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Chapter 8: Problem 2

What is the difference between \(\bar{x}\) and \(\mu\) ? between s and \(\sigma\) ?

Short Answer

Expert verified
\(\bar{x}\) and \(s\) refer to the mean and standard deviation of a sample, respectively, while \(\mu\) and \(\sigma\) refer to the mean and standard deviation of an entire population, respectively.

Step by step solution

01

Understanding \(\bar{x}\) and \(\mu\)

The symbol \(\bar{x}\) represents the sample mean, which is the average of a set of observations drawn from a population. On the other hand, \(\mu\) represents the population mean, which is the average of all observations in the population. So, the difference between them lies in whether we are referring to a sample (subset) or the entire population.
02

Understanding \(s\) and \(\sigma\)

The symbol \(s\) represents the sample standard deviation, which is used to measure the dispersion or variance among the sample data points from the sample mean. Meanwhile, \(\sigma\) represents the population standard deviation, which measures how dispersed or spread out the values in an entire population are from the population mean. So, similar to \(\bar{x}\) and \(\mu\), the difference between \(s\) and \(\sigma\) is that \(s\) refers to a sample (subset) of a population and \(\sigma\) refers to the entire population.

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Key Concepts

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

Sample Mean
The sample mean, denoted as \( \bar{x} \), is essentially the average value of a selected group or subset of data from a larger population. For example, if you take a few leaves from a tree to measure their size, \( \bar{x} \) would be the average size of those particular leaves. Calculating the sample mean involves adding up all the values of the sample data and then dividing by the number of data points. This gives us a snapshot, or a short summary of the sample data:
  • Add together all data points.
  • Divide the total by the number of data points.
This provides a useful overview when we can't measure the whole population.
Population Mean
The population mean, symbolized as \( \mu \), represents the true average of an entire set of data being studied. This could be the average height of all students in a school if every single student鈥檚 height was measured. Unlike the sample mean, the population mean requires data from the entire group we are interested in. Understanding \( \mu \) helps us grasp concepts such as:
  • Central tendency of a population.
  • Effect of each data point equally.
We often have to estimate \( \mu \) because acquiring data from every member of a population can be impractical.
Standard Deviation
Standard deviation is a measure of how spread out numbers are in a data set. It tells us to what degree the values deviate from the mean. In simple terms, it's about understanding how much variation or "scattering" there is in the data around its mean. There are two types:
  • Sample Standard Deviation, \( s \)
  • Population Standard Deviation, \( \sigma \)
The difference lies in whether we are working with a data sample or the full population. Both aim to show how much data values differ but from different perspectives.
Sample
A sample refers to a smaller, manageable version of a larger group or population. It's not feasible to collect data from everyone or everything, so we often use samples to make conclusions about the overall population. Plus, it's:
  • Cost-effective.
  • Time-saving.
  • Practical for large populations.
Remember, the reliability of conclusions drawn from a sample heavily depends on how well the sample represents the population.
Population
The term population refers to the complete set of items or people we are interested in studying. For instance, if we talk about the population of a city, it means every single person living there. Studying an entire population gives the most accurate insights, albeit at higher resource costs. In statistical analyses, populations are:
  • Ideal for complete accuracy.
  • Sometimes too large for practical data collection.
  • The basis for generalizing study findings where possible;
Populations provide a comprehensive view that really captures the full scope of the subject being studied.

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Most popular questions from this chapter

In the library on a university campus, there is a sign in the elevator that indicates a limit of 16 persons. In addition, there is a weight limit of 2500 pounds. Assume that the average weight of students, faculty, and staff on campus is 150 pounds, that the standard deviation is 27 pounds, and that the distribution of weights of individuals on campus is approximately normal. If a random sample of 16 persons from the campus is to be taken: a. What is the expected value of the distribution of the sample mean weight? b. What is the standard deviation of the sampling distribution of the sample mean weight? c. What average weights for a sample of 16 people will result in the total weight exceeding the weight limit of 2500 pounds? d. What is the chance that a random sample of 16 people will exceed the weight limit?

Explain the difference between \(\sigma\) and \(\sigma_{\bar{x}}\) and between \(\mu\) and \(\mu_{\bar{x}}^{-\text {. }}\)

A random sample is to be selected from a population that has a proportion of successes \(p=.65 .\) Determine the mean and standard deviation of the sampling distribution of \(\hat{p}\) for each of the following sample sizes:\(8.23\) A random sample is to be selected from a population that has a proportion of successes \(p=.65 .\) Determine the mean and standard deviation of the sampling distribution of \(\hat{p}\) for each of the following sample sizes: a. \(n=10\) b. \(n=20\) c. \(n=30\) d. \(n=50\) e. \(n=100\) f. \(n=200\)

A manufacturing process is designed to produce bolts with a \(0.5\) -inch diameter. Once each day, a random sample of 36 bolts is selected and the bolt diameters are recorded. If the resulting sample mean is less than \(0.49\) inches or greater than \(0.51\) inches, the process is shut down for adjustment. The standard deviation for diameter is \(0.02\) inches. What is the probability that the manufacturing line will be shut down unnecessarily? (Hint: Find the probability of observing an \(\bar{x}\) in the shutdown range when the true process mean really is \(0.5\) inches.)

A certain chromosome defect occurs in only 1 in 200 adult Caucasian males. A random sample of \(n=100\) adult Caucasian males is to be obtained. a. What is the mean value of the sample proportion \(\hat{p}\), and what is the standard deviation of the sample proportion? b. Does \(\hat{p}\) have approximately a normal distribution in this case? Explain. c. What is the smallest value of \(n\) for which the sampling distribution of \(\hat{p}\) is approximately normal?
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