/*! 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 66 The article gave the following d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 66

The article gave the following data (read from a scatterplot) on \(y=\) glucose concentration \((\mathrm{g} / \mathrm{L})\) and \(x=\) fermentation time (days) for a blend of malt liquor. $$ \begin{array}{rrrrrrrrr} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ y & 74 & 54 & 52 & 51 & 52 & 53 & 58 & 71 \end{array} $$ a. Use the data to calculate the estimated regression line. b. Do the data indicate a linear relationship between \(y\) and \(x\) ? Test using a \(.10\) significance level. c. Using the estimated regression line of Part (a), compute the residuals and construct a plot of the residuals versus \(x\) (that is, of the \((x\), residual \()\) pairs). d. Based on the plot in Part (c), do you think that the simple linear regression model is appropriate for describing the relationship between \(y\) and \(x\) ? Explain.

Short Answer

Expert verified
The regression line, linear relationship and residuals can be determined by using the formulas for calculating the slope, intercept and residuals. The scatter plot of residuals can be made once residuals are calculated. Whether the given model is appropriate or not depends on the visual inspection of this plot. If the residuals scatter randomly, then the linear regression model may be suitable. If there is a noticeable pattern, a different model may be more appropriate.

Step by step solution

01

Calculate Regression Line

In order to find the regression line, we need to calculate the slope and the intercept of the line. We can use the formulas: \( m = \frac{n(\sum {xy}) - (\sum{x})(\sum{y})}{n(\sum{x^2})-(\sum{x})^2} \) for the slope, and then by substititing into \(b = \frac{\sum{y} - m\sum{x}}{n} \), we can find the y-intercept. Substituting the values from the given data we will get the equation for the regression line.
02

Test for Linear Relationship

To determine linear relationship we can use hypothesis testing. Assuming null hypothesis H0: there is no linear relationship (slope is zero) and alternate hypothesis H1: there is linear relationship (slope is not zero). By calculating \( t=\frac{m-0}{SE_m} \) where SE_m is standard error of the slope, we find our t-value which we then compare to the t-distribution table value at the significance level of 0.10. The conclusion about the relationship is reached based on this comparison.
03

Compute the Residuals

The residuals are calculated using the formula: \( e_i = y_i - \hat{y_i} \), where \( y_i \) are the observed values of the dependent variable and \( \hat{y_i} \) are the estimated values of the dependent variable. The estimated values are calculated by substituting the x-values in the regression line equation obtained in step 1.
04

Construct a Plot of Residuals

Once the residuals are calculated, construct a scatter plot of them against the x-values. this gives us a visual sense of the spread of the residuals versus the independent variable.
05

Evaluate Appropriateness of the Model

To assess the appropriateness of the linear regression model, examine the scatter plot made in step 4. If the residuals appear to be randomly scattered around the horizontal axis, then linear regression may be a suitable model. If there is any pattern in the residuals, then we may need to consider a different model.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Regression Analysis
Regression analysis is a statistical technique used to explore the relationship between two or more variables. In this case, we want to understand how glucose concentration changes with fermentation time. This involves finding the best-fitting line through the data points on a scatter plot. This line is known as the regression line and is determined using mathematical formulas for slope and intercept.
The regression line's equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. These values are calculated based on the provided dataset. Using this line allows us to predict the response variable \( y \) for any given value of the independent variable \( x \).
This analysis helps in identifying trends and making data-driven predictions, making it a powerful tool in various fields like economics, biology, and social sciences.
Scatter Plot
A scatter plot is a graphical representation of the relationship between two quantitative variables. This plot displays individual data points, where each point's coordinates represent values of the variables being analyzed. In the context of the exercise, the scatter plot helps visualize glucose concentration against fermentation time.
By examining the scatter plot, one can visually assess any apparent relationship between the variables. For example, one might notice a linear trend or any deviations from it. This visualization is crucial before performing regression analysis, as it provides immediate insights into the data's nature and any potential outliers.
  • Points in a linear trend suggest a potential linear relationship.
  • Randomly scattered points indicate no clear relationship.
The scatter plot is an essential first step in understanding data relationships before delving into more complex statistical analyses.
Hypothesis Testing
Hypothesis testing in regression analysis is used to determine whether there is a significant linear relationship between the variables. We start by setting up two hypotheses:
- **Null hypothesis \( H_0 \):** There is no linear relationship (slope equals zero).
- **Alternative hypothesis \( H_1 \):** There is a linear relationship (slope is not zero).
To test these, we calculate a t-statistic using the slope and its standard error. This t-value is then compared against a critical value from the t-distribution table corresponding to the given significance level (here, 0.10).
If the t-value exceeds the critical value, we reject the null hypothesis, indicating that the linear relationship is statistically significant. Conversely, if it does not exceed the critical value, there is insufficient evidence to claim a linear relationship. This process helps in deciding the validity of the regression model for predicting outcomes.
Residuals
Residuals in regression analysis are the differences between the observed values and the values predicted by the regression line. They are calculated as \( e_i = y_i - \hat{y_i} \), where \( y_i \) is the actual value and \( \hat{y_i} \) is the predicted value.
Residuals serve a crucial role:
  • They help identify how well the regression line fits the data.
  • By plotting residuals against the independent variable, one can check the assumption of homoscedasticity (equal spread of residuals).
A pattern or trend in the residuals plot, such as systematic deviation from zero, can indicate issues with the model, suggesting that a different type of model might be more appropriate. Random scattering around the horizontal axis is ideal, indicating a good fit.
Significance Level
The significance level, often denoted as \( \alpha \), is a threshold used in hypothesis testing to determine the evidence against a null hypothesis. In this context, a significance level of 0.10 was chosen.
This value reflects the probability of making a Type I error, which is rejecting the null hypothesis when it's actually true. A lower significance level means stricter criteria for rejecting the null hypothesis, while a higher level allows for more tolerance in making this error.
Choosing the right significance level is crucial as it impacts the confidence in the results:
  • A common significance level is 0.05, but it can vary depending on the context and field of study.
  • The chosen level should align with the study's goals and the potential consequences of errors.
Overall, the significance level helps guide decisions in testing hypotheses within regression analysis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The accompanying summary quantities for \(x=\) particulate pollution \(\left(\mu \mathrm{g} / \mathrm{m}^{3}\right)\) and \(y=\) luminance \((.01 \mathrm{~cd} /\) \(\mathrm{m}^{2}\) ) were calculated from a representative sample of data that appeared in the article \begin{array}{ccc} n & =15 & \sum x=860 & \sum y=348 \\ \sum x^{2} & =56,700 & \sum y^{2}=8954 & \sum x y=22,265 \end{array} $$ a. Test to see whether there is a positive correlation between particulate pollution and luminance in the population from which the data were selected. b. What proportion of observed variation in luminance can be attributed to the approximate linear relationship between luminance and particulate pollution?

