Question

Let \(x\) denote the time (in minutes) that it takes a fifth-grade student to read a certain passage. Suppose that the mean value and standard deviation of the \(x\) distribution are \(\mu=2\) minutes and \(\sigma=0.8\) minutes, respectively. a. If \(\bar{x}\) is the sample mean time for a random sample of \(n=\) 9 students, where is the \(\bar{x}\) distribution centered, and what is the standard deviation of the \(\bar{x}\) distribution? b. Repeat Part (a) for a sample of size of \(n=20\) and again for a sample of size \(n=100\). How do the centers and spreads of the three \(\bar{x}\) distributions compare to one another? Which sample size would be most likely to result in an \(\bar{x}\) value close to \(\mu\), and why?

Short answer

For all three cases, the sample mean distribution is centered at 2 minutes. The standard deviation of the sample mean decreases as the sample size increases, from 0.27 minutes for n=9, to 0.18 minutes for n=20, and to 0.08 minutes for n=100. Therefore, a sample size of 100 would be the most likely to result in a sample mean value close to the population mean, due to reduced sample-to-sample variability.

Step-by-step solution

Calculate for n=9

First, consider the random sample of 9 students. The sample mean \(\bar{x}\) will be centered at the same value as our population mean, so \(\bar{x}=\mu=2\) minutes. The standard deviation of the sample mean is calculated as \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{0.8}{\sqrt{9}} = \frac{0.8}{3} = 0.27\) minutes.

Calculate for n=20

Next, consider a random sample of 20 students. The sample mean again equals the population mean, so \(\bar{x}=\mu=2\) minutes. The standard deviation of the sample mean is now \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{0.8}{\sqrt{20}} = 0.18\) minutes.

Calculate for n=100

Finally, consider a random sample of 100 students. The sample mean still equals the population mean, so \(\bar{x}=\mu=2\) minutes. The standard deviation of the sample mean is now \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{0.8}{\sqrt{100}} = 0.08\) minutes.

Compare the distributions

All three distributions have the same center value \(\bar{x}=2\) minutes, since this value is determined by the population mean. However, the standard deviation of the sample mean decreases as the sample size increases, with values of 0.27 minutes, 0.18 minutes, and 0.08 minutes for sample sizes of 9, 20, and 100 respectively. Therefore, a larger sample size is more likely to result in a sample mean value closer to the population mean, simply because there is less sample-to-sample variability.

Key concepts

Standard Deviation of the Sample Mean

The standard deviation of the sample mean, often denoted as\(\bar{x}\), plays a pivotal role in understanding the precision of the sample mean as an estimate of the population mean. It is computed using the formula \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.

With the increase in sample size, the standard deviation of the sample mean decreases. This decrease indicates that as we collect more data, our sample mean will typically be closer to the true population mean. Hence, the larger the sample, the more reliable our estimate of the population mean becomes.

Population Mean

In statistics, the population mean, represented by \(\mu\), is the average value of all the measurements in the population. It serves as a measure of central tendency, giving us an overall idea of the typical value within the population. When sampling, regardless of the sample size, the distribution of sample means is expected to be centered around this population mean. This centrality does not tell us about the spread or the variability; it simply indicates where the middle of the sample mean distribution lies.

Sample Size

Sample size, denoted by \(n\), refers to the number of observations collected from a population. The size of the sample has a direct impact on the statistical properties of the sample mean, particularly the standard deviation of the sample mean. As shown in the exercise solutions, increasing sample size results in a decreased standard deviation of the sample mean. But why does this happen?

When the sample size is larger, the sample includes more information about the population, and each individual extreme value has less weight, leading to a reduced chance of an atypical sample mean. This phenomenon is known as the law of large numbers. A larger sample size is thus a key factor in reducing uncertainty about the population mean.

Sampling Variability

Sampling variability, also known as sampling error, is the natural variation in the sample mean from one sample to another. It is influenced by the sample size and the population's natural diversity. When the standard deviation of the sample mean is small, it indicates low sampling variability, meaning that the sample mean is likely to be close to the population mean. As we observed from the step-by-step solution, samples with larger sizes demonstrated lower sampling variability, thereby giving a more consistent estimate of the population mean.

Sampling variability is a crucial concept because it helps in assessing the reliability of statistical estimates. If the sampling variability is high, then different samples could give very different estimates of the population mean, decreasing confidence in our findings.