91Ó°ÊÓ

The shelf life of packaged food depends on many factors. Dry cereal is considered to be a moisturesensitive 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 (days stored at \(73^{\circ} \mathrm{F}\) and \(50 \%\) relative humidity) and \(y=\) moisture content \((\%)\) were recorded. The resulting data are from. \(\begin{array}{rrrrrrrr}x & 0 & 3 & 6 & 8 & 10 & 13 & 16 \\ y & 2.8 & 3.0 & 3.1 & 3.2 & 3.4 & 3.4 & 3.5 \\ x & 20 & 24 & 27 & 30 & 34 & 37 & 41 \\ y & 3.1 & 3.8 & 4.0 & 4.1 & 4.3 & 4.4 & 4.9\end{array}\) a. Summary quantities are \(\sum x=269 \quad \sum y=51 \quad \sum x y=1081.5\) $$ \sum y^{2}=190.78 \quad \sum x^{2}=7745 $$ 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 artide, 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

Calculate the estimated regression line

Use summary quantities to estimate the regression line parameters 'a' (intercept) and 'b' (slope). The formula for 'b' (slope) is: \[ b = \frac{(n \sum xy - \sum x \sum y)}{(n \sum x^2 - (\sum x)^2)} \], and for 'a' (intercept) is \[ a = \bar{y} - b \bar{x} \]. Here, 'n' is the total number of observations, \(\sum xy\) is the sum of the product of x and y, \(\sum x\) and \(\sum y\) are the sum of x and y observations respectively, while \( \bar{x} \) and \( \bar{y} \) represent the mean of x and y observations respectively. Using the given summary quantities, calculate 'a' and 'b'.

02

Evaluate the usefulness of the model

To assess whether the model is useful for predicting moisture content based on shelf time, conduct a significance test. The null hypothesis (H0) is that the shelf time does not affect the moisture content (b=0), while the alternative hypothesis (H1) is that it does (b≠0). Calculate the t-statistic and p-value and interpret the results. If the p-value is small (typically ≤ 0.05), reject the null hypothesis and conclude that the model is useful.

03

Calculate the prediction interval

Find a 95% prediction interval for the moisture content of cereal that has been on the shelf for 30 days. The formula for the prediction interval is \[ \hat{y} ± t*\sqrt{MSE*(1+1/n+(x-\bar{x})^2/\sum(x-\bar{x})^2)} \], where \(\hat{y}\) is the predicted value for a given x, t is the t-value for the desired level of confidence, n is the total number of observations, x is the value for which the prediction is being done, and MSE is the mean squared error. Here, assume that MSE is constant and can be estimated from the data.

04

Explain the acceptability of the cereal box

Based on the 95% prediction interval calculated in step 3, evaluate whether a cereal box with a shelf life of 30 days will have acceptable moisture content. If the upper limit of the prediction interval is below or equivalent to the threshold (4.1%), then the cereal box is acceptable. Otherwise, it is not.

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Key Concepts

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

Regression Line

A regression line is a fundamental concept in linear regression, which represents the relationship between two variables. In this case, we're trying to predict the moisture content (y) based on the number of days the cereal has been stored (x).
This is known as the 'best-fit' line, and it's derived from the equation of a straight line: \[ y = a + bx \] where 'a' is the intercept, and 'b' is the slope of the line.
The slope 'b' tells us how much y changes for a one-unit change in x. To calculate these values, we use a set of summary statistics derived from the data.

• To find the slope 'b', the formula used is: \[ b = \frac{(n \sum xy - \sum x \sum y)}{(n \sum x^2 - (\sum x)^2)} \]
• For the intercept 'a', the formula is: \[ a = \bar{y} - b \bar{x} \]

With these calculations, we can plot the regression line, which helps in making predictions and understanding the trend of the data.

Prediction Interval

A prediction interval gives us a range of values for the dependent variable, in this case, moisture content, where we expect an individual new observation to fall with a specified probability, such as 95%.
This is particularly useful when you're making predictions for future data points and you want to understand how precise your predictions are.
The formula for a prediction interval is given by:\[ \hat{y} \pm t*\sqrt{MSE*(1+1/n+(x-\bar{x})^2/\sum(x-\bar{x})^2)} \] where:

• \(\hat{y}\) is the predicted value at a specific value of x
• \(t\) is the t-value correlating to the confidence level chosen
• MSE is the mean squared error
• \(n\) is the number of observations

This interval accounts for variability not only in the dataset but also in the prediction itself. In other words, it represents how confident we are in predicting the moisture content for a cereal box that has been on the shelf for a specified number of days.

Significance Test

A significance test in linear regression helps us decide whether the predictor variable, which is 'time on shelf' in this scenario, significantly impacts the response variable, 'moisture content'.
It essentially checks if there's a meaningful relationship between the two variables.
For this, we establish two hypotheses:

• Null hypothesis (\(H_0\)): The slope \(b\) is zero, meaning the shelf time does not affect moisture content.
• Alternative hypothesis (\(H_1\)): The slope \(b\) is not zero, indicating an effect of shelf time on moisture content.

We'll use a t-statistic to determine significance:\[ t = \frac{b}{SE_b} \] where \(SE_b\) is the standard error of the slope.
If the computed p-value from this t-statistic is less than a critical value (usually 0.05 for a 5% significance level), we reject the null hypothesis in favor of the alternative, suggesting the shelf time is a significant predictor for moisture content.
This allows us to be confident in using the regression model for predictions.