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Chapter 21: Problem 20

Do positive or negative messages have a greater effect on behavior? Forty-two subjects were randomly assigned to one of two treatment groups, 21 per group. In one group, subjects read an introduction about a nonprofit organization followed by a negative message encouraging subjects to sign a petition for cleaner lakes. In the other group, subjects read the same introduction followed by a positive message encouraging subjects to sign the petition. All subjects were then tested to determine how strongly they intended to sign the petition, with higher scores indicating greater intent. To compare the mean scores for the two groups of subjects using the two-sample \(t\) procedures with the conservative Option 2, the correct degrees of freedom is (a) 20 . (b) 40 . (c) 41 .

Short Answer

Expert verified
The correct degrees of freedom is 40 (option b).

Step by step solution

01

Understanding Two-Sample t-Test

First, recognize that we are dealing with a two-sample t-test since we are comparing the means of two independent groups (positive vs. negative messages). Each group has 21 subjects.
02

Determining Sample Sizes

Identify the sample size for each group. In this case, both the groups have 21 subjects, so the sizes are equal: \( n_1 = 21 \) and \( n_2 = 21 \).
03

Calculate Degrees of Freedom

To find the degrees of freedom (df) using the conservative Option 2 for a two-sample t-test with equal sample sizes, use the formula \( n_1 + n_2 - 2 \). This formula accounts for both groups' contributions without complex calculations.
04

Plug Values Into the Formula

Substitute the known values for \( n_1 \) and \( n_2 \) into the degrees of freedom formula: \( df = 21 + 21 - 2 \).
05

Compute Degrees of Freedom

Simplify the expression to find the degrees of freedom: \( df = 42 - 2 = 40 \). Hence, the answer is (b) 40.

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Key Concepts

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

Degrees of Freedom
Understanding the concept of "Degrees of Freedom" is crucial for statistical analysis, especially when dealing with a two-sample t-test. Degrees of Freedom (df) represent the number of independent values or quantities that can vary in the analysis after certain constraints or conditions are applied.In the context of a two-sample t-test, degrees of freedom are calculated to evaluate the variability between the two groups.This variability helps to determine the significance of the difference in means, which is central to validating the hypothesis.In simpler terms, consider degrees of freedom as the amount of information we have about our data that is "free" to vary when estimating statistical parameters.
For our problem:
  • We have two groups with 21 subjects each.
  • The degrees of freedom using the conservative formula (Option 2) is calculated as: \[ df = n_1 + n_2 - 2 \]
  • Inserting our sample sizes: \[ df = 21 + 21 - 2 = 40 \]
Statistical Significance
The term "Statistical Significance" refers to the likelihood that the observed difference or relationship in a study is not due to random chance. In our two-sample t-test, we are looking to see if the difference between the effects of positive and negative messages on subjects' intentions is statistically significant. To determine statistical significance:
  • A null hypothesis is usually established, indicating no difference between the groups.
  • A p-value is calculated during the t-test; a common threshold for significance is a p-value less than 0.05.
  • If the resultant p-value is below this threshold, we can reject the null hypothesis and state that the difference observed is statistically significant.
This indicates that the observed effect (whether positive or negative messages have a greater impact) is unlikely to have occurred by random chance alone.
Comparative Analysis
Comparative Analysis is a key statistical tool used to evaluate the differences between two or more groups based on specific variables. In this exercise, we are comparing two groups of subjects to understand the impact of positive versus negative messaging on their intention to sign a petition. Here's how we perform a comparative analysis using a two-sample t-test:
  • The main goal is to see if there's a significant difference in the mean scores of the groups.
  • Each individual鈥檚 reaction or score represents their intent after receiving different types of messages.
  • By comparing the means of these scores, we can comprehend if the message type affected the subjects' behaviors.
Comparative analysis in this context helps to highlight how different interventions (positive or negative messages) might alter behavior, providing insights for behavioral studies and strategic communications.
Behavioral Study
A "Behavioral Study" refers to research conducted to observe and analyze how individuals react to different stimuli or situations. In this exercise, the goal is to understand how positive versus negative messages impact behavior, specifically the intent to sign a petition for cleaner lakes. Key factors in a behavioral study include:
  • The type of message (positive or negative) each group receives is the independent variable.
  • The intent score after reading the message is the dependent variable, reflecting the subject's behavior.
  • Random assignment of subjects to each group ensures unbiased results, increasing the reliability of the study's findings.
Behavioral studies like this provide valuable insights into human psychology and can inform strategies in communication, marketing, and advocacy efforts.
They help decode underlying patterns and influences in decision-making, shedding light on how people can be motivated to take action.

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Most popular questions from this chapter

Do movie trailers for eventual box-office hit movies produce greater brain activity than trailers for eventual box-office failures? \({ }^{8}\) Seven subjects viewed movie trailers for eventual box-office hits (boxoffice revenues of more than \(\$ 80\) million) and their brain activity was measured using electroencephalography (EEG). Similarly, 11 subjects viewed movie trailers for eventual box-office failures (box-office revenues of less than \(\$ 20\) million) and their brain activity was also measured. Brain activity was reported as a standardized score. Here are the standardized scores: $$ \begin{array}{ccccccccccc} &&&&{\text { Box-office hits }} \\ \hline-0.06 & -0.06 & -0.04 & 0.30 & 0.32 & 0.54 & 0.72 & & & & \\ &&&&{\text { Box-office failures }} \\ \hline-0.20 & -0.18 & -0.27 & -0.25 & -0.13 & -0.07 & 0.00 & 0.09 & 0.11 & 0.22 & 0.62 \\ \hline \end{array} $$ (a) Make stemplots to investigate the shapes of the distributions. The brain activities for both box-office hits and failures are skewed to the right and the brain activity for box-office failures has a high outlier. Researchers were uncertain whether this movie should have been classified as a box-office failure, and because of this uncertainty, we believe there is some justification for removing the outlier. (b) We suspect that brain activity for box-office hits is larger than that for boxoffice failures. Do the data (with the outlier removed) support this suspicion?

