information in our data, which could lead to unreliable predictions.