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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

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, \(n=16, t=1.6\) b. Upper-tailed test, \(n=14, t=3.2\) c. Lower-tailed test, \(n=20, t=-5.1\) d. Two-tailed test, \(n=16, t=6.3\)

Suppose that the population mean value of interpupillary distance (the distance between the pupils of the left and right eyes) for adult males is \(65 \mathrm{~mm}\) and that the population standard deviation is \(5 \mathrm{~mm}\). a. If the distribution of interpupillary distance is normal and a random sample of \(n=25\) adult males is to be selected, what is the probability that the sample mean distance \(\bar{x}\) for these 25 will be between 64 and \(67 \mathrm{~mm}\) ? At least \(68 \mathrm{~mm}\) ? b. Suppose that a random sample of 100 adult males is to be selected. Without assuming that interpupillary distance is normally distributed, what is the approximate probability that the sample mean distance will be between 64 and 67 \(\mathrm{mm}\) ? At least \(68 \mathrm{~mm} ?\)

The authors of the paper "Serum Zinc Levels of Cord Blood: Relation to Birth Weight and Gestational Period" (Journal of Trace Elements in Medicine and Biology [2015]: \(180-183)\) carried out a study of zinc levels of low-birth- weight babies and normal-birth-weight babies. For a sample of 50 lowbirth- weight babies, the sample mean zinc level was 17.00 and the standard error \(\left(\frac{s}{\sqrt{n}}\right)\) was \(0.43 .\) For a sample of 73 normal- birth-weight babies, the sample mean zinc level was 18.16 and the standard error was 0.32 . Explain why the two standard errors are not the same.

USA TODAY reported that the average amount of money spent on coffee drinks each month is \(\$ 78.00\) (USA Snapshot, November 4, 2016). a. Suppose that this estimate was based on a representative sample of 20 adult Americans. Would you recommend using the one-sample \(t\) confidence interval to estimate the population mean amount spent on coffee for the population of all adult Americans? Explain why or why not. b. If the sample size had been 200 , would you recommend using the one-sample \(t\) confidence interval to estimate the population mean amount spent on coffee for the population of all adult Americans? Explain why or why not.

Students in a representative sample of 65 first-year students selected from a large university in England participated in a study of academic procrastination ("Study Goals and Procrastination Tendencies at Different Stages of the Undergraduate Degree," Studies in Higher Education [2016]: 2028-2043). Each student in the sample completed the Tuckman Procrastination Scale, which measures procrastination tendencies. Scores on this scale can range from 16 to \(64,\) with scores over 40 indicating higher levels of procrastination. For the 65 first-year students in this study, the mean score on the procrastination scale was 37.02 and the standard deviation was 6.44 . a. Construct a \(95 \%\) confidence interval estimate of \(\mu,\) the mean procrastination scale for first-year students at this college. (Hint: See Example 12.7.) b. Based on your interval, is 40 a plausible value for the population mean score? What does this imply about the population of first-year students?
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