Question

In an online psychology experiment sponsored by the University of Mississippi, researchers asked study participants to respond to various stimuli. Participants were randomly assigned to one of three treatment groups: Group $1,$ the simple group, required to respond as quickly as possible after a stimulus was presented; Group $2,$ the go/ no-go group, required to respond to a particular stimulus while disregarding other stimuli; and Group $3,$ the choice group, required to respond differently depending on the stimuli presented. The researcher felt that age may be a factor in determining reaction time, so she organized the experimental units by age and obtained the following data: $$ \begin{array}{ccc} & \text { Stimulus } & \\ \hline \text { Simple } & \text { Go/No-Go } & \text { Choice } \\ \hline 0.248 & 0.338 & 0.586 \\ \hline 0.428 & 0.631 & 0.364 \\ \hline 0.191 & 0.485 & 0.626 \\ \hline 0.303 & 0.389 & 0.858 \\ \hline 0.467 & 0.629 & 0.529 \\ \hline 0.494 & 0.585 & 0.520 \\ \hline 0.384 & 0.782 & 0.854 \\ \hline 0.567 & 0.529 & 0.509 \\ \hline 0.302 & 0.495 & 0.700 \\ \hline \end{array} $$ (a) What type of factorial design is this? How many replications are there within each cell? (b) Normal probability plots indicate that it is reasonable to believe that the data come from populations that are normally distributed. Verify the requirement of equal population variances. (c) Determine if there is significant interaction between stimulus and age. (d) If there is no significant interaction, determine whether there is significant difference in the means for the three types of stimulus. If there is no significant interaction, determine whether there is significant difference in the means for the three categories of age. (e) The residuals are normally distributed. Verify this. (f) Draw an interaction plot of the data to support the results of parts (c) and (d).

Name: Problem 18, Chapter 13: Statistics Informed Decisions Using Data
Uploaded: 2024-01-22T09:03:23-08:00
Channel: Shu Naito

   In an online psychology experiment sponsored by the University of Mississippi, researchers asked study participants to respond to various stimuli. Participants were randomly assigned to one of three treatment groups:
Group $1,$ the simple group, required to respond as quickly as possible after a stimulus was presented; Group $2,$ the go/ no-go group, required to respond to a particular stimulus while disregarding other stimuli; and Group $3,$ the choice group, required to respond differently depending on the stimuli presented. The researcher felt that age may be a factor in determining reaction time, so she organized the experimental units by age and obtained the following data:
$$
\begin{array}{ccc} 
& \text { Stimulus } & \\
\hline \text { Simple } & \text { Go/No-Go } & \text { Choice } \\
\hline 0.248 & 0.338 & 0.586 \\
\hline 0.428 & 0.631 & 0.364 \\
\hline 0.191 & 0.485 & 0.626 \\
\hline 0.303 & 0.389 & 0.858 \\
\hline 0.467 & 0.629 & 0.529 \\
\hline 0.494 & 0.585 & 0.520 \\
\hline 0.384 & 0.782 & 0.854 \\
\hline 0.567 & 0.529 & 0.509 \\
\hline 0.302 & 0.495 & 0.700 \\
\hline
\end{array}
$$
(a) What type of factorial design is this? How many replications are there within each cell?
(b) Normal probability plots indicate that it is reasonable to believe that the data come from populations that are normally distributed. Verify the requirement of equal population variances.
(c) Determine if there is significant interaction between stimulus and age.
(d) If there is no significant interaction, determine whether there is significant difference in the means for the three types of stimulus. If there is no significant interaction, determine whether there is significant difference in the means for the three categories of age.
(e) The residuals are normally distributed. Verify this.
(f) Draw an interaction plot of the data to support the results of parts (c) and (d).

