/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 76 No tortilla chip lover likes sog... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 76

No tortilla chip lover likes soggy chips, so it is important to find characteristics of the production process that produce chips with an appealing texture. The accompanying data on \(x=\) frying time (in seconds) and \(y=\) moisture content \((\%)\) appeared in the paper, "Thermal and Physical Properties of Tortilla Chips as a Function of Frying Time" (journal of Food Processing and Preservation [1995]: \(175-189\) ): \(\begin{array}{lrrrrrrrr}\text { Frying time }(x): & 5 & 10 & 15 & 20 & 25 & 30 & 45 & 60 \\ \text { Moisture } & 16.3 & 9.7 & 8.1 & 4.2 & 3.4 & 2.9 & 1.9 & 1.3\end{array}\) content \((y)\) : a. Construct a scatterplot of these data. Does the relationship between moisture content and frying time appear to be linear? b. Transform the \(y\) values using \(y^{\prime}=\log (y)\) and construct a scatterplot of the \(\left(x, y^{\prime}\right)\) pairs. Does this scatterplot look more nearly linear than the one in Part (a)? c. Find the equation of the least-squares line that describes the relationship between \(y^{\prime}\) and \(x\). d. Use the least-squares line from Part (c) to predict moisture content for a frying time of 35 minutes.

Short Answer

Expert verified
The main steps to solve this task involve creating scatterplots from the given and transformed data, identifying if they show a linear relationship, calculating the equation of the best-fit line for the transformed data, and using this line to predict the moisture content for given a frying time. Note that software or a calculator would typically be used to handle the graphical and numerical computations involved.

Step by step solution

01

Create a scatterplot for the given data

First, graph the given pairs of \((x,y)\) using any statistical software. Identify each pair as a point in your graph, with the x-values (frying time) on the horizontal axis and the y-values (moisture content) on the vertical axis.
02

Analyze the scatterplot

Squint at the positions of the points to determine if they seem to form a line that could be drawn through them. If they do, then it is likely that there is a linear relationship between x and y.
03

Transform the y-values

Calculate the natural logarithm for each y-value. The transformed y-values \(y'\) will now be used for the rest of the exercise.
04

Create a new scatterplot using the transformed y-values

Just as in Step 1, create a new scatterplot. This time, the points should consist of the pairs \((x,y')\). Then, assess if this plot shows a linear relationship as done in Step 2.
05

Find the least-squares line

Assuming a linear relationship after Step 4, use a statistical method to determine the equation of the line that fits the transformed data \(y'\). This line will have the form \(y' = ax+b\) where a is the slope and b is the intercept.
06

Use least-squares line to predict moisture content

Substitute the x-value of 35 into the equation from Step 5 to predict the transformed moisture content corresponding to a frying time of 35 min. Use the natural exponent (i.e., the inverse of natural logarithm) to transform the predicted \(y'\) value back and get the predicted moisture content in its original scale.

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Key Concepts

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

Scatterplot
A scatterplot is a simple yet powerful graphical tool to visualize the relationship between two variables. In our tortilla chip exercise, we have data on frying time (in seconds) and moisture content (in percentage). By creating a scatterplot, we can place each data pair as a point with the frying time on the x-axis and moisture content on the y-axis.

This visual representation helps us look for patterns. Specifically, we're interested in seeing if these points form any sort of predictable pattern, like a line. If the points appear to follow a linear pattern, it might suggest a straight-line relationship between frying time and moisture content. For those new to scatterplots, remember:
  • Poisition each data point using its x and y values.
  • Look for clusters, or a general trend (upwards or downwards) of the points.
  • Patterns often suggest a mathematical relationship that can be explored further.
This visual tool is a great first step in regression analysis to determine relationships between data points.
Least-Squares Line
The least-squares line is fundamental in regression analysis. After plotting your data, if the points suggest a linear relationship, you can draw a line that best fits all the data points. The goal of the least-squares method is to minimize the sum of the squares of the vertical distances of the points from the line.

This line has an equation like: \[ y = ax + b \]where:
  • \( a \) is the slope, indicating how much \( y \) changes for a change in \( x \).
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
This line helps in precise predictions. For example, in our exercise, once we identify the least-squares line using the transformed \( y' \) values, it provides us with a predictive tool to estimate moisture content for frying times not in the given data set.

Understanding the underlying mathematics of the least-squares concept not only strengthens analytical skills but ensures you're making informed predictions that are backed by data.
Logarithmic Transformation
A logarithmic transformation is a powerful technique used to linearize data that isn't linear at first glance. In our tortilla chip moisture content exercise, the initial scatterplot of moisture content and frying time might not be linear. To remedy this, we apply a logarithmic transformation to the y-values, converting them into \( y' \) values where \( y' = \log(y) \).

This transformation can help in:
  • Straightening the relationship between variables in the scatterplot.
  • Making patterns in data clearer and reducing skewness.
  • Stabilizing variance across levels of the independent variable.
After applying logarithmic transformation, if the scatterplot of \( x \) and \( y' \) looks more linear, it indicates that the transformation was effective. Finally, with this linear relationship, we can use regression techniques to find a line of best fit and make predictions.

Remember, while applying logarithmic transformation, you change the scale of the y-values, and any predictions in logarithm form must be converted back using the exponential function to interpret in the original context.

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Most popular questions from this chapter

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