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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 \(s\) represent sample mean and sample standard deviation respectively, while \(\mu\) and \(\sigma\) represent population mean and population standard deviation respectively.

Step by step solution

01

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

The symbol \(\bar{x}\) denotes the sample mean, which is calculated by summing all the data points in a sample and dividing by the number of points in that sample. \(\mu\), on the other hand, denotes the population mean, which is calculated by summing all data points in an entire population and dividing by the number of points in the population.
02

Differentiate between \(\bar{x}\) and \(\mu\)

The main difference between \(\bar{x}\) and \(\mu\) is the data groups they represent. \(\bar{x}\) is used when the data represents a sample from the larger population, while \(\mu\) is used when the data encompasses the entire population.
03

Define \(s\) and \(\sigma\)

The symbol \(s\) denotes the sample standard deviation, a measure of the dispersion or spread of data points in a sample. It is calculated considering the mean of the sample. \(\sigma\), on the other hand, is the population standard deviation, a measure of the spread of data points in a population. It is computed using the population mean.
04

Differentiate between \(s\) and \(\sigma\)

The primary difference between \(s\) and \(\sigma\) is the data groups they represent. \(s\) is used for the standard deviation of a sample, while \(\sigma\) is used for the standard deviation of an entire population.

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Key Concepts

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

Sample Mean (\bar{x})
The sample mean, symbolized as \(\bar{x}\), plays a crucial role when dealing with statistics derived from a subset of a larger group or population. For instance, if you wanted to know about average test scores but couldn't review every single student's paper, you would take the scores of a few, a 'sample'. Calculating the sample mean involves adding their scores and dividing by the total number of scores you've collected. This figure gives you an estimate, a glimpse, into the average of the entire group without the need for exhaustive data collection.

Population Mean (\text{\textmu})
On the opposite end, we have the population mean, denoted as \(\mu\), which is the average calculated by considering every single individual within a full set. This means if you wanted to calculate the true average test score, you'd add up the scores from every student and divide by the total number of students. The population mean is used when it's possible to include all data points, and it provides the most accurate reflection of the data set as a whole.

  • Understanding the population mean is essential when accuracy is non-negotiable.
  • It is impractical for large populations due to time and resource constraints.
Sample Standard Deviation (s)
Turning our attention to variability within our data, the sample standard deviation, referred to with the small letter \(s\), quantifies how spread out our sample data points are around their mean (\bar{x}). Imagine that all students use a particular study strategy, and you want to know how consistent their test score results are. You gather scores from a handful of these students; calculating the standard deviation will tell you how varied their scores are from your calculated sample mean. It's a valuable measure that conveys the diversity within a sample set.

Population Standard Deviation (\text{\textsigma})
Similar to what we discussed with the mean, the population standard deviation, represented by the Greek letter \(\sigma\), measures the dispersion of data points in a full population relative to the population mean (\text{\textmu}). Continuing with our example, if it was possible to compile the results of every student using the study strategy, and you determined their average results, \(\sigma\) would tell you how much individual scores deviate from this average. It's an intensive process but crucial for accurate and comprehensive variability analysis.

  • Population standard deviation is necessary for precise data representation when dealing with entire populations.
  • It provides the bedrock for conclusions about data spread on a massive scale.

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Most popular questions from this chapter

Consider a population consisting of the following five values, which represent the number of DVD rentals during the academic year for each of five housemates: \(\begin{array}{lllll}8 & 14 & 16 & 10 & 11\end{array}\) a. Compute the mean of this population. b. Select a random sample of size 2 by writing the five numbers in this population on slips of paper, mixing them, and then selecting two. Compute the mean of your sample. c. Repeatedly select samples of size 2 , and compute the \(\bar{x}\) value for each sample until you have the \(\bar{x}\) values for 25 samples. d. Construct a density histogram using the \(25 \bar{x}\) values. Are most of the \(\bar{x}\) values near the population mean? Do the \(\bar{x}\) values differ a lot from sample to sample, or do they tend to be similar?

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

Let \(x\) denote the time (in minutes) that it takes a fifth-grade student to read a certain passage. Suppose that the mean value and standard deviation of \(x\) are \(\mu=\) 2 minutes and \(\sigma=0.8\) minute, respectively. a. If \(\bar{x}\) is the sample mean time for a random sample of \(n=9\) students, where is the \(\bar{x}\) distribution centered, and how much does it spread out about the center (as described by its standard deviation)? \(\mu_{\mathrm{x}}=2, \sigma_{\mathrm{x}}=0.267\) b. Repeat Part (a) for a sample of size of \(n=20\) and again for a sample of size \(n=100\). How do the centers and spreads of the three \(\bar{x}\) distributions compare to one another? Which sample size would be most likely to result in an \(\bar{x}\) value close to \(\mu\), and why?

Consider the following population: \(\\{1,2,3,4\\} .\) Note that the population mean is \(\mu=\frac{1+2+3+4}{4}=2.5\) a. Suppose that a random sample of size 2 is to be selected without replacement from this population. There are 12 possible samples (provided that the order in which observations are selected is taken into account): \(\begin{array}{rrrrrr}1,2 & 1,3 & 1,4 & 2,1 & 2,3 & 2,4 \\ 3,1 & 3,2 & 3,4 & 4,1 & 4,2 & 4,3\end{array}\) Compute the sample mean for each of the 12 possible samples. Use this information to construct the sampling distribution of \(\bar{x}\). (Display the sampling distribution as a density histogram.) b. Suppose that a random sample of size 2 is to be selected, but this time sampling will be done with replacement. Using a method similar to that of Part (a), construct the sampling distribution of \(\bar{x}\). (Hint: There are 16 different possible samples in this case.) c. In what ways are the two sampling distributions of Parts (a) and (b) similar? In what ways are they different?

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 \(\bar{x}\) sampling distribution? Describe the shape of the \(\bar{x}\) sampling distribution. \(\mu_{\bar{\lambda}}=40 \sigma_{\bar{x}}=0.625\) 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 \(0.7 ?\)
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