Question

An important indicator of lung function is forced expiratory volume (FEV), which is the volume of air that a person can expire in one second. Dr. Hernandez plans to measure FEV in a random sample of \(n\) young women from a certain population, and to use the sample mean \(\bar{y}\) as an estimate of the population mean. Let \(E\) be the event that Hernandez's sample mean will be within \(\pm 100 \mathrm{ml}\) of the population mean. Assume that the population distribution is normal with mean \(3,000 \mathrm{ml}\) and standard deviation \(400 \mathrm{ml}^{3}\) Find \(\operatorname{Pr}\\{E\\}\) if (a) \(n=15\) (b) \(n=60\) (c) How does \(\operatorname{Pr}\\{E\\}\) depend on the sample size? That is, as \(n\) increases, does \(\operatorname{Pr}\\{E\\}\) increase, decrease, or stay the same?

Short answer

For both sample sizes, calculate the standard error, find the corresponding z-scores, and determine the probabilities using the standard normal distribution. As the sample size increases, the standard error decreases, the z-scores get smaller, and thus the probability of the event \(E\) increases.

Step-by-step solution

Identify the Known Parameters

First, identify the known parameters from the problem which include the population mean (\(\text{\(\mu\)} = 3000\) ml), the population standard deviation (\(\text{\(\sigma\)} = 400\) ml), and the desired margin of error (\(E = \pm 100\) ml). The problem statement also assumes the population distribution is normal, which allows the use of normal distribution properties.

Calculate the Standard Error for Each Sample Size

Using the standard deviation and the sample size, calculate the standard error (SE) for each sample size: \(n=15\) and \(n=60\). The standard error is given by \(SE = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the standard deviation and \(n\) is the sample size.

Calculate the Z-Score for Each Sample Size

Determine the z-score that corresponds to being within \(\pm 100\) ml of the population mean. Use the formula \(z = \frac{E}{SE}\), where \(E\) is the margin of error and \(SE\) is the standard error. Calculate the z-score for each sample size \(n=15\) and \(n=60\).

Find the Probability for Each Sample Size

Find the probability \(\Pr{E}\) that the sample mean is within \(\pm 100\) ml of the population mean by looking up the z-score in a standard normal distribution table or using a calculator with normal distribution functions. Since we need the probability that the sample mean will fall within the range around the population mean, look for the cumulative probabilities at both ends of the range, and subtract the lower cumulative probability from the higher one.

Analyze the Probability's Dependence on Sample Size

Analyzing the formula for standard error, \(SE = \frac{\sigma}{\sqrt{n}}\), it is evident that as the sample size \(n\) increases, the standard error decreases. Since the margin of error \(E\) is fixed, a smaller standard error results in a smaller z-score, which corresponds to a higher probability in the standard normal distribution. Therefore, as the sample size increases, \(\Pr{E}\) also increases.

Key concepts

Standard Error Calculation

Understanding the standard error (SE) is crucial when dealing with statistics, especially when estimating population parameters from sample data. The standard error quantifies the variability of sample mean estimates from one sample to another. If you imagine taking multiple samples from a population and calculating the mean for each sample, the standard error tells how spread out these means would be.

In mathematical terms, the standard error is given by the formula: \[\begin{equation}SE = \frac{\sigma}{\sqrt{n}},\end{equation}\]where \(\sigma\) is the standard deviation of the population, which shows how much the values in the population deviate from the population mean, and \(n\) is the sample size. The relationship between SE and \(n\) is inverse; as your sample size increases, the standard error decreases. This means the larger your sample, the more precise your estimate of the population mean (\(\bar{y}\)) will be. Importantly, calculating the standard error correctly is the backbone of creating confidence intervals and conducting hypothesis tests for the mean.

Normal Distribution in Statistics

The normal distribution, often called the bell curve due to its shape, is a cornerstone of statistical analysis. It describes how the values of a variable are distributed in a way that most of the values cluster around a central mean and taper off symmetrically towards the extremes.

The characteristics of a normal distribution are determined by its mean (\(\mu\)) and standard deviation (\(\sigma\)). The mean indicates where the center of the distribution lies, while the standard deviation tells us how tightly the values are grouped around that mean. In the context of forced expiratory volume (FEV), assuming a normal distribution means that most individuals' FEV will be near the population mean, with fewer individuals having extremely high or low values.

The significance of the normal distribution lies in its predictability. Knowing that FEV follows a normal distribution allows us to use tools such as z-scores to determine how likely it is that a random individual's FEV—or a sample's mean FEV—will fall within a certain range. This predictability extends to the calculation of probabilities, such as determining \(\Pr{E}\), the probability of the sample mean being within a specific margin from the population mean.

Probability and Sample Size Relationship

The interplay between probability and sample size is fundamental in statistics, impacting the precision of results and the confidence we can have in our findings. The probability that an estimate falls within a certain range of the true population parameter can be thought of as the confidence level. The event \(E\) outlined in the exercise, representing the sample mean being within \(\pm 100\) ml of the population mean, is directly tied to the sample size.

As highlighted by the provided solution, an increase in sample size yields a decrease in the standard error. The smaller standard error, with a fixed margin of error, leads to a smaller z-score. In a normal distribution, smaller z-scores correspond to points closer to the mean, which implies a greater area under the curve—or higher probability—of the sample mean falling within the desired margin from the population mean. Therefore, increasing the sample size increases this probability, enhancing the reliability of the sample mean as an estimate of the population mean.

To summarize, as the sample size \(n\) increases, not only does the standard error decrease, leading to more precise estimates, but the probability \(\Pr{E}\) of the sample mean falling within a given range also increases, improving our confidence in the estimation. This relationship is a key consideration when designing studies and interpreting statistical results.