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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 Farly 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. Twenty-seven infants participated, with 12 exposed to a human model and 15 exposed to the doll. Summary values are at the top of the following page. 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

We are testing whether the mean number of imitations is higher for infants who watch a human model than for infants who watch a doll. Thus, the null hypothesis (H0) is that the mean number of imitations is same for both the groups. The alternative hypothesis (Ha) is that the mean number of imitations is higher for infants who watch a human model. Thus, \(H0: \mu_{Human} = \mu_{Doll}\); \(Ha: \mu_{Human} > \mu_{Doll}\).

02

Calculate the Test Statistic

The test statistic for comparing two means can be calculated using the formula: \(Z = \frac{{\overline{X}_{Human} - \overline{X}_{Doll}}} {{\sqrt{\frac{S_{Human}^2}{n_{Human}} + \frac{S_{Doll}^2}{n_{Doll}}}}}\), where \(\overline{X}_{Human}\) and \(\overline{X}_{Doll}\) are the sample means, \(S_{Human}^2\) and \(S_{Doll}^2\) are the sample variances and \(n_{Human}\) and \(n_{Doll}\) are the sample sizes.

03

Find the P-value

The P-value can be found using the standard normal (Z) distribution table or a calculator that can evaluate P-values. The P-value is the probability of getting a Z as extreme as our calculated Z when the null hypothesis is true. For a one-tailed test (since we are testing whether the mean for human group is higher), we look at the area under the curve beyond our calculated Z.

04

Make a Decision and Interpret

We compare the P-value to the given significance level (0.01). If the P-value is less than 0.01, we reject the null hypothesis. If it is greater, we do not reject. Then we interpret our findings in the context of the problem.

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Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis is a statement that indicates no effect or no difference. It's the starting assumption that any observed differences are due to random chance. In our infant imitation study, the null hypothesis, denoted as \(H_0\), states that the mean number of imitations is the same for both infants watching a human model and those watching a doll. In mathematical terms, this is expressed as \( \mu_{\text{Human}} = \mu_{\text{Doll}} \). The null hypothesis serves as a baseline that the test seeks to challenge by determining if there is enough evidence to suggest a real effect or difference.

Alternative Hypothesis

The alternative hypothesis is the counter-claim to the null hypothesis. In hypothesis testing, it's what you seek evidence for through your data and analysis. Unlike the null hypothesis that assumes no difference, the alternative hypothesis suggests that there is indeed a statistically significant difference. In the context of this study, the alternative hypothesis suggests that infants who observe a human model imitate more frequently than those observing a doll. It is represented mathematically as \( \mu_{\text{Human}} > \mu_{\text{Doll}} \). This directional hypothesis indicates a specific expectation that guides the testing process.

Significance Level

The significance level, often denoted by \(\alpha\), is the threshold used to decide whether the observed data are unlikely under the assumption of the null hypothesis. In this study, a significance level of \(0.01\) is used.

• This means there is a 1% risk of concluding that there is an effect when there is none, generally considered a very stringent criterion for evidence.
• A smaller \(\alpha\) means stricter requirements for evidence.

By choosing a low significance level, the researchers ensure that findings of a significant difference are robust and not just by random variability.

P-value

The P-value is a critical component of hypothesis testing, representing the probability of observing your data—or something more extreme—assuming the null hypothesis is true. In simple terms, it helps decide whether to reject the null hypothesis. For this exercise, once the test statistic is calculated, you compare the P-value to the significance level:

• If the P-value is less than the significance level (0.01 here), you reject the null hypothesis.
• If it is greater, you fail to reject the null hypothesis.

A low P-value indicates that the observed data would be very unlikely if the null hypothesis were true, suggesting that the alternative hypothesis may be more plausible.

Test Statistic

The test statistic is a standardized value that is calculated from sample data during a hypothesis test. It measures how far the sample mean deviates from the null hypothesis mean, relative to the variability of the sample data. In this study, the test statistic is calculated using the formula: \[Z = \frac{{\overline{X}_{\text{Human}} - \overline{X}_{\text{Doll}}}}{{\sqrt{\frac{S_{\text{Human}}^2}{n_{\text{Human}}} + \frac{S_{\text{Doll}}^2}{n_{\text{Doll}}}}}}\]Where:

• \( \overline{X}_{\text{Human}} \) and \( \overline{X}_{\text{Doll}} \) are the sample means of the human and doll groups respectively.
• \( S_{\text{Human}}^2 \) and \( S_{\text{Doll}}^2 \) are the sample variances.
• \( n_{\text{Human}} \) and \( n_{\text{Doll}} \) are the sample sizes.

The calculated \( Z \)-value is then used to find the P-value, determining the statistical significance of the findings.