Question

As part of the National Health and Nutrition Examination Survey (NHANES), hemoglobin levels were checked for a sample of 1139 men age 70 and over. 56 The sample mean was \(145.3 \mathrm{~g} / \mathrm{l}\) and the standard deviation was \(12.87 \mathrm{~g} /\) l. (a) Calculate the standard error of the mean. (b) Assuming that the hemoglobin level follows a normal distribution, construct a \(99 \%\) confidence interval for the population mean of hemoglobin levels.

Short answer

The standard error of the mean is approximately \(0.381 g/l\), and the 99% confidence interval for the population mean of hemoglobin levels is \(145.3 \pm (2.576 \times 0.381)\), which results in \((145.3 - 0.9816) to (145.3 + 0.9816)\) or \((144.3184 g/l, 146.2816 g/l)\).

Step-by-step solution

Calculating the standard error of the mean

The standard error of the mean (SEM) is calculated by dividing the sample standard deviation by the square root of the sample size. The formula to use is: SEM = \(\frac{s}{\sqrt{n}}\), where \(s\) is the standard deviation and \(n\) is the sample size. In this case, the standard deviation is \(12.87 g/l\) and the sample size is 1139. Thus, the calculation is SEM = \(\frac{12.87}{\sqrt{1139}}\).

Finding the critical Z-value for a 99% confidence interval

To find the critical Z-value for a 99% confidence interval, we need to look at the z-table or use a calculator that provides the z-value corresponding to the desired level of confidence. Usually for a 99% confidence level, the z value is approximately 2.576.

Constructing the 99% confidence interval for the population mean

The confidence interval is calculated with the formula: \(\text{CI} = \bar{x} \pm (z \times SEM)\), where \(\bar{x}\) is the sample mean, \(z\) is the critical value from the z-distribution, and SEM is the standard error of the mean. Here, the sample mean \(\bar{x}\) is \(145.3 g/l\). The critical z-value for a 99% confidence interval is 2.576, and the SEM has been calculated in Step 1. Thus, plug these values into the confidence interval formula to obtain the range.

Key concepts

Standard Error of the Mean

The standard error of the mean (SEM) is a crucial statistic that measures the accuracy with which a sample mean represents a population mean. In essence, it's an indication of the variability you would expect in sample means if you were to repeat the sampling process many times. Mathematically, it's calculated by dividing the sample standard deviation by the square root of the sample size, resulting in the formula SEM = \(\frac{s}{\sqrt{n}}\), where \(s\) represents the sample standard deviation and \(n\) the sample size. Lower SEM values imply more precision in the estimation of the population mean—meaning the sample mean is most likely close to the true population mean. In our exercise example, a lower SEM would suggest strong evidence that the mean hemoglobin level calculated is representative of the population mean for men aged 70 and above.

Normal Distribution

A normal distribution, also known as the Gaussian distribution, is a probability distribution symmetrically centered around the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. The characteristic 'bell curve' shape of a normal distribution is ubiquitous in statistics and is a key assumption in constructing confidence intervals for a population mean. In our example, the assumption that hemoglobin levels follow a normal distribution allows for the use of z-scores and the calculation of a confidence interval that is expected to contain the true population mean a certain percentage of the time, provided that the sampling method is consistent and random.

Sample Standard Deviation

The sample standard deviation measures the amount of variability or spread in a set of sample data. In other words, it gives us an idea of how spread out the measurements of our sample are from the sample mean. The larger the standard deviation, the more spread out our data points are. It's an essential component when calculating the SEM and is derived from the formula \(s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n}(x_i - \bar{x})^2}\), where \(x_i\) are the individual samples, \(\bar{x}\) is the sample mean, and \(n\) is the sample size. The sample standard deviation is critical for estimating the precision and reliability of the sample mean as an estimate of the population mean.

Sample Size

Sample size, denoted by \(n\), is the number of individual data points collected from a population. It plays a vital role in statistical analyses, particularly in the calculation of the SEM and subsequently the confidence intervals. Larger sample sizes generally lead to a lower SEM, suggesting more precise estimates of the population mean. Additionally, large sample sizes can help mitigate the influence of outliers or abnormal observations, leading to a more robust estimation of the population characteristics. In the case of hemoglobin levels in our problem, a sample size of 1139 men is significantly large, providing confidence that the SEM and associated confidence interval are reliable estimates.

Z-Value

The Z-value or Z-score in the context of confidence intervals is the number of standard errors away from the sample mean a particular point is. It is a key component when determining how wide the confidence interval should be for a certain level of confidence. For a 99% confidence interval, the Z-value corresponds to the Z-score that includes 99% of the data in a standard normal distribution, with roughly 0.5% of the data on each tail end of the distribution being excluded. In practice, a Z-value of approximately 2.576 is used for a 99% confidence level, which implies that we expect our confidence interval to contain the true population mean 99% of the time, assuming a perfectly normal distribution and random sampling.

Population Mean

The population mean is the average of all measurements in a total population. It is a fixed value, although often unknown, and the objective of many statistical studies is to estimate this value based on sample data. The confidence interval we construct gives us a range within which we are a certain percentage confident the true population mean lies. It is crucial to understand that the population mean is a parameter that is estimated by the sample mean, which is a statistic calculated from the sampled data. In this NHANES study, we are estimating the true average hemoglobin level for men age 70 and over based on our sample mean and the associated confidence interval.