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Chapter 12: Problem 13

Explain the difference between \(\bar{x}\) and \(\mu_{\vec{x}}\).

Short Answer

Expert verified
In summary, \(\bar{x}\) and \(\mu_{\vec{x}}\) both represent the mean of a set of numbers, but \(\bar{x}\) is the sample mean, which estimates the true population mean, while \(\mu_{\vec{x}}\) is the actual population mean. The primary differences are the context and usage, with \(\bar{x}\) used for samples and \(\mu_{\vec{x}}\) for entire populations, and the fact that \(\bar{x}\) is usually calculated when \(\mu_{\vec{x}}\) is unknown or impractical to obtain.

Step by step solution

01

Definition of \(\bar{x}\)

\(\bar{x}\) refers to the sample mean, which is the average value of a sample from a population. It is a statistic used to estimate the true population mean, and it is calculated by summing the values in the sample and dividing by the number of items in the sample. The formula for \(\bar{x}\) is given by: \[ \bar{x} =\frac{1}{n} \sum_{i=1}^n x_i \] where \(x_i\) are the individual sample values and \(n\) is the sample size.
02

Definition of \(\mu_{\vec{x}}\)

\(\mu_{\vec{x}}\) is the population mean, which is the average value of an entire population. It represents the true average value of any randomly chosen item from the population. The formula for \(\mu_{\vec{x}}\) is given by: \[ \mu_{\vec{x}} =\frac{1}{N} \sum_{i=1}^N x_i \] where \(x_i\) are the individual values of the entire population and \(N\) is the population size.
03

Differences between \(\bar{x}\) and \(\mu_{\vec{x}}\)

1. Context: \(\bar{x}\) is used when dealing with a sample (a subset of the population), while \(\mu_{\vec{x}}\) is used when working with an entire population. 2. Usage: \(\bar{x}\) is an estimate of the true population mean (\(\mu_{\vec{x}}\)) when it is not possible or practical to compute the mean from the entire population. 3. Sample size vs. Population size: In calculating \(\bar{x}\), we divide by the sample size \(n\), whereas for \(\mu_{\vec{x}}\), we divide by the population size \(N\). 4. Unknown Information: Usually, \(\mu_{\vec{x}}\) is unknown, and we need to estimate it using the sample mean (\(\bar{x}\)). In summary, \(\bar{x}\) and \(\mu_{\vec{x}}\) both refer to the average of a set of numbers, but \(\bar{x}\) is the sample mean representing an estimate of the true population mean, while \(\mu_{\vec{x}}\) is the population mean representing the average value of 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 concept of the "Sample Mean" is essential in statistics. It is denoted by the symbol \( \bar{x} \) and represents the average of a sample, which is a subset of a larger population. To calculate the sample mean, add up all the individual sample values and divide the total by the number of values in the sample.
For example, if you have a sample of five heights in centimeters: 150, 160, 170, 180, and 190, the sample mean \( \bar{x} \) is calculated as:
  • Add up all the heights: 150 + 160 + 170 + 180 + 190 = 850
  • Divide by the number of heights: \( \frac{850}{5} = 170 \)
The sample mean helps us estimate the characteristics of a larger group without assessing each individual in the group. It is a powerful tool because it allows us to draw conclusions about the population with limited data.
Population Mean
The "Population Mean," symbolized by \( \mu_{\vec{x}} \), is the true average of every individual in an entire population. Unlike the sample mean, which only includes a subset, the population mean considers all individuals. Calculating the population mean involves summing up all the individual values of the population and dividing by the population size \( N \).
For instance, if every student's height in a school is recorded, the population mean is the average of all these heights. It accurately reflects the average within the full dataset. However, in most real-world scenarios, obtaining data from every individual in a population is impractical due to constraints like time, cost, and logistics.
  • In surveys, the population mean is often estimated using the sample mean when complete data collection is not feasible.
  • The population mean is a fixed value, but due to difficulty in data collection, it's often unknown, leading to reliance on estimation.
Estimation
Estimation in statistics refers to the process of making inferences about a population parameter based on sample data. When a population mean \( \mu_{\vec{x}} \) is unknown, the sample mean \( \bar{x} \) is often used to estimate it.
This approach is common because it is usually impossible to access or measure every element of a large population. Therefore, researchers use techniques like point estimation or interval estimation to infer population characteristics.
  • Point Estimation: Provides a single value estimate of a population parameter, such as the sample mean being used to infer the population mean.
  • Interval Estimation: Offers a range of values, called confidence intervals, that likely include the population parameter.
It鈥檚 important to select a representative sample for estimation to be accurate, minimizing bias and ensuring reliability of the results.
Sample Size
"Sample Size" refers to the number of observations or data points collected from a population to form a sample. The size of a sample is critical because it influences the accuracy of estimations made about the population.
A larger sample size generally leads to more reliable and precise estimates of the population mean and reduces the margin of error. However, larger samples can also come with increased costs and time requirements.
  • A small sample may lead to imprecise estimates, as it might not capture the diversity within the full population.
  • An appropriate sample size balances the need for accuracy versus the resources available for conducting the study.
Choosing the correct sample size involves statistical planning and consideration of factors like population variability and desired confidence levels. This planning ensures that the gained insights truly represent the whole population.

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Most popular questions from this chapter

Acrylic bone cement is sometimes used in hip and knee replacements to secure an artificial joint in place. The force required to break an acrylic bone cement bond was measured for six specimens, and the resulting mean and standard deviation were 306.09 Newtons and 41.97 Newtons, respectively. Assuming that it is reasonable to believe that breaking force has a distribution that is approximately normal, use a confidence interval to estimate the mean breaking force for acrylic bone cement.

Give as much information as you can about the \(P\) -value of a \(t\) test in each of the following situations: a. Two-tailed test, \(\mathrm{df}=9, t=0.73\) b. Upper-tailed test, \(\mathrm{df}=10, t=0.5\) c. Lower-tailed test, \(n=20, t=-2.1\) d. Two-tailed test, \(n=40, t=1.7\)

A random sample is selected from a population with mean \(\mu=60\) and standard deviation \(\sigma=3\). Determine the mean and standard deviation of the sampling distribution of \(\bar{x}\) for each of the following sample sizes: a. \(n=6\) d. \(n=75\) b. \(n=18\) e. \(n=200\) c. \(n=42\) f. \(n=400\)

Suppose that a random sample of size 64 is to be selected from a population with mean 40 and standard deviation 5. a. What are the mean and standard deviation of the sampling distribution of \(\bar{x}\) ? Describe the shape of the sampling distribution of \(\bar{x}\). b. What is the approximate probability that \(\bar{x}\) will be within 0.5 of the population mean \(\mu\) ? c. What is the approximate probability that \(\bar{x}\) will differ from \(\mu\) by more than \(0.7 ?\)

Suppose that a random sample of size 100 is to be drawn from a population with standard deviation 10 . a. What is the probability that the sample mean will be within 20 of the value of \(\mu\) ? b. For this example \((n=100, \sigma=10)\), complete each of the following statements by calculating the appropriate value: i. Approximately \(95 \%\) of the time, \(\bar{x}\) will be within of \(\mu\). ii. Approximately \(0.3 \%\) of the time, \(\bar{x}\) will be farther than from \(\mu\).
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