Question

The authors of the paper "Short-Term Health and Economic Benefits of Smoking Cessation: Low Birth Weight" (Pediatrics [1999]: \(1312-1320\) ) investigated the medical cost associated with babies born to mothers who smoke. The paper included estimates of mean medical cost for low-birth-weight babies for different ethnic groups. For a sample of 654 Hispanic low-birth- weight babies, the mean medical cost was \(\$ 55,007\) and the standard error \((s / \sqrt{n})\) was \(\$ 3011 .\) For a sample of 13 Native American low-birth- weight babies, the mean and standard error were \(\$ 73,418\) and \(\$ 29,577\), respectively. Explain why the two standard errors are so different.

Short answer

The standard errors for the samples of Hispanic and Native American low-birth-weight babies are so different due to the impact of sample size and data dispersion on standard error calculations. The larger sample size and potentially less dispersion in the Hispanic babies sample led to smaller standard error, while the smaller sample size and potentially more dispersed data in the Native American babies sample resulted in a larger standard error.

Step-by-step solution

Understanding Standard Error

The standard error is a measure of the statistical accuracy of an estimate, equal to the standard deviation of the theoretical distribution of a large population of such estimates. It is usually calculated by the formula: \(SE = s / \sqrt{n}\), where 's' is the approximation of the population standard deviation (sample standard deviation) and 'n' is the sample size.

Analyzing the Hispanic Sample

For the sample of 654 Hispanic low-birth-weight babies, the mean medical cost was \$55007 and the standard error was \$3011. The large sample size would contribute to a smaller standard error, assuming the data distribution is not widely dispersed.

Analyzing the Native American Sample

In contrast, for the sample of 13 Native American low-birth-weight babies, the mean and standard error were \$73418 and \$29577, respectively. The relatively small sample size would result in a larger standard error as standard error is inversely proportional to the square root of the sample size. In addition, a high standard error relative to the mean could indicate a wider distribution of the data.

Comparing the Two Samples

Comparing the standard errors for the two samples, it is clear that the sample size impacts the standard error calculations significantly. Small sample size and possibly a wide dispersion of costs in the case of the Native American babies led to a larger standard error, while the larger sample size and more clustered distribution resulted in a smaller standard error for the Hispanic babies.

Key concepts

Statistical Accuracy

Statistical accuracy refers to how closely a sample statistic estimates the true population parameter. In the context of the provided exercise, the statistic in question is the mean medical cost for low-birth-weight babies belonging to different ethnic groups. The standard error (SE) is intimately connected to statistical accuracy since it provides a measure of the variability of the sample mean around the population mean.

Mathematically, the standard error is computed using the formula: \( SE = \frac{s}{\sqrt{n}} \), where \(s\) is the sample standard deviation, and \(n\) is the sample size. A lower standard error indicates a more precise estimate of the population mean, suggesting that the sample mean is a more reliable indicator of the true population mean. In the exercise, the standard error helps understand the precision of the mean medical costs estimated for the two samples. The smaller standard error for the Hispanic sample compared to the Native American sample suggests that the estimate for the Hispanic babies' medical costs is statistically more accurate.

Sample Size Effect

The impact of sample size on statistical inference cannot be overstated, and it is highlighted in the exercise as a key factor contributing to the difference in standard errors between the two samples. The formula for standard error incorporates the sample size in the denominator, \( SE = \frac{s}{\sqrt{n}} \), which means that as the sample size increases, the standard error decreases. This relationship implies that the larger the sample, the closer our sample mean is likely to be to the true population mean.

In the provided calculations, the comparatively large sample of 654 Hispanic babies results in a smaller standard error, enhancing the statistical accuracy of the mean medical cost estimate. Conversely, the small sample of 13 Native American babies leads to a much larger standard error, indicating less confidence in the precision of the mean estimate for that group. It is crucial for students to understand that increasing the sample size is one of the most direct methods to improve the precision of sample estimates and reduce statistical error.

Data Distribution

The distribution of data in a sample influences the calculation of standard error and the interpretation of statistical results. Standard error assumes that the data points are distributed in a particular manner, typically along a bell-shaped curve known as the normal distribution for many biological and social phenomena. The degree to which the data is spread out affects the standard deviation, and consequently, the standard error.

A wide spread in data, indicating higher variability, will result in a larger standard deviation and thus a larger standard error. Conversely, if the data points are clustered closely around the mean, this suggests less variability, leading to a smaller standard deviation and a smaller standard error. In the exercise, we're hinted that the Native American low-birth-weight babies' data might be more widely dispersed than the Hispanic babies' data, which could partially explain the larger standard error despite the substantial influence of the small sample size. Understanding the shape and spread of the data distribution helps in comprehending the variation within a sample and the reliability of the sample mean as an estimate of the population mean.