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Chapter 7: Problem 10

How does the value of \(\sigma_{\bar{x}}\) change as the sample size increases? Explain.

Short Answer

Expert verified
The value of the standard deviation of the sample mean, \(\sigma_{\bar{x}}\), decreases as the sample size increases. This is because the standard error is inversely proportional to the square root of the sample size. So, a larger sample size results in a smaller standard error.

Step by step solution

01

Understand the concept of Standard Deviation

The standard deviation measures the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.
02

Understand the concept of Standard Deviation of the Sample Mean (known as the Standard Error)

The standard deviation of the sample mean, also called the standard error, shows how much the sample mean is likely to vary from the population mean. The formula to compute the standard deviation of the sample mean is: \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\) where \(\sigma\) is the population standard deviation and \(n\) is the sample size.
03

Explain the effect of sample size on the standard error

From the formula, we see that the standard error is inversely proportional to the square root of the sample size. As \(n\) (sample size) increases, the denominator of the formula becomes larger, which reduces the value of \(\sigma_{\bar{x}}\). Therefore, as the sample size increases, the standard error (or the standard deviation of the sample mean) decreases.

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Key Concepts

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

Sample Size
The term "sample size" refers to the number of observations or data points collected in a study or experiment. It plays a crucial role in determining the accuracy and reliability of statistical results.
  • A larger sample size tends to provide more accurate estimates of the population parameters, such as the population mean or standard deviation.
  • The sample size affects the precision of the sample mean. As it increases, the variability of the sample mean tends to decrease.
  • It impacts the standard error, which decreases as the sample size grows. Due to this, the conclusions drawn from larger samples are generally more reliable.
When you increase the sample size, you enhance the robustness of your findings, allowing for a more precise estimation of the population characteristics.
Understanding sample size is vital because it directly influences the level of confidence you can have in the results of your analysis.
Standard Deviation
Standard deviation is a statistical measurement that indicates the spread or dispersion of a dataset around its mean. It is a crucial concept for understanding data variability.
  • A low standard deviation means that the data points are closer to the mean, suggesting less variability within the dataset.
  • A high standard deviation implies a wide range of values, indicating greater variability and a spread-out dataset.
To calculate standard deviation, the data is represented by a mean (\(ar{x}\)), and the deviation of each point from this mean is computed.
The standard deviation is then derived from the square root of the average squared deviations from the mean.
This provides insight into how data values diverge, helping to assess data consistency and predictability. With knowledge of standard deviation, you can ascertain the overall distribution pattern of your data.
Population Mean
The population mean is the average of all measurements in a population. It reflects the central tendency of the entire data set. While it serves as one of the core parameters in statistics, finding the actual population mean can be challenging.
  • The population mean is denoted by the symbol \(\mu\) and is seen as an ultimate benchmark against which sample means are compared.
  • It provides a baseline for understanding other statistical measures like variance and standard deviation.
  • When the population size is large, achieving an accurate estimate of the population mean through sampling becomes essential.
A sample mean attempts to approximate the population mean.
Random sampling helps in estimating the population mean with some degree of accuracy by providing a smaller representation of the larger population.
Understanding the population mean is fundamental as it provides a basis for inferring statistical results about the population from which the sample is drawn.

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Most popular questions from this chapter

Briefly explain the meaning of a population probability distribution and a sampling distribution. Give an example of each.

In a January 2014 survey conducted by the Associated PressWe TV, \(68 \%\) of American adults said that owning a home is the most important thing or \(a\) very important but not the most important thing (opportunityagenda.org). Assume that this percentage is true for the current population of American adults. Let \(\hat{p}\) be the proportion in a random sample of 1000 American adults who hold the above opinion. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\) and describe its shape.

What is the estimator of the population proportion? Is this estimator an unbiased estimator of \(p ?\) Explain why or why not.

When is an estimator said to be consistent? Is the sample mean, \(\bar{x}\), a consistent estimator of \(\mu ?\) Explain.

The times that college students spend studying per week have a distribution that is skewed to the right with a mean of \(8.4\) hours and a standard deviation of \(2.7\) hours. Find the probability that the mean time spent studying per week for a random sample of 45 students would be a. between 8 and 9 hours b. less than 8 hours
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