Service without a smile. “Service with a smile” is a slogan that many businesses adhere to. However, some jobs (e.g., judges, law enforcement officers, and pollsters) require neutrality when dealing with the public. An organization will typically provide “display rules” to guide employees on what emotions they should use when interacting with the public. A Journal of Applied Psychology (Vol. 96, 2011) study compared the results of surveys conducted using two different types of display rules: positive (requiring a strong display of positive emotions) and neutral (maintaining neutral emotions at all times). In this designed experiment, $\mathbf{145}$undergraduate students were randomly assigned to either a positive display rule condition$\mathbf{\left(}{\mathbf{\text{n}}}_{\mathbf{\text{1}}}\mathbf{\text{=78}}\mathbf{\right)}$or a neutral display rule condition$\mathbf{\left(}{\mathbf{\text{n}}}_{\mathbf{\text{2}}}\mathbf{\text{=67}}\mathbf{\right)}$. Each participant was trained to conduct the survey using the display rules. As a manipulation check, the researchers asked each participant to rate, on a scale of $\mathbf{1}$= “strongly agree” to$\mathbf{5}$= “strongly disagree,” the statement, “This task requires me to be neutral in my expressions.”

a. If the manipulation of the participants was successful, which group should have the larger mean response? Explain.

b. The data for the study (simulated based on information provided in the journal article) are listed in the table above. Access the data and run an analysis to determine if the manipulation was successful. Conduct a test of hypothesis using$\mathbf{\text{α=0.05}}$ .

c. What assumptions, if any, are required for the inference from the test to be valid?

The data is given below

Positive Display Rule: $\begin{array}{l}\mathbf{\text{2433334444444444444544444444444444455555555}}\\ \mathbf{\text{55555555555555555555555555555555555}}\end{array}$

Neutral Display Rule: $\begin{array}{l}\mathbf{\text{3321211122122232212222212222221222222232122}}\\ \mathbf{\text{212122322222222222122222}}\end{array}$

Step by step solution

01

Step-by-Step SolutionStep 1: Calculate the means and standard deviation of both groups

The mean of students displaying Positive Rule =$4.4872$and the mean of students displaying Neutral Rule = role="math" localid="1652701137112" $1.8955$

The standard deviation of students displaying Positive Rule = $0.6595$and the standard deviation of students displaying Neutral Rule = $0.4965$

02

(a) Find the group having a larger mean response 

The mean response of the positive and neutral groups is$4.4872$ and$1.8955$ respectively.

Thus, if the manipulation of the participants is successful, then the positive emotions group has the larger mean response than the neutral emotions group.

Hence, the corresponding members of the groups of positive emotions must disagree with the given statement, which results in a higher response.

03

(b) Conduct z-test

Null Hypothesis: The manipulation was not successful. $H_0:{\mu }_1-{\mu }_2=0$

Alternate Hypothesis: The manipulation was successful $H_a:{\mu }_1-{\mu }_2>0$

Let the confidence level be $90\%$.

Level of significance $\left(\alpha \right)=0.10$

So,role="math" localid="1652701587818" $\frac{\alpha }{2}=0.05$and $z_{0.05}=1.645$

$$\begin{gathered} z=\frac{\overline{x_1}-{\overline{x}}_2}{\sqrt{\left(\frac{{{\sigma }_1}^2}{n_1}+\frac{{{\sigma }_2}^2}{n_2}\right)}} \\ =\frac{4.4872-1.8955}{\sqrt{\left(\frac{{\left(0.6595\right)}^2}{78}+\frac{{\left(0.4965\right)}^2}{67}\right)}} \\ =\frac{2.5917}{0.0962} \\ =26.940 \end{gathered}$$

The critical value is$1.645$

As the value role="math" localid="1652701582886" $z$is more than the critical value, the null hypothesis should be rejected.

Therefore, the data provide sufficient evidence to indicate that${\mu }_1-{\mu }_2>0$ .

04

(c) State the assumption

The assumption is that independent random samples are selected for each population.

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