Statistics Informed Decisions Using Data

Michael Sullivan III 5th Edition

Chapter 13, Problem 18

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Solved by Expert Shu Naito

01/22/2024

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There are 3 replications within each cell, as there are 3 data points for each combination of stimulus and age. Show more…

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In an online psychology experiment sponsored by the University of Mississippi, researchers asked study participants to respond to various stimuli. Participants were randomly assigned to one of three treatment groups: Group $1,$ the simple group, required to respond as quickly as possible after a stimulus was presented; Group $2,$ the go/ no-go group, required to respond to a particular stimulus while disregarding other stimuli; and Group $3,$ the choice group, required to respond differently depending on the stimuli presented. The researcher felt that age may be a factor in determining reaction time, so she organized the experimental units by age and obtained the following data: $$ \begin{array}{ccc} & \text { Stimulus } & \\ \hline \text { Simple } & \text { Go/No-Go } & \text { Choice } \\ \hline 0.248 & 0.338 & 0.586 \\ \hline 0.428 & 0.631 & 0.364 \\ \hline 0.191 & 0.485 & 0.626 \\ \hline 0.303 & 0.389 & 0.858 \\ \hline 0.467 & 0.629 & 0.529 \\ \hline 0.494 & 0.585 & 0.520 \\ \hline 0.384 & 0.782 & 0.854 \\ \hline 0.567 & 0.529 & 0.509 \\ \hline 0.302 & 0.495 & 0.700 \\ \hline \end{array} $$ (a) What type of factorial design is this? How many replications are there within each cell? (b) Normal probability plots indicate that it is reasonable to believe that the data come from populations that are normally distributed. Verify the requirement of equal population variances. (c) Determine if there is significant interaction between stimulus and age. (d) If there is no significant interaction, determine whether there is significant difference in the means for the three types of stimulus. If there is no significant interaction, determine whether there is significant difference in the means for the three categories of age. (e) The residuals are normally distributed. Verify this. (f) Draw an interaction plot of the data to support the results of parts (c) and (d).

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Key Concepts

Residual Analysis

Residual analysis involves examining the differences between observed and predicted values to assess whether the assumptions underlying the statistical model are satisfied. In particular, examining residuals for normality is critical in confirming that errors are normally distributed, a necessary condition for valid inference in many parametric tests, including ANOVA. Techniques such as normal probability plots are commonly used to visually assess this assumption.

Interaction Plots

Interaction plots are graphical representations that display the means of the dependent variable for different combinations of factors. They are used to visually assess the presence or absence of interaction effects between factors. If the lines in an interaction plot are parallel, this generally suggests no interaction. Conversely, non-parallel lines may indicate that the effect of one factor depends on the level of another, highlighting the need for further analysis of interaction effects.

Main Effects

Main effects refer to the individual impact of each factor on the dependent variable, ignoring the interaction with other factors. In factorial experiments, after assessing and potentially ruling out significant interaction effects, the main effects can be examined to understand how each factor independently influences the outcome. This forms the basis for drawing conclusions about the overall effect of a factor on the measured response.

Factorial Design

A factorial design is an experimental strategy that involves two or more factors, each with multiple levels, where all possible combinations of factor levels are investigated. This design allows researchers not only to assess the individual effects (main effects) of each factor, but also to explore potential interactions between factors, where the effect of one factor may depend on the level of another factor.

Homogeneity of Variance

The assumption of homogeneity of variance, also referred to as equal variances, is a key requirement in many parametric tests such as ANOVA. It assumes that the different groups or treatments have similar variability. Verifying this assumption typically involves graphical methods like normal probability plots or statistical tests such as Levene's or Bartlett's test, which help determine whether the differences in variances across groups are statistically insignificant.

Replication

Replication in the context of a factorial design refers to the number of times each treatment combination is independently repeated. Replications provide an estimate of experimental error and increase the reliability of conclusions drawn from the study. They are crucial for detecting differences between treatment effects and for validating the assumptions underlying the statistical analysis.

Interaction Effects

Interaction effects occur when the effect of one independent variable on the dependent variable differs depending on the level of another independent variable. Assessing interaction effects is essential because significant interactions can indicate that the simple main effects (effects of one factor alone) do not provide a complete understanding of the combined influence of the factors. In the context of factorial designs, exploring interactions is fundamental to understanding the dynamics between factors.

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