Question

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.

Short answer

The solution can be reached by conducting a one-tailed t-test. However, specific calculations cannot be made without the actual data. If the p-value obtained is less than the significance level of 0.01, then there is sufficient evidence to accept the alternative hypothesis that infants are more likely to imitate actions modeled by a human than a doll.

Step-by-step solution

Formulating Hypotheses

The null hypothesis (\(H_0\)) is that there is no difference in the mean number of imitations of very young infants who watch a human model than those who watch a doll; mathematically, \(μ1 = μ2\). The Alternative hypothesis (\(H_1\)) is that the mean number of imitations is higher for infants who watch a human model than for infants who watch a doll; mathematically, \(μ1 > μ2\). Here, \(μ1\) and \(μ2\) represent the population means of imitations for infants who watched a human and a doll respectively.

Conducting t-test

Since the problem does not provide specific data, we can represent the conducting t-test step generally. To perform the t-test, use the given values of the results from both groups in the formula for the t-statistic \(t_{observed} = \frac{\bar{x}_1 - \bar{x}_2}{√{(s_1^2/n_1 + s_2^2/n_2)}}\), where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means of the human model and doll groups respectively, \(s_1^2\) and \(s_2^2\) are the sample variances of the two groups respectively, and \(n_1\) and \(n_2\) are the sample sizes of the two groups respectively. This will give us the observed t-statistic value.

Calculating Degrees of Freedom

Degrees of freedom is calculated by adding the number of items in each group and subtracting 2 (\(df = n_1 + n_2 - 2\)). The degree of freedom is used to determine the critical value from the t-distribution table.

Determining the p-value

The p-value is the probability that the results of the experiment occurred by chance. Using the calculated t-statistic and degree of freedom, look up the corresponding p-value in the t-distribution table.

Making the Decision

If the p-value is less than the significance level (0.01 in this case), then it can be concluded that the null hypothesis can be rejected and there is sufficient evidence to say that the mean number of imitations by infants who viewed a human model is greater than those who watched a doll.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is crucial in hypothesis testing. The null hypothesis, denoted as \(H_0\), represents a statement of no effect or no difference. It is the default assumption that there is no relationship between two measured phenomena. In the context of our exercise, the null hypothesis suggests that the mean number of imitations by infants is the same whether they observed a human or a doll, mathematically stated as \(μ1 = μ2\).

The alternative hypothesis, denoted as \(H_1\) or \(H_a\), is a statement that contradicts the null hypothesis. It posits there is an effect or a difference. For the given exercise, the alternative hypothesis claims that infants who observed a human model have a higher mean number of imitations compared to those who observed a doll, formally written as \(μ1 > μ2\). This hypothesis is directional, as it specifies that one mean is greater than the other, rather than simply being different.

When we perform hypothesis testing, we aim to collect evidence to support the alternative hypothesis while simultaneously evaluating the likelihood of the null hypothesis being true. This sets the stage for employing statistical tools to make an informed decision based on data.

T-test Statistical Analysis

The t-test is a statistical analysis used to determine if there is a significant difference between the means of two groups, which may be related in certain features. The t-test takes into account the sample size and variation within the data to help researchers decide if their findings can be generalized to the larger population.

In our exercise, to compare the mean number of imitations between infants exposed to human models and those exposed to dolls, we would use an independent t-test. This test is appropriate when comparing two independent groups, as in our case with two distinct sets of infants. The formula to calculate the observed t-value is: \(t_{observed} = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{(s_1^2/n_1 + s_2^2/n_2)}}\), where \(

\bar{x}_1

\) and \(

\bar{x}_2

\) are the sample means, \(

s_1^2

\) and \(

s_2^2

\) are the sample variances, and \(n_1\) and \(n_2\) are the sample sizes of the human and doll groups, respectively.

The t-test analyzes the size of the difference between groups relative to the variation in the data. A high t-value can indicate a significant difference between group means if the value is higher than what would be expected by random chance.

P-value Significance

The p-value is a fundamental concept in statistical hypothesis testing, offering a measure of the strength of evidence against the null hypothesis. It helps researchers decide whether their results are statistically significant.

The p-value is the probability of obtaining test results at least as extreme as the ones observed during the test, assuming that the null hypothesis is true. So, a low p-value indicates that the observed data are unlikely under the null hypothesis. In the context of our exercise, if the p-value is less than the chosen significance level (0.01, indicating a 1% threshold for declaring a finding statistically significant), we reject the null hypothesis. This would suggest that there is a significant difference in the mean number of imitations between infants exposed to human models and those exposed to dolls.

Researchers typically set a threshold before the study, known as the alpha level (\(α\)), to determine the cut-off for significance. Common alpha levels are 0.05, 0.01, or 0.001. When we say that a result is statistically significant at the 0.01 level, we are confidently stating that there is only a 1% chance that the result is due to random variation and not because of the factor we are testing.

Degrees of Freedom Calculation

Degrees of freedom (df) are essential in the context of t-tests, as they are used to determine the critical values from the t-distribution, which then allow us to calculate the p-value.

The degrees of freedom can be thought of as the number of independent pieces of information that go into the estimation of a parameter. When conducting an independent t-test, as in our infant imitation study, the degrees of freedom are calculated based on the sample size of the two groups being compared. The formula for an independent t-test is \( df = n_1 + n_2 - 2 \), where \(n_1\) and \(n_2\) are the sample sizes of the two groups.

Once the degrees of freedom are calculated, we use this value to reference the correct distribution — the t-distribution — which helps in finding the critical value that corresponds to our significance level. By comparing the observed t-value to this critical value, or by using the degrees of freedom to get the p-value from a t-distribution table or software, we determine whether the results are statistically significant. Thus, degrees of freedom not only influence the shape of the t-distribution but also the accuracy of our significance testing.