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Chapter 8: Problem 11

Determine \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\) from the given parameters of the population and the sample size. \(\mu=52, \sigma=10, n=21\)

Short Answer

Expert verified
\(\mu_{\bar{x}} = 52\), \(\sigma_{\bar{x}} \approx 2.18\)

Step by step solution

01

Identify Given Parameters

First, identify the given parameters from the problem. Here, we have the population mean \(\mu = 52\), the population standard deviation \(\sigma = 10\), and the sample size \(\ n = 21\).
02

Determine the Mean of the Sample Mean Distribution

The mean of the sample mean distribution \(\mu_{\bar{x}}\) is equal to the mean of the population \(\mu\). Therefore, \(\mu_{\bar{x}} = 52\).
03

Calculate the Standard Error of the Mean

The standard error of the mean \(\sigma_{\bar{x}}\) can be calculated using the formula \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]. Plugging in the given values: \[ \sigma_{\bar{x}} = \frac{10}{\sqrt{21}} \approx 2.18 \]

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Key Concepts

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

Population Mean
The population mean, denoted by \(\mu\), represents the average value of a particular characteristic in a given population. It is a vital statistic because it offers insights into the central tendency of the population data.

For example, in our exercise, the population mean \(\mu\) is 52. This implies that if we were to calculate the average of all individuals in the population, the result would be 52. Understanding the population mean is crucial because it acts as a benchmark for various statistical analyses.

Whenever we take samples from this population, we would expect the average value of each sample to be around 52, although not necessarily exactly 52 due to random variations in sampling.
Standard Deviation
The standard deviation, represented by \(\sigma\), measures the dispersion or spread of a set of data points in a population. In simpler terms, it tells us how much individual data points in the population deviate from the population mean.

In our exercise, the standard deviation \(\sigma\) is 10. This means that on average, the data points in the population differ from the mean by 10 units. A smaller standard deviation would indicate that the data points are closely clustered around the mean, while a larger standard deviation would mean they are more spread out.

Understanding standard deviation is crucial because it helps in comparing the variability or consistency within different datasets. It also lays the foundation for calculating the standard error of the mean, a concept we will discuss next.
Standard Error of the Mean
The standard error of the mean, denoted as \(\sigma_{\bar{x}}\), represents the standard deviation of the sampling distribution of the sample mean. It provides an idea of how much the sample mean is expected to vary from the population mean when multiple samples are taken from the same population.

In our exercise, to calculate the standard error of the mean, we use the formula \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{\}} \].

Given the population standard deviation \(\sigma = 10\) and the sample size \(\ n = 21\), we can plug these values into the formula:
\[ \sigma_{\bar{x}} = \frac{10}{\sqrt{21}} \approx 2.18 \]

This calculation tells us that the sample mean will be around 2.18 units away from the population mean on average. The standard error of the mean helps in understanding the preciseness of the sample mean as an estimate of the population mean. Smaller values indicate more reliable estimates.

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Most popular questions from this chapter

The quality-control manager of a Long John Silver's restaurant wants to analyze the length of time that a car spends at the drive-through window waiting for an order. It is determined that the mean time spent at the window is 59.3 seconds with a standard deviation of 13.1 seconds. The distribution of time at the window is skewed right (data based on information provided by Danica Williams, student at Joliet Junior College). (a) To obtain probabilities regarding a sample mean using the normal model, what size sample is required? (b) The quality-control manager wishes to use a new delivery system designed to get cars through the drive-through system faster. A random sample of 40 cars results in a sample mean time spent at the window of 56.8 seconds. What is the probability of obtaining a sample mean of 56.8 seconds or less, assuming that the population mean is 59.3 seconds? Do you think that the new system is effective? (c) Treat the next 50 cars that arrive as a simple random sample. There is a \(15 \%\) chance the mean time will be at or below ____ seconds

What happens to the standard deviation of \(\hat{p}\) as the sample size increases? If the sample size is increased by a factor of 4 what happens to the standard deviation of \(\hat{p} ?\)

We assume that we are obtaining simple random samples from infinite populations when obtaining sampling distributions. If the size of the population is finite, we technically need a finite population correction factor. However, if the sample size is small relative to the size of the population, this factor can be ignored. Explain what an "infinite population" is. What is the finite population correction factor? How small must the sample size be relative to the size of the population so that we can ignore the factor? Finally, explain why the factor can be ignored for such samples.

Election Prediction Exit polling is a popular technique used to determine the outcome of an election prior to results being tallied. Suppose a referendum to increase funding for education is on the ballot in a large town (voting population over 100,000 ). An exit poll of 310 voters finds that 164 voted for the referendum. How likely are the results of your sample if the population proportion of voters in the town in favor of the referendum is 0.49 ? Based on your result, comment on the dangers of using exit polling to call elections. Include a discussion of the potential nonsampling error that could disrupt your findings.

Determine \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\) from the given parameters of the population and the sample size. \(\mu=80, \sigma=14, n=49\)
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