Question

An article in USA Today (October 19,2010 ) described a study to investigate how young children learn. Sixty-four toddlers age 18 months participated in the study. The toddlers were allowed to play in a lab equipped with toys and which had a robot that was hidden behind a screen. The article states: "After allowing the infants playtime, the team removed the screen and let the children see the robot. In some tests, an adult talked to the robot and played with it. In others the adult ignored the robot. After the adult left the room, the robot beeped and then turned its head to look at a toy to the side of the infant." The researchers planned to see if the resulting data supported the claim that children are more likely to follow the robot's gaze to the toy when they see an adult play with the robot than when they see an adult ignore the robot.

Short answer

Without specific data from the experiment, a concrete answer cannot be given. However, through outlining how to conduct hypothesis testing, one would review the gathered data and conduct statistical tests to see if there is a significant difference in the children's behavior based on whether the adult interacted with the robot or not. The final conclusion would be made based on the resulting p-value compared to the significance level.

Step-by-step solution

Formulate the Null and Alternative Hypotheses

The null hypothesis (H0) could be: children are not more likely to follow the robot's gaze to the toy when they see an adult play with the robot, while the alternative hypothesis (H1) would be: children are more likely to follow the robot's gaze to the toy when they see an adult play with the robot.

Collect & Analyze the Data

Next, one would need to look at the data from the experiment, specifically tracking whether each child followed the robot's gaze or not in both conditions - when an adult has interacted with the robot and when they haven't.

Compute the Test Statistic

This would likely be a chi-square test or a z-test to measure if the difference in proportions in both groups (children following the gaze when adult interacted and children following the gaze when adult ignored) is statistically significant.

Make a Decision

With the calculated test statistic and its corresponding p-value, one must compare this p-value to the selected significance level (usually 0.05). If the p-value is less than the significance level, then the null hypothesis is rejected, if not, then one does not have enough evidence to reject the null hypothesis.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is essential in any statistical analysis. The null hypothesis, denoted as H₀, represents a default state or a position that indicates no effect or no difference. In the context of the child learning study, the null hypothesis would be that the toddlers are equally likely to follow the robot's gaze to the toy irrespective of whether an adult played with the robot or not.

On the other hand, the alternative hypothesis, symbolized as H₁ or H_a, is what researchers want to demonstrate. It represents a new theory or an expectation of a meaningful difference. In this case, the alternative hypothesis is that the toddlers are more likely to follow the robot's gaze to the toy after witnessing an adult interacting with the robot.

• The null hypothesis serves as a starting point for statistical testing;
• The alternative hypothesis is what the study aims to support with evidence.

Formulating these hypotheses correctly is a crucial step in any experimental design as they help in constructing the structure for data analysis and statistical testing.

Data Analysis

After setting the hypotheses, data analysis comes into play. Data analysis refers to the process of inspecting, cleansing, transforming, and modeling data to discover useful information, inform conclusions, and support decision-making. In the robot gaze study, researchers need to meticulously track each toddler's response: did the child follow the gaze of the robot or not, and under which condition did this response occur?

The analysis will involve comparing the proportion of toddlers who followed the gaze in both conditions - with adult interaction and without. This comparison can be done through various statistical methods, helping to determine if there is a significant difference in the toddlers' behavior based on the adult's interaction with the robot.

• Data must be thoroughly collected and checked for accuracy;
• Analysis methods must be chosen based on the nature of the data and the hypotheses.

Statistical software or even manual computation can assist in organizing and interpreting the data, leading to conclusions that can either refute or support the initial hypotheses.

Test Statistic

The test statistic is a crucial component that bridges data analysis and decision-making. It is a standardized value that is calculated from sample data during a hypothesis test. Essentially, it's a tool that helps determine whether the observed data deviates significantly from what was expected under the null hypothesis.

In the context of the child learning study, a test statistic will quantify the difference between the observed behaviors of the toddlers in the two scenarios: adult playing with the robot vs. adult ignoring the robot. Depending on the type of data and the hypotheses, the study might use a chi-square test statistic, a z-test, or another appropriate statistical test.

• The test statistic reflects how far the data departs from the null hypothesis;
• The choice of statistic depends on the experimental design and data type.

Once we have the test statistic, it'll help us determine the p-value, which is used to draw conclusions about the hypotheses.

Significance Level

The significance level, often denoted as α, is a threshold that determines when we reject the null hypothesis. It quantifies the risk level we are willing to take for incorrectly rejecting H₀ when it is actually true (a Type I error). Commonly, a significance level of 0.05 is used, implying a 5% risk of a Type I error.

In the example of our toddler study, if the p-value obtained from the test statistic is less than the chosen significance level, it suggests that the observed data is highly unlikely to occur if the null hypothesis were true. Thereby, we would reject the null hypothesis in favor of the alternative.

• Lower significance levels mean lower chances of committing a Type I error;
• The significance level is pre-set before analyzing data to avoid biased decisions.

The significance level is a critical concept as it provides a criterion for making a binary decision in hypothesis testing – whether to reject the null hypothesis or not given the evidence presented by the data.