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Chapter 7: Problem 65

For a population, \(N=12,000\) and \(p=.71 .\) A random sample of 900 elements selected from this population gave \(\hat{p}=.66 .\) Find the sampling error.

Short Answer

Expert verified
The sampling error is 0.05.

Step by step solution

01

Identify the given values

Firstly, identify the given values in the problem. So, the population size (N) is 12,000, the population proportion (p) is .71, the sample size (n) is 900, and the sample proportion (\(\hat{p}\)) is .66.
02

Formulate the formula for Sampling Error

The formula for the Sampling Error is \(\|p - \hat{p}\|\) where \(p\) is the population proportion, \(\hat{p}\) is the sample proportion.
03

Substitute the values into the formula

Substitute the given values into the formula: \(\|0.71 - 0.66\|\)
04

Solve the equation

Subtract to get the sampling error. The result is 0.05.

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Key Concepts

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

Population Proportion
Population proportion, denoted by the symbol \(p\), represents the ratio of elements in a population that possess a certain characteristic. To illustrate, in a survey of 12,000 individuals, if 71% are left-handed, then the population proportion is 0.71. It's a vital figure in statistics as it helps us understand the makeup of a whole population. This proportion is used as a benchmark to determine how accurate a sample is at representing the population.
The more precise the population proportion, the easier it becomes to infer the behaviors or traits of a large group based on smaller samples.
Sample Proportion
The sample proportion, denoted as \(\hat{p}\), is the equivalent of the population proportion but for the sample only. It reflects the percentage of sample elements exhibiting a specific trait. If a sample of 900 people from a city of 12,000 has 594 left-handed individuals, then the sample proportion is 0.66 (594 divided by 900).
While it's aimed to mimic the population proportion, variations can exist due to sampling errors, which result from the randomness inherent in sample selection.
  • Sample proportion provides a snapshot of data used to estimate the population proportion.
  • It serves as a foundation for making inferences about the broader population.
  • Discrepancies between sample and population proportions lead to sampling error, crucial for statistical analysis.
Sampling Techniques
Sampling techniques are strategies employed to select a subset of individuals from a larger population. They are essential to ensure the sample accurately reflects the population, reducing biases and errors.
Common techniques include:
  • Simple Random Sampling: Involves randomly selecting samples, ensuring every individual has an equal chance of selection.
  • Stratified Sampling: The population is divided into distinct strata, and samples are taken from each stratum proportional to their size in the population.
  • Systematic Sampling: Selects every \(k\)-th individual from a list after a random start.
Employing appropriate sampling techniques ensures more reliable and valid data, crucial for accurate statistical analysis.
Sample Size
The sample size represents the number of individuals from the population chosen to represent the population itself. In statistics, the sample size is crucial for the accuracy of conclusions drawn from sample analysis.
A larger sample size generally provides a more accurate estimation of the population parameters. It's akin to taking more snapshots of a phenomenon, creating a clearer picture. However, it should be balanced against cost and feasibility.
  • Larger sample sizes tend to yield results closer to the actual population parameters, reducing sampling error.
  • The sample size must be adequate to ensure statistical power and precision in estimates.
  • Determining the optimal sample size is a critical step in study design, influenced by desired confidence levels and margin of error.
Understanding the implications of sample size assists researchers in making well-informed decisions and deriving meaningful insights from data.

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Most popular questions from this chapter

For a population, \(\mu=46\) and \(\sigma=10\). a. For a sample selected from this population, \(\mu_{i}=46\) and \(\sigma_{\bar{z}}=2.0 .\) Find the sample size. Assume \(n / N \leq .05\). b. For a sample selected from this population, \(\mu_{\bar{x}}=46\) and \(\sigma_{\bar{x}}=1.6 .\) Find the sample size. Assume \(n / N \simeq 05\).

In an article by Laroche et al. (The Journal of the American Board of Family Medicine \(2007 ; 20: 9-15\) ), the average daily fat intake of U.S. adults with children in the household is \(91.4\) grams, with a standard deviation of \(93.25\) grams. These results are based on a sample of 3714 adults. Suppose that these results hold true for the current population distribution of daily fat intake of such adults, and that this distribution is strongly skewed to the right. Let \(\bar{x}\) be the average daily fat intake of 20 randomly selected U.S. adults with children in the household. Find the mean and the standard deviation of the sampling distribution of \(\bar{x}\). Do the same for a random sample of size \(75 .\) How do the shapes of the sampling distributions differ for the two sample sizes?

The amounts of electricity bills for all households in a city have a skewed probability distribution with a mean of \(\$ 140\) and a standard deviation of \(\$ 30\). Find the probability that the mean amount of electric bills for a random sample of 75 households selected from this city will be \(\mathrm{a}_{2}\) between \(\$ 132\) and \(\$ 136\) b. within \(\$ 6\) of the population mean \(c\), more than the population mean by at least \(\$ 4\)

Suppose the incomes of all people in the United States who own hybrid (gas and electric) automobiles are normally distributed with a mean of \(\$ 78,000\) and a standard deviation of \(\$ 8300\). Let \(\bar{x}\) be the mean income of a random sample of 50 such owners. Calculate the mean and standard deviation of \(\bar{x}\) and describe the shape of its sampling distribution.

According to the central limit theorem, the sampling distribution of \(\hat{p}\) is approximately normal when the sample is large. What is considered a large sample in the case of the proportion? Briefly explain.
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