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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

In a population of 5000 subjects, 600 possess a certain characteristic. In a sample of 120 subjects selected from this population, 18 possess the same characteristic. What are the values of the population and sample proportions?

If all possible samples of the same (large) size are selected from a population, what percentage of all the sample means will be within \(2.5\) standard deviations \(\left(\sigma_{\bar{x}}\right)\) of the population mean?

Consider a large population with \(\mu=60\) and \(\sigma=10\). Assuming \(n / N \leq .05\), find the mean and standard deviation of the sample mean, \(\bar{x}\), for a sample size of a. 18 b. 90

Mong Corporation makes auto batteries. The company claims that \(80 \%\) of its LL70 batteries are good for 70 months or longer. Assume that this claim is true. Let \(\hat{p}\) be the proportion in a sample of 100 such batteries that are good for 70 months or longer. a. What is the probability that this sample proportion is within \(.05\) of the population proportion? b. What is the probability that this sample proportion is less than the population proportion by \(.06\) or more? c. What is the probability that this sample proportion is greater than the population proportion by \(.07\) or more?

Consider a large population with \(p=.21\). Assuming \(n / N \leq .05\), find the mean and standard deviation of the sample proportion \(\hat{p}\) for a sample size of a. 400 b. 750
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