91Ó°ÊÓ

Do iPads Help Kindergartners Learn: A Subtest The Auburn, Maine, school district conducted an early literacy experiment in the fall of 2011 . In September, half of the kindergarten classes were randomly assigned iPads (the intervention group) while the other half of the classes got them in December (the control group.) Kids were tested in September and December and the study measures the average difference in score gains between the control and intervention group. \(^{41}\) The experimenters tested whether the mean score for the intervention group was higher on the HRSIW subtest (Hearing and Recording Sounds in Words) than the mean score for the control group. (a) State the null and alternative hypotheses of the test and define any relevant parameters. (b) The p-value for the test is 0.02 . State the conclusion of the test in context. Are the results statistically significant at the \(5 \%\) level? (c) The effect size was about two points, which means the mean score for the intervention group was approximately two points higher than the mean score for the control group on this subtest. A school board member argues, "While these results might be statistically significant, they may not be practically significant." What does she mean by this in this context?

Step by step solution

01

Null and Alternative Hypotheses

The null hypothesis (H0) typically proposes that there is no effect of the intervention. In this context, it means there is no difference in the mean scores between the intervention group (those with iPads) and the control group (those without iPads initially). So, H0: \(\mu_{int} = \mu_{ctrl}\), where \(\mu_{int}\) and \(\mu_{ctrl}\) are the population mean scores for the intervention and control groups respectively.The alternative hypothesis (H1) proposes the contrary, specifically that there is an effect. In this case, it would be that the intervention group's mean is higher than the control group's. Therefore, H1: \(\mu_{int} > \mu_{ctrl}\)

02

Interpreting the P-value

The p-value is a measure of how extreme the data are. In this case, a p-value of 0.02 suggests that the likelihood of seeing the observed data (or more extreme), given that the null hypothesis is true, is just 0.02, or 2%.

03

Statistical Significance

A result is typically considered statistically significant if the p-value is less than a predetermined threshold (commonly 5%). Since the given p-value of 0.02 is less than 0.05, the result is statistically significant at the 5% level. This means that there is significant evidence to reject the null hypothesis in favor of the alternative hypothesis - suggesting that the iPads had a significant effect on the mean score of kindergarteners.

04

Statistical Significance vs. Practical Significance

Practical significance refers to the magnitude of the difference, whether it's large enough to be of value in a practical sense. The effect size here is about two points. This implies that while there may be a statistically significant improvement, whether this improvement is practically significant (i.e., large enough to bring substantial difference or value in real world application) will depend on the conventional standards, professional judgement or the context of usage. The school board member is pointing out that while statistically there is an improvement, it may not bring substantial difference or value in practical usage or application.

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Key Concepts

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

Null and Alternative Hypotheses

In statistical hypothesis testing, we start with two competing statements: the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis is a general statement or default position that suggests there is no difference or effect. It is essentially the position that any kind of difference or significance you see in your data is due to random chance. In the context of the kindergarten iPad study, the null hypothesis would be that the average scores of children who used iPads and those who didn't are the same; symbolically, this is represented as \( H_0: \mu_{int} = \mu_{ctrl} \).

The alternative hypothesis suggests that there is a meaningful effect or difference, challenging the status quo or norm represented by the null hypothesis. In our exercise, the alternative hypothesis posits that using iPads leads to higher average scores, expressed as \( H_1: \mu_{int} > \mu_{ctrl} \). Establishing clear hypotheses is crucial as it guides the statistical tests and analysis that will follow.

P-value Interpretation

The p-value is a critical concept in understanding the results of hypothesis tests. It represents the probability of observing your data, or something more extreme, if the null hypothesis is true. Thus, a smaller p-value indicates that your observed data would be unlikely under the assumption of the null hypothesis.

In our experiment with iPads, a p-value of 0.02 implies that there is a 2% chance of obtaining the observed difference in average scores assuming the null hypothesis is true (i.e., assuming there is actually no difference between the groups). With this small p-value, we conclude that the data observed is rare enough under the null hypothesis to challenge its credibility. However, it's important to note that a p-value does not measure the probability that the null hypothesis is true or false, nor does it indicate the size of an effect or its practical significance.

Statistical vs Practical Significance

It is possible for a result to be statistically significant but not practically significant. Statistical significance refers to the likelihood that a result or effect seen in data is not due to chance. In our example, with a p-value of 0.02 (which is below the conventional threshold of 0.05), we can say that the iPad intervention had a statistically significant effect on kindergartners' scores.

However, practical significance is about whether the size of the effect is large enough to be meaningful in the real world. In this study, the effect consisted of an average score increase of two points for the intervention group. Whether this two-point increase is meaningful depends on the context, such as how educational outcomes are measured and valued. This is what the school board member meant by the improvement not being practically significant: while the statistical test shows an effect exists, it might not be large enough to matter for policy or decision-making.

Effect Size

Effect size measures the magnitude of the difference between groups and provides extra insight beyond p-values. A p-value only informs us about the likelihood of the data given the null hypothesis but tells us nothing about the size of the effect.

In the iPad study, the effect size was defined as about two points more on average for the intervention group compared to the control group. Knowing the effect size helps educators and stakeholders understand the practical implications of the intervention. A small effect size, despite being statistically significant, might not warrant changes in policy or merit extensive resources. Conversely, a large effect size, even if not statistically significant, might indicate an important practical learning improvement.

Effect sizes help illustrate the real-world significance of study findings, offering a more comprehensive picture than p-values alone.