Give a brief answer, comment, or explanation for each of the following. a. What is the difference between \(e_{1}, e_{2}, \ldots, e_{n}\) and the \(n\) residuals? b. The simple linear regression model states that \(y=\alpha+\beta x\). c. Does it make sense to test hypotheses about \(b\) ? d. SSResid is always positive. e. A student reported that a data set consisting of \(n=6\) observations yielded residuals \(2,0,5,3,0\), and 1 from the least-squares line. f. A research report included the following summary quantities obtained from a simple linear regression analysis: \(\sum(y-\bar{y})^{2}=615 \quad \sum(y-\hat{y})^{2}=731\)

\(13.26\) In anthropological studies, an important characteristic of fossils is cranial capacity. Frequently skulls are at least partially decomposed, so it is necessary to use other characteristics to obtain information about capacity. One such measure that has been used is the length of the lambda- opisthion chord. The article reported the accompanying data for \(n=7\) Homo erectus fossils. \(\begin{array}{llllllll}x \text { (chord } & 78 & 75 & 78 & 81 & 84 & 86 & 87\end{array}\) length in \(\mathrm{mm}\) ) \(\begin{array}{llllllll}\text { (capacity } & 850 & 775 & 750 & 975 & 915 & 1015 & 1030\end{array}\) in \(\mathrm{cm}^{3}\) ) Suppose that from previous evidence, anthropologists had believed that for each \(1-\mathrm{mm}\) increase in chord length, cranial capacity would be expected to increase by \(20 \mathrm{~cm}^{3}\). Do these new experimental data strongly contradict prior belief?

The data of Exercise \(13.25\), in which \(x=\) milk temperature and \(y=\) milk \(\mathrm{pH}\), yield $$ \begin{aligned} &n=16 \quad \bar{x}=42.375 \quad S_{x x}=7325.75 \\ &b=-.00730608 \quad a=6.843345 \quad s_{e}=.0356 \end{aligned} $$ a. Obtain a \(95 \%\) confidence interval for \(\alpha+\beta(40)\), the mean milk \(\mathrm{pH}\) when the milk temperature is \(40^{\circ} \mathrm{C} .\) b. Calculate a \(99 \%\) confidence interval for the mean milk \(\mathrm{pH}\) when the milk temperature is \(35^{\circ} \mathrm{C} .\) c. Would you recommend using the data to calculate a \(95 \%\) confidence interval for the mean \(\mathrm{pH}\) when the temperature is \(90^{\circ} \mathrm{C}\) ? Why or why not?

A subset of data read from a graph that appeared in the paper "Decreased Brain Volume in Adults with Childhood Lead Exposure" (Public Library of Science Medicine [May 27, 2008]: ell2) was used to produce the following Minitab output, where \(x=\) mean childhood blood lead level \((\mu \mathrm{g} / \mathrm{dL})\) and \(y=\) brain volume change (percentage). (See Exercise \(13.19\) for a more complete description of the study described in this paper) Regression Analysis: Response versus Mean Blood Lead Level The regression equation is Response \(=-0.00179-0.00210\) Mean Blood Lead Level \(\begin{array}{lrrrr}\text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & -0.001790 & 0.008303 & -0.22 & 0.830 \\ \text { Mean Blood Lead Level } & -0.0021007 & 0.0005743 & -3.66 & 0.000\end{array}\) a. What is the equation of the estimated regression line? b. For this dataset, \(n=100, \bar{x}=11.5, s_{e}=0.032\), and \(S_{x x}=1764 .\) Estimate the mean brain volume change for people with a childhood blood lead level of \(20 \mu \mathrm{g} / \mathrm{dL}\), using a \(90 \%\) confidence interval. c. Construct a \(90 \%\) prediction interval for brain volume change for a person with a childhood blood lead level of \(20 \mu \mathrm{g} / \mathrm{dL}\). d. Explain the difference in interpretation of the intervals computed in Parts (b) and (c).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.