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 of the Hispanic and Native American samples are vastly different due to the difference in their respective sample sizes and potentially the variability within each group. The larger sample size of the Hispanic group gives it a smaller standard error due to a more accurate estimation of the population mean, while the Native American sample has a smaller size and potentially more variability, which contribute to its larger standard error.

Step-by-step solution

Understanding Standard Error

Standard error is a measure of how much a sample mean estimates the population mean. A smaller standard error indicates the sample mean is a more accurate reflection of the actual population mean. The formula for standard error is SE = s / sqrt(n), where s is the standard deviation of the sample and n is the number of observations in the sample. As such, the standard error relies heavily on the sample size and the variability within the sample.

Comparing the Samples

In this exercise, two different samples are taken: one sample of 654 Hispanic low-birth-weight babies and one sample of 13 Native American low-birth-weight babies. The larger the sample size, the smaller the standard error, because the estimate becomes closer to the population mean. Thus, we would expect the Hispanic sample to have a smaller standard error due to its larger size. However, the standard error also depends on the variability within the sample. A higher variability within the sample would mean a larger standard deviation and thus a larger standard error.

Analyzing the Difference in Standard Errors

The standard error for the Hispanic sample is $3011, while it is noticeably higher for the Native American sample at $29577. Even though the Hispanic sample is much larger, the standard error is much smaller. This discrepancy can be explained by the much smaller sample size of the Native American group and possibly a greater variability within the Native American sample compared to the Hispanic sample. Thus, the smaller sample size and the potentially higher variability within the Native American sample can explain the larger standard error.

Key concepts

Sample Mean Estimation

The process of sample mean estimation is at the core of statistical analysis. It involves using the average value from a smaller, selected group, known as a sample, to estimate the average value for a larger group or population. In our exercise, the mean medical costs for two samples of low-birth-weight babies were estimated to: \(\$55,007\) for Hispanic babies, based on a sample of 654, and \(\$73,418\) for Native American babies, from a much smaller sample of 13.

Why is the estimate significant? Well, when researchers cannot measure every individual in an entire population due to resource constraints, they take a manageable number of observations from a subset. This sample mean serves as a point estimate of the population mean—the true average we’re trying to estimate. The accuracy of this estimate, inferred from the sample data, is reflected in its standard error, which brings us to the importance of understanding sample size and variability.

Population Mean

The population mean is the average of all measurements or values in a population. It represents the expected value that you would obtain if you could measure every single member of the entire group without error or bias. When we mention the term 'population', we're talking about the full set of data from which a sample is drawn. In statistical terminology, 'population' does not only refer to people but to any collection of observations for which we wish to make inferences.

In the medical cost study, we're looking to estimate the mean cost associated with low-birth-weight babies for different ethnic groups, with the ultimate goal of arriving at the population mean. However, since we’ve only sampled parts of the population, our estimates might not perfectly match the true population mean. This is where understanding standard error and its relationship to sample size and variability becomes crucial.

Sample Size

Sample size, denoted by \(n\), is the number of observations included in the statistical sample. It's a pillar in producing a reliable estimate of the population mean. The larger the sample size, the less variability there is likely to be in the estimate of the population mean, thus the more confidence we can have in our sample mean estimation.

In the provided exercise, there’s a notable contrast between the sample sizes: 654 Hispanic babies versus only 13 Native American babies. The standard error of the Hispanic sample is significantly lower due to the sample's size. The law of large numbers tells us that as a sample size increases, the sample mean gets closer to the population mean, resulting in a smaller standard error, assuming the sample is random and representative. This is precisely why large-scale studies are often favored in research: they are more likely to produce results that accurately reflect the true attributes of the population.

Sample Variability

Sample variability refers to the range of different values, or variance, within a sample. High variability indicates that the sample values are spread out widely from each other and from the sample mean. Conversely, low variability means that the data points are closer to each other and the sample mean. Variability is measured using standard deviation (\(s\)).

Returning to our exercise, while the sample size had a significant effect on the standard error (SE), sample variability is another aspect to be scrutinized. If the 13 Native American babies had a wide range of medical costs, this would lead to a high standard deviation, consequently inflating the standard error. When we pair a small sample size with high variability, we end up with a less precise estimate of the population mean, as witnessed by the larger standard error in the Native American sample.