91Ó°ÊÓ

Exercises 25 to 28 refer to the following setting. Does the color in which words are printed affect your ability to read them? Do the words themselves affect your ability to name the color in which they are printed? Mr. Starnes designed a study to investigate these questions using the 16 students in his AP \(^{\text {R }}\) Statistics class as subjects. Each student performed two tasks in a random order while a partner timed: ( 1 ) read 32 words aloud as quickly as possible, and ( 2 ) say the color in which each of 32 words is printed as quickly as possible. Try both tasks for yourself using the word list below $$ \begin{array}{llll} \text { YELLOW } & \text { RED } & \text { BLUE } & \text { GREEN } \\ \text { RED } & \text { GREEN } & \text { YELLOW } & \text { YELLOW } \\ \text { GREEN } & \text { RED } & \text { BLUE } & \text { BLUE } \\ \text { YELLOW } & \text { BLUE } & \text { GREEN } & \text { RED } \\ \text { BLUE } & \text { YELLOW } & \text { RED } & \text { RED } \\ \text { RED } & \text { BLUE } & \text { YELLOW } & \text { GREN } \\ \text { BLUE } & \text { GREEN } & \text { GREEN } & \text { BLUE } \\ \text { GREEN } & \text { YELLOW } & \text { RED } & \text { YELLOW } \end{array} $$ Color words (3.1,3.2,12.1) Can we use a student's word task time to predict his or her color task time? (a) Make an appropriate scatterplot to help answer this question. Describe what you see. (b) Use your calculator to find the equation of the leastsquares regression line. Define any symbols you use. (c) Find and interpret the residual for the student who completed the word task in 9 seconds. (d) Assume that the conditions for performing inference about the slope of the true regression line are met. The \(P\) -value for a test of \(H_{0}: \beta=0\) versus \(H_{a}: \beta>0\) is \(0.0215 .\) Explain what this value means in context.

Step by step solution

01

Create a Scatterplot

To create a scatterplot, first denote each student's word task time as the independent variable (x) and the color task time as the dependent variable (y). Plot each student's data point with their word task time on the x-axis and their color task time on the y-axis. A visual inspection of the scatterplot helps to identify any correlation between the two variables.

02

Describe the Scatterplot

Look at the scatterplot to determine the form of the relationship. If the points seem to cluster around a straight line, this indicates a linear relationship. Identify whether the relationship is positive (as word task time increases, color task time also increases) or negative, and note the strength (how tightly the points cluster around the line).

03

Find the Least-Squares Regression Line

Using a calculator or statistical software, input the data for word task time and color task time to calculate the least squares regression line. The equation will be of the form \( y = a + bx \), where \( y \) is the predicted color task time, \( x \) is the word task time, \( a \) is the y-intercept, and \( b \) is the slope of the line.

04

Define Symbols

In the regression equation \( y = a + bx \), \( y \) represents the predicted time to complete the color task, \( x \) represents the time to complete the word task, \( a \) is the y-intercept (the predicted color task time when the word task time is zero), and \( b \) is the slope (the expected change in color task time for each additional second taken on the word task).

05

Calculate the Residual for 9 Seconds

Find the residual for the student who completed the word task in 9 seconds. Using the regression line equation, calculate the predicted color task time when \( x = 9 \). The residual is the actual color task time minus the predicted color task time.

06

Interpret the Residual

A positive residual means the actual color task time was longer than predicted, while a negative residual means it was shorter. Interpret the residual value to understand if the model overestimated or underestimated the student's color task time.

07

Explain the P-value

The hypothesis test is: \(H_{0}: \beta=0\) (no association between word and color task times) versus \(H_{a}: \beta>0\) (positive association exists). A \(P\)-value of 0.0215 indicates that there is a 2.15% chance of observing the data assuming the null hypothesis is true. Since this \(P\)-value is typically below common significance levels (e.g., 0.05), it suggests rejecting the null hypothesis, indicating evidence of a positive association between word and color task times.

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 91Ó°ÊÓ!

Key Concepts

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

Regression Analysis

Regression analysis is a powerful statistical method used to examine the relationship between two or more variables. In the exercise with students' word and color task times, we aim to see if we can predict one task's duration based on the other. First, we describe the relationship using a scatterplot. Then, we determine the best-fitting line, known as the least squares regression line. This line's equation takes the form: - \( y = a + bx \) - \(y\) is the predicted value, - \(x\) is the independent variable (word task time here), - \(a\) is the y-intercept, - \(b\) is the slope. This slope (\(b\)) tells us how much the dependent variable (color task time) is expected to change with a one-unit change in the independent variable. Finding these parameters is crucial for making predictions and understanding correlations between the word and color task times.

Scatterplot

A scatterplot is a type of graph used to visually display and assess the relationship between two numerical variables. Each point represents a pair of values, one from each variable, plotted on a two-dimensional graph. In this exercise, each point on the scatterplot represents a student's respective word task and color task times. - The x-axis typically shows the independent variable (here, word task time). - The y-axis represents the dependent variable (color task time). When we plot these data points, we're able to visually inspect whether there's a trend or correlation. If the data points cluster around an increasing line, it suggests a positive correlation. Conversely, if they cluster around a decreasing line, it points to a negative correlation. The scatterplot provides a quick visual cue about the nature and strength of the relationship, whether linear or otherwise.

Hypothesis Testing

Hypothesis testing is a statistical method that helps you decide whether your data supports a specific hypothesis. In the context of this exercise, we're testing whether there's a significant relationship between the word and color task times. The hypotheses are defined as:- Null hypothesis \(H_{0}: \beta=0\): Suggests no association exists between the two times.- Alternative hypothesis \(H_{a}: \beta>0\): Suggests a positive association exists.A critical element of hypothesis testing is the \(P\)-value, which quantifies the probability of observing the given data, assuming the null hypothesis is true. A low \(P\)-value (commonly below 0.05) suggests that the observed data is unlikely under the null hypothesis, leading us to reject the null hypothesis. For this exercise, a \(P\)-value of 0.0215 suggests we have evidence that a positive correlation between the task times exists.

Residuals

Residuals are a key concept in regression analysis, representing the difference between the observed value and the predicted value from the regression line. Understanding residuals helps determine how well your regression model fits the data. In this exercise, after calculating the least squares regression line, we can compute residuals for each student.- **Formula**: \(\text{Residual} = \text{Observed value} - \text{Predicted value}\)A positive residual indicates that the actual task time was longer than predicted, suggesting an underestimation by the model. Conversely, a negative residual means the actual task was completed faster than predicted, pointing to overestimation by the model. Analyzing residuals can highlight patterns, suggesting areas where the model might be improved, or uncovering anomalies within your data.