In the last 10 years, several authors have stated that people are miserable in their jobs and have become increasingly unhappy over time. Job satisfaction scores from participants in the General Social Survey \((\mathrm{GSS})\) in 1975 and 2006 were studied. Higher scores indicate greater satisfaction, with \(2.5\) being a neutral score. Here are summaries of scores for 1975 and \(2006:^{16}\) $$ \begin{array}{lccc} \hline \text { Year } & \text { Sample Size } & \text { Mean } & \text { Standard Deviation } \\ \hline 1975 & 1165 & 3.37 & 0.81 \\ \hline 2006 & 2177 & 3.32 & 0.80 \\ \hline \end{array} $$ (a) Is there a significant decrease in mean scores from 1975-2006? What do these data show about job satisfaction in 1975 compared to 2006? (b) The paper from which the data came includes several years from \(1972-\) 2006. 1975 was the year with the highest mean score. What, if any, effect does this information have on your assessment of whether job satisfaction has decreased over time?

In the 2015 NAEP sample of 12th-graders in the United States, the mean mathematics scores were 150 for female students and 153 for male students. To see if this difference is statistically significant, you would use (a) the two-sample \(t\) test. (b) the matched pairs \(t\) test. (c) the one-sample \(t\) test.

Height and the big picture. Forty-six college students were randomly divided into two groups of size 23. One group was asked to imagine being on the upper floor of a tall building and the other on the lowest floor. Participants were then asked to choose between a job that required more detail orientation versus a job that required a more big-picture orientation. They rated their job preferences on an 11-point scale, with higher numbers corresponding to a greater preference for the big-picture job. Here are the summary statistics: 17 $$ \begin{array}{lccc} \hline \text { Group } & \text { Group Size } & \text { Mean } & \text { Standard Deviation } \\ \hline \text { Low } & 23 & 4.61 & 3.08 \\ \hline \text { High } & 23 & 6.68 & 3.45 \\ \hline \end{array} $$ (a) What degrees of freedom would you use in the conservative two-sample \(t\) procedures to compare the lower and higher floor groups? (b) What is the two-sample \(t\) test statistic for comparing the mean job preference ratings for the two groups? (c) Test the null hypothesis of no difference between the two population means against the two-sided alternative. Use your statistic from part (b) with degrees of freedom from part (a).

Although painful experiences are involved in social rituals in many parts of the world, little is known about the social effects of pain. Will sharing a painful experience in a small group lead to greater bonding of group members than sharing a similar nonpainful experience? Fiftyfour university students in South Wales were divided at random into a pain group containing 27 students and a no-pain group containing the remaining 27 students. Pain was induced by two tasks. In the first task, students submerged their hands in freezing water for as long as possible, moving metal balls at the bottom of the vessel into a submerged container. In the second task, students performed a standing wall squat with back straight and knees at 90 degrees for as long as possible. The no-pain group completed the first task using room temperature water for 90 seconds and the second task by balancing on one foot for 60 seconds, changing feet if necessary. In both the pain and no-pain settings, the students completed the tasks in small groups which typically consisted of four students and contained similar levels of group interaction. Afterward, each student completed a questionnaire to create a bonding score based on responses to seven statements such as, "I feel the participants in this study have a lot in common," or "I feel I can trust the other participants." Each response was scored on a five-point scale ( 1 = strongly disagree, 5 = strongly agree), and the scores on the seven statements were averaged to create a bonding score for each subject. Here are the bonding scores for the subjects in the two groups: \({ }^{26}\) $$ \begin{array}{llllllll} \hline \text { No-pain group: } & 3.43 & 4.86 & 1.71 & 1.71 & 3.86 & 3.14 & 4.14 \\ & 3.14 & 4.43 & 3.71 & 3.00 & 3.14 & 4.14 & 4.29 \\ & 2.43 & 2.71 & 4.43 & 3.43 & 1.29 & 1.29 & 3.00 \\ & 3.00 & 2.86 & 2.14 & 4.71 & 1.00 & 3.71 & \\ \hline \text { Pain group: } & 4.71 & 4.86 & 4.14 & 1.29 & 2.29 & 4.43 & 3.57 \\\ & 4.43 & 3.57 & 3.43 & 4.14 & 3.86 & 4.57 & 4.57 \\ & 4.29 & 1.43 & 4.29 & 3.57 & 3.57 & 3.43 & 2.29 \\ & 4.00 & 4.43 & 4.71 & 4.71 & 2.14 & 3.57 & \\ \hline \end{array} $$ Do the data show that sharing a painful experience in a small group leads to higher bonding scores for group members than sharing a similar nonpainful experience?
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