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Chapter 8: Problem 11

Explain what it means when we say the value of a sample statistic varies from sample to sample.

Short Answer

Expert verified
The value of a sample statistic varying from sample to sample means that if different samples are taken from the same population and the same statistic is computed for each, it is likely to get different results each time. This variability is inherent due to sampling and forms the basis of statistical inference.

Step by step solution

01

Definition of Sample Statistic

A sample statistic is a numerical value that describes a characteristic of a sample. Samples are subsets of a population, and a sample statistic gives information about some aspect of this sample. Examples include mean, median, mode and standard deviation.
02

Variability of Sample Statistic

When we say that the value of a sample statistic varies from sample to sample, it means that if we were to take multiple samples from the same population, and compute the same statistic for each, we would likely get different results each time. This is primarily due to the fact that each sample, unless it is a census including all members of the population, only contains a subset of the data from the population.
03

Implication of the Variability

The variability of a sample statistic from sample to sample is important because it introduces uncertainty. This concept forms the basis of statistical inference. The goal of statistical inference is to make conclusions about an entire population based on findings from a smaller sample. Understanding the variability in the sample statistics helps counter this uncertainty.

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Most popular questions from this chapter

Suppose that a particular candidate for public office is favored by \(48 \%\) of all registered voters in the district. A polling organization will take a random sample of 500 of these voters and will use \(\hat{p}\), the sample proportion, to estimate \(p\). a. Show that \(\sigma_{p}\), the standard deviation of \(\hat{p}\), is equal to \(0.0223 .\) b. If for a different sample size, \(\sigma_{p}=0.0500\), would you expect more or less sample-to-sample variability in the sample proportions than when \(n=500 ?\) c. Is the sample size that resulted in \(\sigma_{\rho}=0.0500\) larger than 500 or smaller than \(500 ?\) Explain your reasoning.

A random sample is to be selected from a population that has a proportion of successes \(p=0.25\). Determine the mean and standard deviation of the sampling distribution of \(\hat{p}\) for each of the following sample sizes: a. \(n=10\) d. \(n=50\) b. \(n=20\) e. \(n=100\) c. \(n=30\) f. \(n=200\)

For which of the following combinations of sample size and population proportion would the standard deviation of \(\hat{p}\) be smallest? $$ \begin{array}{ll} n=40 & p=0.3 \\ n=60 & p=0.4 \\ n=100 & p=0.5 \end{array} $$

A random sample of size 300 is to be selected from a population. Determine the mean and standard deviation of the sampling distribution of \(\hat{p}\) for each of the following population proportions. a. \(p=0.20\) b. \(p=0.45\) c. \(p=0.70\) d. \(p=0.90\)

A certain chromosome defect occurs in only 1 in 200 adult Caucasian males. A random sample of 100 adult Caucasian males will be selected. The proportion of men in this sample who have the defect, \(\hat{p},\) will be calculated. a. What are the mean and standard deviation of the sampling distribution of \(\hat{p}\) ? b. Is the sampling distribution of \(\hat{p}\) approximately normal? Explain. c. What is the smallest value of \(n\) for which the sampling distribution of \(\hat{p}\) is approximately normal?
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