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Chapter 6: Problem 90

Compute the standard error for sample means from a population with mean \(\mu=100\) and standard deviation \(\sigma=25\) for sample sizes of \(n=30, n=200\), and \(n=1000\). What effect does increasing the sample size have on the standard error? Using this information about the effect on the standard error, discuss the effect of increasing the sample size on the accuracy of using a sample mean to estimate a population mean.

Short Answer

Expert verified
It can be concluded that increasing the sample size results in smaller standard error, leading to more accurate estimates. Hence, a larger sample is more likely to closely approximate the true population mean.

Step by step solution

01

Compute Standard Error for n=30

The standard error (SE) is computed using the formula \(\text{SE} = \frac{\sigma}{\sqrt{n}}\) where \( \sigma \) is the standard deviation and \( n \) is the sample size. Plugging in the values given, we get \( \text{SE} = \frac{25}{\sqrt{30}} \)
02

Compute Standard Error for n=200

Using the same formula, the SE for a sample size of 200 is \( \text{SE} = \frac{25}{\sqrt{200}} \)
03

Compute Standard Error for n=1000

Similarly, for a sample size of 1000, the SE is \( \text{SE} = \frac{25}{\sqrt{1000}} \)
04

Analyze effect of increasing sample size on SE

As is evident from the calculated SEs, the SE decreases with the increase in sample size. What this means is that the estimates based on larger samples tend to be more accurate than those based on smaller ones. By increasing the sample size, the sample mean becomes a more accurate predictor of the population mean.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Sample Size
When discussing statistics, the term *sample size* is often encountered. It refers to the number of observations or data points collected in a sample. In statistical analyses, choosing the appropriate sample size is crucial. A well-chosen sample size ensures that the results are reliable and insights are valid.

In the context of the standard error calculation, the sample size is represented by the symbol \(n\). The relationship between sample size and standard error is inversely proportional. This means that as you increase your sample size, the standard error decreases, leading to more reliable estimations. Having a larger sample size reduces the uncertainty in how well the sample mean approximates the population mean.
  • Bigger sample sizes generally offer more accurate and less variable results.
  • A small sample size might give an estimate that is too unstable or variable.
Choosing the right sample size is a balance between cost, time, and the need for accuracy. Often, statisticians will use power analysis to determine the optimal sample size before beginning a study.
What is Population Mean?
The *population mean* is a key measure in statistics that represents the average of a set of characteristics or values of an entire population. It is denoted by the Greek letter \(\mu\) and is one of the most fundamental parameters of interest in statistical analysis.

Understanding population mean is crucial because it serves as the reference point for statistical inference. Whenever we collect data from a sample, our target is to accurately estimate the population mean using this sample. However, because collecting data from an entire population is often impractical, we rely on sample means, which are used to make inferences about the population mean.
  • If the sample is truly representative, the sample mean outputs are expected to hover around the population mean.
  • The accuracy of the sample mean as an estimator of the population mean improves with increased sample size, because the standard error gets smaller.
Demystifying Sampling Distribution
A *sampling distribution* is a probability distribution of a statistic obtained through a large number of samples drawn from a specific population. In practical terms, it describes the spread of various sample means for a given sample size.

The beauty of sampling distributions lies in their ability to give us insights into how sample means behave. According to the Central Limit Theorem, as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the population distribution's shape.
  • The standard deviation of a sampling distribution is known as the standard error (SE), which decreases with an increased sample size.
  • The decreasing standard error means that the sample mean becomes a better estimator of the population mean as the sample size grows.
Understanding sampling distributions allows statisticians to make predictions and form conclusions about population parameters based on sample data. It shows us the power of using samples to draw meaningful conclusions about populations.

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Most popular questions from this chapter

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A sample with \(n=12, \bar{x}=7.6,\) and \(s=1.6\)
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