91Ó°ÊÓ

Are very young infants more likely to imitate actions that are modeled by a person or simulated by an object? This question was the basis of a research study summarized in the article "The Role of Person and Object in Eliciting Early Imitation" (Journal of Experimental Child Psychology [1991]: 423-433). One action examined was mouth opening. This action was modeled repeatedly by either a person or a doll, and the number of times that the infant imitated the behavior was recorded. Twentyseven infants participated, with 12 exposed to a human model and 15 exposed to the doll. Summary values are given here. Is there sufficient evidence to conclude that the mean number of imitations is higher for infants who watch a human model than for infants who watch a doll? Test the relevant hypotheses using a .01 significance level.

Step by step solution

01

Formulate the Hypotheses

The null hypothesis (H0) assumes no significant difference between the means of imitations by the two groups, i.e., the mean number of imitations of infants exposed to a human model and a doll are equal. The alternative hypothesis (HA) is that the mean number of imitations of infants exposed to a human model is greater. A one-tailed test is considered here because the problem is interested in whether the mean of the first group is higher.

02

Consider the Significance Level

The problem suggests a .01 significance level. This level of 0.01 means that, if the null hypothesis is true, there's a 1% chance that the sample data will fall in the critical area of the one-tailed test statistic distribution.

03

Calculate the Test Statistic

A t-test is applied here because we're comparing two means and we're likely not knowing the population standard deviations. The t-statistic is given by \(t = \frac{ (\bar{X}_{human} - \bar{X}_{doll}) - (µ_{human} - µ_{doll}) }{ \sqrt{ \frac{s_{human}^2 }{N_{human}} + \frac{s_{doll}^2 } {N_{doll}} } }\), where \(\bar{X}\) is the sample mean, µ is the assumed population mean under the null hypothesis, s is the sample standard deviation, and N is the sample size.

04

Determine the Critical Value

The critical value is the t-value that delineates the critical region boundary, where if the test statistic falls into this region, the null hypothesis is rejected. In a one-tailed test with significance level 0.01 and degree of freedom given by the total sample size minus 2, we can obtain the critical value from the t-distribution table.

05

Make a Decision

If the calculated t value is greater than the critical value, reject the null hypothesis, meaning there is sufficient evidence to conclude the mean number of imitations is higher for infants who watched a human model. Else, do not reject the null hypothesis and conclude there isn’t enough evidence to support the claim.

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Key Concepts

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

T-Test

The T-Test is a statistical method used to determine if there is a significant difference between the means of two groups. In the context of the exercise, a T-Test helps assess whether infants imitate more when exposed to a human model compared to a doll. The choice of a T-Test is suitable when the data sample sizes are small, as is often the case in psychological studies with infants.

The calculation involves comparing the observed difference between groups to what might be observed under the null hypothesis, which assumes no difference. The T-Test formula we use is \[t = \frac{(\bar{X}_{human} - \bar{X}_{doll})}{\sqrt{ \frac{s_{human}^2}{N_{human}} + \frac{s_{doll}^2}{N_{doll}} }}\]

• \(\bar{X}\) stands for the sample mean.
• \(s\) is the sample standard deviation.
• \(N\) represents the number of subjects in each group.

The T-Test gives us a test statistic that we can compare against a critical value to make our decision.

Significance Level

The significance level, often denoted as \(\alpha\), is a crucial concept in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true. In the exercise, a significance level of 0.01 is used. This means there's only a 1% risk of concluding that there is a difference between the two groups when there isn't one.

This low significance level indicates the researcher wants to be very confident about their results. In practical terms, it makes the test more stringent. To make a decision, you compare the p-value obtained from the T-test with the significance level. If the p-value is smaller than 0.01, you reject the null hypothesis.

• Low \(\alpha\) means you need stronger evidence to support the alternative hypothesis.
• Reflects the trade-off between Type I error (rejecting a true null hypothesis) and Type II error (failing to reject a false null hypothesis).

Null Hypothesis

The null hypothesis, denoted as \(H_0\), is the statement that there is no effect or no difference. For the exercise concerning infant imitation, the null hypothesis claims that the mean imitations by infants watching a human model are equal to those watching a doll.

Formulating the null hypothesis is the first step in the hypothesis testing process. It's a statement of "no effect" or "no difference." The null hypothesis is what we test against our data. It provides a baseline measure, allowing us to judge whether observed data significantly deviate from expectations under this assumption.

• \(H_0: \mu_{human} = \mu_{doll}\)
• Hypothesis testing aims to see if data provide enough evidence against \(H_0\).
• Rejecting \(H_0\) suggests that there's substantial evidence for the alternative hypothesis.

Alternative Hypothesis

The alternative hypothesis, represented by \(H_A\), is what we want to prove. In contrast to the null hypothesis, it states that there is an effect or a difference. For our exercise, the alternative hypothesis is that infants imitate more frequently when observing a human model than a doll.

This hypothesis is critical because it defines the research question's direction. In this case, the hypothesis is one-tailed because we are only checking if the mean for the human model group is greater.

• \(H_A: \mu_{human} > \mu_{doll}\)
• One-tailed tests are directional; they focus on one specific outcome (greater or lesser).
• Provides a framework for the research to conclude if there's directional support.

Expanding on this hypothesis is vital as it carries the implication of discovering new psychological insights about imitation behaviors in infants.