Question

A comprehensive study conducted by the National Institute of Child Health and Human Development tracked more than 1000 children from an early age through elementary school (New york Times, November 1,2005\()\). The study concluded that children who spent more than 30 hours a week in child care before entering school tended to score higher in math and reading when they were in the third grade. The researchers cautioned that the findings should not be a cause for alarm because the effects of child care were found to be small. Explain how the difference between the sample mean math score for third graders who spent long hours in child care and the known overall mean for third graders could be small but the researchers could still reach the conclusion that the mean for the child care group is significantly higher than the overall mean for third graders.

Short answer

Even though the difference between the sample mean score for third graders, who spent long hours in child care and the known overall mean for third graders, is small, researchers can still conclude that the mean for the child care group is significantly higher than the overall mean for third graders. This conclusion is made possible by statistical significance testing, which allows researchers to determine whether the observed difference is likely to have occurred by random chance. If the likelihood is low, the difference is deemed significant, indicating our observed difference is likely a true difference.

Step-by-step solution

Understanding the basic idea of statistical significance

Statistical significance is a term used often in research studies to comment on the strength of the results. Essentially, it tells you how certain you can be that the results are not just random chance. If a result is statistically significant, it means that the likelihood of the result having occurred by chance is very low. Therefore, even if the difference between two means is small, the difference could still be considered statistically significant if the likelihood of that difference occurring by chance is very low.

Apply the concept to the scenario

For the study in question, the researchers have noticed a small difference between the mean scores of two groups of children - those who spent more than 30 hours a week in day care and the rest of the third graders. This difference, though minor, could be significant. That is, the small difference in the sample mean math score might not be due to chance and might indicate a true difference between the two groups. Hence, the researchers can conclude that the mean for the child care group is significantly higher.

Understand the significance testing

The researchers used a significance test, a technique that enables them to determine the likelihood that the observed difference in mean scores occurred by chance. While the difference might be small, if it is statistically significant, it increases our confidence that the observed difference wasn't simply due to random variation, but rather reflects a true difference in math score means between the two groups. It is this significance that allows researchers to claim that the difference in the mean scores is statistically significant, despite being small.

Key concepts

Sample Mean

The sample mean is a critical statistic that represents the average of a set of data points from a sample of a larger population. In the context of the National Institute of Child Health and Human Development study, the researchers would have calculated the sample mean math score for third graders who spent more than 30 hours a week in child care. This mean is simply the sum of all the math scores from the children in this group divided by the total number of scores.

The importance of the sample mean lies in its role as an estimate of the population mean. In this case, the population would refer to all third graders who spent extensive hours in child care. When examining the effects of child care hours on academic performance, comparing the sample mean to the known overall mean of all third graders provides insights into whether child care hours might influence math scores.

Significance Test

A significance test is a statistical method used to determine the likelihood that a result observed in a study is not due to random chance. This test usually involves generating a p-value, which reflects the probability that the observed effect, such as a difference in means, would occur if there actually were no effect in the larger population.

In our study, the significance test would measure how likely it is that the observed difference between the sample mean math score of the child care group and the overall mean for third graders could happen by chance. If the test yields a p-value lower than a preset threshold (commonly 0.05), the researchers would reject the null hypothesis – that there is no difference – and conclude that the difference in math scores is statistically significant.

Research Study Analysis

Research study analysis involves the critical examination of all elements of a study, from the methods used to collect data to the statistical techniques applied for interpretation. In the case of the child care study, the research analysis would consider the sample size, sampling method, controls introduced to prevent bias, and the significance tests used.

This process is essential to validate the findings of the study. For example, a large sample size and random sampling can increase the reliability of the sample mean as a representation of the population. Researchers must carefully analyze these factors to draw accurate conclusions about the impact of child care hours on math scores and to ensure that the observed results are indeed meaningful and not an artifact of poor study design or methodology.

Math Score Comparison

Math score comparison in the context of this research means comparing the average math scores of two distinct groups - children with extensive child care hours and the general population of third graders. When researchers notice a difference in the sample mean scores between these groups, it's vital to interpret the difference carefully.

Even if the actual numerical difference is small, it can be statistically significant if it is unlikely to be due to chance. Falling into the trap of considering only the magnitude without acknowledging the significance can mislead stakeholders. It is the role of research to not only observe and measure these differences but to also use statistical tests to infer whether they reflect a real-world effect large enough to warrant attention or intervention.