Question

Babies born extremely prematurely run the risk of various neurological problems and tend to have lower IQ and verbal ability scores than babies that are not premature. The article "Premature Babies May Recover Intelligence, Study Says" (San Luis Obispo Tribune, February 12,2003 ) summarized medical research that suggests that the deficits observed at an early age may decrease as children age. Children who were born prematurely were given a test of verbal ability at age 3 and again at age 8 . The test is scaled so that a score of 100 would be average for normal-birth-weight children. Data for 50 children who were born prematurely were used to generate the accompanying Minitab output, where Age 3 represents the verbal ability score at age 3 and Age8 represents the verbal ability score at age \(8 .\) Use the Minitab output to determine if there is convincing evidence that the mean verbal ability score for children born prematurely increases between age 3 and age 8 . You can assume that it is reasonable to regard the sample of 50 children as a random sample from the population of all children born prematurely. $$ \begin{aligned} &\text { Paired T-Test and Cl: Age8, Age3 }\\\ &\begin{array}{l} \text { Paired } T \text { for } \text { Age8 - Age3 } \\ \begin{array}{rrrrr} & \text { N } & \text { Mean } & \text { StDev } & \text { Se Mean } \\ \text { Age8 } & 50 & 97.21 & 16.97 & 2.40 \\ \text { Age3 } & 50 & 87.30 & 13.84 & 1.96 \\ \text { Difference } & 50 & 9.91 & 22.11 & 3.13 \\ \text { T-Test of mean difference } & =0(\mathrm{vs}>0): \text { T-Value }=3.17 \\ \text { P-Value }=0.001 & & & \end{array} \end{array} \end{aligned} $$

Short answer

There is statistically significant evidence to suggest that children born prematurely experience an increase in verbal ability score between the ages 3 and 8, thus supporting the study's claim that deficits observed at an early age may decrease as children age.

Step-by-step solution

Understand the t-test report

In the given paired t-test report, the t-value is 3.17 and the p-value is 0.001. The t-test is used to determine whether the mean difference of two sets of data is significantly different from zero. P-value in a hypothesis test is used as a criterion for rejection or non-rejection of null hypothesis.

Formulate the null and alternative hypotheses

In this case, we test the null hypothesis \( H_0 : \mu_d = 0 \) (Mean difference in the scores is zero or they are same at both ages) against the alternative hypothesis \( H_a : \mu_d > 0 \) (Mean difference in the scores is greater than zero or the scores have improved).

Interpreting the t-test

The t-value of 3.17 means that the difference in means is 3.17 times as large as what would be expected by chance variation (the standard error of the difference). If p-value is less than the significance level (0.05), we reject the null hypothesis.

Checking the p-value to arrive at a conclusion

Given that the p-value is 0.001, which is less than the significance level of 0.05, we can reject the null hypothesis and infer that there is a significant increase in the verbal ability score from age 3 to age 8. Therefore, the study's claim is supported by this statistical analysis.

Key concepts

Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics used to determine if there's evidence in a sample of data to conclude that a certain condition holds for the entire population. It starts by assuming the null hypothesis (\( H_0 \)) is true, which in our scenario presupposes that the mean verbal ability score of premature babies at age 3 is equal to their score at age 8.

An alternative hypothesis (\( H_a \)or \( H_1 \) is formulated to represent a condition that contradicts the null hypothesis: for instance, it could state that the mean score at age 8 is higher than at age 3. Through the paired t-test, as in the provided exercise, we calculate a p-value that quantifies the probability of observing the data or something more extreme if the null hypothesis were true.

A low p-value (typically below 0.05) suggests that the observed data is unlikely under the null hypothesis, prompting analysts to reject \( H_0 \) in favor of \( H_a \). Hence, a p-value of 0.001 is strong evidence against the null hypothesis, pointing toward a real increase in verbal ability scores as these children aged.

Verbal Ability Score Analysis

Verbal ability score analysis involves examining and interpreting the scores from tests that measure verbal skills. In the exercise example, premature children's verbal ability scores were examined at age 3 and again at age 8. Here, we employ statistical techniques like the paired t-test to analyze these scores over time.

The paired t-test is particularly useful in this context because it looks at two related samples; the same group of children tested at different ages. It assesses whether their mean scores at one time point are significantly different from another. An important consideration is the mean difference, which is the average change in scores for the sample. In our case, a mean difference of 9.91 suggests an improvement. Coupled with the p-value, this analysis indicates whether the change in verbal ability scores is statistically significant.

Neurological Development in Premature Babies

Premature babies often face challenges that can affect their neurological development. This development is a complex process that can be influenced by a range of factors, including the level of care they receive and the environmental and genetic influences they're subject to post-birth.

Premature infants are at a higher risk of developmental delays, which might impact their cognitive abilities, including verbal skills. Standardized tests, such as the one employed in the exercise, can track these abilities and offer insights into their developmental trajectory.

Studies and analyses like the one referenced in the exercise are crucial for understanding how premature infants develop over time. They also serve as a basis for developing interventions aimed at mitigating the developmental deficits that could be a consequence of their early birth.