Abstract
In this chapter, we take an in-depth look at the operational end of statistical analysis. Statistical analysis primarily involves selecting parts of populations (known as samples) and analyzing them in order to make inferences about the populations. Inferences made about a population by using sample data are widespread in business, economics, and finance. For example, the A. C. Nielsen Company infers the number of people who watch each television show on the basis of a sample of TV viewers. The use of political polls to project election winners is another example of statistical inference. And when you fill out a warranty card on an appliance you have bought, you are often asked to provide information about yourself that the warrantor compiles (and probably sells to someone who will later try to convince you to buy a magazine subscription). These data are also sample data.
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Notes
- 1.
This is because
$$\begin{array}{llll}E\left( {\overline{X}} \right) & =E\left({\frac{1}{n}\sum\limits_{i=1}^n {{X_i}} } \right) \\&=\frac{1}{n}\left[ {E\left( {{X_1}} \right)+E\left( {{X_2}}\right)+\cdots +E\left( {{X_n}} \right)} \right] \\&=\frac{1}{n}(n\mu )=\mu \end{array}$$ - 2.
X 1, X 2, …, X n are independent of each other, so we can use Eq. 6.31 in Chap. 6 to obtain
$$ \begin{array}{lll} \mathrm{ Var}\left( {\sum\limits_{i=1}^n {{X_i}} } \right)&= \;\mathrm{ Var}\left( {{X_1}} \right)+\mathrm{Var}\left( {{X_2}} \right)+\cdots +\mathrm{ Var}\left( {{X_n}}\right) \\ & = \; n\sigma_X^2\end{array} $$Therefore,
$$ \frac{1}{{{n^2}}}\mathrm{ Var}\left( {\sum\limits_{i=1}^n {{X_i}} } \right)=\frac{1}{{{n^2}}}\left( {n\sigma_x^2} \right)=\frac{{^{{\sigma_X^2}}}}{n} $$Because \( \sigma_X^2 \) generally is not known, it can be estimated by \( s_X^2 \), the sample variance:
$$ s_X^2=\frac{{\sum\limits_{i=1}^n {{{{\left( {{X_i}-\overline{X}} \right)}}^2}} }}{n-1 } $$ - 3.
We encountered this issue in Chap. 6, where we found that the hypergeometric distribution considered the population size N but the binomial distribution did not. Equation 6.15 can be redefined as
$$ \left[ {\begin{array}{ccc} {\mathrm{ Variance}\;\mathrm{ of}\;\mathrm{ hypergeometric}} \\{\mathrm{ random}\;\mathrm{ variable}} \\\end{array}} \right]=\left[ {\begin{array}{ccc} {\mathrm{ Variance}\;\mathrm{ of}\;\mathrm{ corresponding}} \\{\mathrm{ binomial}\;\mathrm{ random}\;\mathrm{ variable}} \\\end{array}} \right]\cdot \left[ {\frac{N-n }{N-1 }} \right] $$ - 4.
Random samples from a uniform distribution for sample size n = 2, 5, 10, 25, and 50 are presented in Appendix 1.
- 5.
Bailey A.D. Jr.: Statistical Auditing: Review, Concepts and Problems, pp. 138–42. New York, Harcourt, Brace Jovanovich (1981)
- 6.
- 7.
If the population standard deviation is not known, then we can use the information on sample mean and sample variance to do a similar analysis. This kind of analysis will be done in Sect. 9.3.
- 8.
Sloan F.A., Lorant J.H.:The role of patient waiting time: Evidence from physicians’ practices. J. Bus., October, 486–507 (1977)
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Appendix 1: Sampling Distribution from a Uniform Population Distribution
Appendix 1: Sampling Distribution from a Uniform Population Distribution
To show how sample size can affect the shape and standard deviation of a sample distribution, consider samples of size n = 2, 5, 10, 25, and 50 taken from the uniform distribution shown in Fig. 8.9.
Uniform distribution from 0 to 1
To generate different random samples with different sample sizes, we use the MINITAB random variable generator with uniform distribution. Portions of this output are shown in Fig. 8.1b in the text discussion. First we generate 40 random samples with a sample size of 2. Similarly, we generate 40 random samples for n = 5, n = 10, n = 25, and n = 50.
Forty sample means for sample sizes equal to 2, 5, 10, 25, and 50 are presented in Table 8.13. