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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

The report "Majoring in Money: How American College Students Manage Their Finances" (SallieMae, \(2016,\) www.news/salliemae.com, retrieved December 24,2106\()\) includes data from a survey of college students. Each person in a representative sample of 793 college students was asked if they had one or more credit cards and if so, whether they paid their balance in full each month. There were 500 who paid in full each month. For this sample of 500 students, the sample mean credit card balance was reported to be \(\$ 825 .\) The sample standard deviation of the credit card balances for these 500 students was not reported, but for purposes of this exercises, suppose that it was \(\$ 200\). Is there convincing evidence that college students who pay their credit card balance in full each month have mean balance that is lower than \(\$ 906\), the value reported for all college students with credit cards? Carry out a hypothesis test using a significance level of 0.01 .

USA TODAY (October 14,2016 ) reported that Americans spend 4.1 hours per weekday checking work e-mail. This was an estimate based on a survey of 1004 white-collar workers in the United States. a. Suppose that you would like to know if there is evidence that the mean time spent checking work e-mail for white-collar workers in the United States is more than half of the 8 -hour work day. What would you need to assume about the sample in order to use the given sample data to answer this question? b. Given that any concerns about the sample were satisfactorily addressed, carry out a test to decide if there is evidence that the mean time spent checking work e-mail for white-collar workers in the United States is more than half of the 8 -hour work day. Suppose that the sample standard deviation was \(s=1.3\) hours.

The formula used to calculate a confidence interval for the mean of a normal population when \(n\) is small is $$ \bar{x} \pm(t \text { critical value }) \frac{s}{\sqrt{n}} $$ What is the appropriate \(t\) critical value for each of the following confidence levels and sample sizes? a. \(95 \%\) confidence, \(n=15\) b. \(99 \%\) confidence, \(n=20\) c. \(90 \%\) confidence, \(n=26\)

The formula used to calculate a confidence interval for the mean of a normal population is $$ \bar{x} \pm(t \text { critical value }) \frac{s}{\sqrt{n}} $$ What is the appropriate \(t\) critical value for each of the following confidence levels and sample sizes? a. \(90 \%\) confidence, \(n=12\) b. \(90 \%\) confidence, \(n=25\) c. \(95 \%\) confidence, \(n=10\)

The Economist collects data each year on the price of a Big Mac in various countries around the world. A sample of McDonald's restaurants in Europe in July 2016 resulted in the following Big Mac prices (after conversion to U.S. dollars): \(\begin{array}{llll}4.44 & 3.15 & 2.42 & 3.96\end{array}\) \(\begin{array}{llll}4.51 & 4.17 & 3.69 & 4.62\end{array}\) \(\begin{array}{lll}3.80 & 3.36 & 3.85\end{array}\) The mean price of a Big Mac in the U.S. in July 2016 was \$5.04. For purposes of this exercise, you can assume it is reasonable to regard the sample as representative of European McDonald's restaurants. Does the sample provide convincing evidence that the mean July 2016 price of a Big Mac in Europe is less than the reported U.S. price? Test the relevant hypotheses using \(\alpha=0.05 .\) (Hint: See Example 12.12.)
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