Question

Babies born extremely prematurely run the risk of various neurological problems and tend to have lower IQ and verbal ability scores than babies who 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 deficit 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 that are consistent with summary quantities given in the paper 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 information in the Minitab output to construct and interpret a \(95 \%\) confidence interval for the change in mean verbal ability score from age 3 to 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. Paired T-Test and Cl: Age8, Age3 Paired \(\mathrm{T}\) for Age8 - Age3 \(\begin{array}{lrrrr} & \mathrm{N} & \text { Mean } & \text { StDev } & \text { Se Mean } \\ \text { Age8 } & 50 & 97.21 & 16.97 & 2.40 \\ \text { Age3 } & 50 & 87.30 & 13.84 & 1.96 \\ \text { Difference } & 50 & 9.91 & 22.11 & 3.13\end{array}\)

Short answer

The 95% confidence interval for the mean difference in verbal ability scores from age 3 to age 8 is calculated as 3.39 to 16.43. This interval does not contain zero, indicating a significant improvement in the children's verbal ability scores as they grow from age 3 to age 8.

Step-by-step solution

Understanding Confidence Interval

A confidence interval is an estimate of an interval in statistics that may contain a population parameter. The unknown population parameter's best estimate is the sample mean. The confidence interval puts the estimate into a range that likely contains the population parameter. Here, the goal is to compute the confidence interval for the mean difference of verbal ability score from age 3 to age 8.

Computing Confidence Interval

For a 95% confidence interval and the sample data provided, the confidence interval can be calculated using the formula: sample mean ± (1.96 x standard error). The standard error for the difference is given as 3.13 and the mean difference is 9.91. Plugging these values in the formula, we get the confidence interval as: 9.91 ± (1.96 x 3.13).

Calculating Upper and Lower Limits

Now the actual numbers for the confidence interval are needed. Calculate the upper and lower limits of the confidence interval by performing the addition and subtraction. Lower Limit = 9.91 - (1.96 x 3.13) and Upper Limit = 9.91 + (1.96 x 3.13).

Interpretation of the Confidence Interval

The calculation gives the range for the change in mean verbal ability score from age 3 to age 8. If the confidence interval contains zero, it would mean that there is no significant difference in the verbal ability scores at the two ages. If the interval does not contain zero, it indicates a significant difference. The higher the confidence level (here 95%), the more certain we can be about the accuracy of the estimated range.

Key concepts

Verbal Ability Score

The verbal ability score is a standardized assessment metric often used to measure language skills, including vocabulary, reading comprehension, and verbal fluency. In the context of educational and psychological studies, tracking changes in verbal ability scores can help researchers and educators understand cognitive development or the impact of interventions. For premature babies, as indicated by the exercise, these scores can highlight the progress in their verbal communication skills as they grow from age 3 to age 8. Improvements in scores could suggest positive developmental trajectories or the efficacy of therapeutic measures, while stagnant or declining scores might signal a need for additional support.

To ensure that the improvements in the scores are statistically reliable and not due to chance, the analysis often involves comparing them at different ages or conditions, employing statistical tests such as the paired t-test, to ascertain the significance of the developmental changes.

Paired t-test

The paired t-test is a statistical procedure used to determine whether the mean difference between two sets of observations is significant. This test is particularly useful when comparing two related samples, such as the same group of individuals measured at two points in time. It accounts for the inherent link between the paired observations, which is key to reducing variability and obtaining a more precise estimate of the effects being studied.

In the provided Minitab output from our exercise, the paired t-test compares the verbal ability scores of children at age 3 and then again at age 8. The test reveals changes in scores that are not just random fluctuations but are potentially meaningful reflections of cognitive growth or recovery. The t-test provides the statistical backbone to support or refute hypotheses regarding the developmental progress of prematurely born children.

Statistical Significance

Statistical significance is a term used to describe a result that is not likely to occur by chance. This concept helps researchers discern whether the findings in their study can be attributed to a specific hypothesis or if they are simply a random occurrence. When a finding is statistically significant, it implies that the observed effects are strong enough to be unlikely due to random error.

In the context of our exercise, the significance of the change in the verbal ability score is crucial. It helps to conclude whether the observed average improvement is real or may have occurred randomly. The confidence interval constructed in the solution steps assists in this determination—if the interval does not include the value of zero, it suggests that the change is significant at the 95% confidence level, thereby supporting the notion that the verbal capabilities of these children have indeed improved significantly over time.

Standard Error

The standard error of the mean (often abbreviated as SEM) is a measure of how far the sample mean of the data is likely to be from the true population mean. It is derived from the standard deviation of the sample and the square root of the sample size. Essentially, the standard error quantifies uncertainty: a smaller standard error suggests more precise estimates of the population mean.

The role of standard error in our exercise is critical for calculating the confidence interval of the mean difference in verbal ability scores. It helps gauge the reliability of the mean difference by considering both the amount of variation in the scores and the number of observations in the study. A confidence interval that factors in the standard error provides a range within which we can be fairly confident that the true mean difference lies.

Sample Mean

The sample mean is the average value of a set of observations and serves as an estimate of the population mean. It is a central measure of central tendency and provides a snapshot of the typical value in a dataset. The mean is highly relevant in research as it often represents the expected value against which individual observations can be compared.

In the Minitab output for our exercise, the sample mean of the difference reflects the average change in the verbal ability score of the study's participants from age 3 to age 8. This average change is what researchers are interested in estimating to understand the general trend in how premature children's verbal abilities develop as they grow older. The sample mean is pivotal to constructing confidence intervals and conducting the paired t-test, making it a cornerstone of statistical analysis in this case study.