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Chapter 4: Problem 163

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. \(^{46}\) 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?

Short Answer

Expert verified
The null hypothesis (\(H_0: \mu_{I} \leq \mu_{C}\)) is rejected because the p-value (0.02) is less than 0.05, indicating a statistically significant difference favoring the intervention group. However, the practical significance of the 2-point effect size is debatable, as it may not lead to a meaningful improvement in learning considering the costs involved.

Step by step solution

01

Define Null and Alternative Hypotheses

In a statistical test, the null hypothesis is the statement being tested, often suggesting 'no effect' or 'no difference'. In contrast, the alternative hypothesis is the statement that is accepted if the null hypothesis is rejected. For this study, the null hypothesis (\(H_0\)) states that the mean score for the intervention group (with iPads) is not higher than the control group (without iPads). That is, \(H_0: \mu_{I} \leq \mu_{C}\), where \(\mu_I\) is the population mean score for the intervention group, and \(\mu_C\) is the mean for the control group. The alternative hypothesis (\(H_1\) or \(H_a\)) posits that the mean score for the intervention group is higher than that of the control group, i.e., \(H_a: \mu_{I} > \mu_{C}\).
02

Interpret the p-value

The p-value, which stands at 0.02 in this case, represents the probability of finding the observed results when the null hypothesis is true. Here, the smaller the p-value (<0.05 typically), the stronger the evidence to reject the null hypothesis. Since 0.02 < 0.05, the results are statistically significant at the 5% level, and we reject the null hypothesis. This suggests that the intervention group (the one that used iPads) scored significantly higher on the HRSIW subtest than the control group (that didn't use iPads).
03

Discuss the Effect Size and Practical Significance

The effect size refers to how much of a difference there is between the two groups, which in this case is 2 points. While this shows statistical significance, its practical or real-world significance might be questioned. When the school board member suggests that the results might not be 'practically significant,' they mean that the 2-point difference, even though statistically significant, may not have a substantial or meaningful impact on the students' learning in a practical sense. For example, it may not be worth the cost of buying iPads for all kindergartners for a 2-point increase in scores.

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Key Concepts

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

Null and Alternative Hypotheses
Understanding the null and alternative hypotheses is a cornerstone of statistical testing. In educational experiments like the one conducted to determine if iPads help kindergartners learn, we begin by establishing two contrasting statements. The null hypothesis (\(H_0\)) represents a position of skepticism, holding that the new teaching tool, in this case, the use of iPads, does not result in any difference in children's learning outcomes when compared to traditional methods. Formally, we can express this as \(H_0: \mu_{I} \leq \mu_{C}\), where \mu_{I}\ denotes the average score for the intervention group (with iPads) and \mu_{C}\ for the control group (without iPads).

On the flip side is the alternative hypothesis (\(H_1\) or \(H_a\)), which embodies the optimistic view that the iPads do indeed have a positive effect on learning, hence \(H_a: \mu_{I} > \mu_{C}\). This hypothesis is what researchers hope to support through their experiment. Crafting these hypotheses correctly is vital as they define the framework within which statistical evidence is evaluated, and conclusions are drawn.
P-value Interpretation
When dealing with p-values, we're assessing the strength of evidence against the null hypothesis. In our iPad learning study, a p-value of 0.02 signals the probability of observing results at least as extreme as those measured in the experiment鈥攁ssuming the null hypothesis is true. A p-value of 0.02 is quite low, which in the realm of statistics, whispers to us that the intervention group's higher scores are unlikely to be due to just chance alone.

A commonly employed threshold to ascertain 'statistical significance' is a p-value of less than 0.05. Since our p-value (0.02) dips below this cutoff, we have substantial grounds to reject the null hypothesis. This means we accept the alternative hypothesis that iPads have a positive impact on HRSIW subtest scores. It's crucial for students to appreciate the subtleties involved in p-value interpretation鈥攊t's not about the probability of the hypothesis being true or false, but about the data's compatibility with the assumed model of no effect.
Practical Significance
While the statistical significance is the mathematician's verdict, practical significance is the educator's concern. It's about the so what? of the findings. The effect size鈥攁 2-point average difference in this study鈥攎ight not scream significance from a practical standpoint. While statistically significant, the school board member challenges if this slight score bump justifies the expense and effort of integrating iPads into the classroom.

This nuance of practical significance compels us to consider if the educational intervention makes a meaningful difference in a real-world setting. It's not only about measuring if the effect exists but also if the effect matters. When the school board member questions the practicality, they're invoking a cost-benefit analysis of sorts: Is the educational gain worth the resource investment? It behooves educators and policymakers to always weigh the statistical findings against the tangible benefits when making decisions that impact curriculums and, by extension, students' futures.

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Most popular questions from this chapter

Interpreting a P-value In each case, indicate whether the statement is a proper interpretation of what a p-value measures. (a) The probability the null hypothesis \(H_{0}\) is true. (b) The probability that the alternative hypothesis \(H_{a}\) is true. (c) The probability of seeing data as extreme as the sample, when the null hypothesis \(H_{0}\) is true. (d) The probability of making a Type I error if the null hypothesis \(H_{0}\) is true. (e) The probability of making a Type II error if the alternative hypothesis \(H_{a}\) is true.

Indicate whether the analysis involves a statistical test. If it does involve a statistical test, state the population parameter(s) of interest and the null and alternative hypotheses. Polling 1000 people in a large community to determine if there is evidence for the claim that the percentage of people in the community living in a mobile home is greater than \(10 \%\)

Measuring the Impact of Great Teachers An education study in Tennessee in the 1980 s (known as Project Star) randomly assigned 12,000 students to kindergarten classes, with the result that all classes had fairly similar socioeconomic mixes of students. \({ }^{17}\) The students are now about 30 years old, and the study is ongoing. In each case below, assume that we are conducting a test to compare performance of students taught by outstanding kindergarten teachers with performance of students taught by mediocre kindergarten teachers. What does the quoted information tell us about whether the p-value is relatively large or relatively small in a test for the indicated effect? (a) On the tests at the end of the kindergarten school year, "some classes did far better than others. The differences were too big to be explained by randomness." (b) By junior high and high school, the effect appears to be gone: "Children who had excellent early schooling do little better on tests than similar children who did not." (c) The newest results, reported in July 2010 by economist Chetty, show that the effects seem to re-emerge in adulthood. The students who were in a classroom that made significant gains in kindergarten were significantly "more likely to go to college,... less likely to become single parents, ... more likely to be saving for retirement,... Perhaps most striking, they were earning more." (Economists Chetty and Saez estimate that a standout kindergarten teacher is worth about \(\$ 320,000\) a year in increased future earnings of one class of students. If you had an outstanding grade-school teacher, consider sending a thank you note!)

4.20 Taste Test A taste test is conducted between two brands of diet cola, Brand \(A\) and Brand \(B\), to determine if there is evidence that more people prefer Brand A. A total of 100 people participate in the taste test. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Give an example of possible sample results that would provide strong evidence that more people prefer Brand A. (Give your results as number choosing Brand \(\mathrm{A}\) and number choosing Brand B.) (c) Give an example of possible sample results that would provide no evidence to support the claim that more people prefer Brand A. (d) Give an example of possible sample results for which the results would be inconclusive: The sample provides some evidence that Brand \(\mathrm{A}\) is preferred but the evidence is not strong.

For each situation described in Exercises 4.93 to 4.98 , indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level \((\) such as \(\alpha=0.01)\) A pharmaceutical company is testing to see whether its new drug is significantly better than the existing drug on the market. It is more expensive than the existing drug. Which significance level would the company prefer? Which significance level would the consumer prefer?
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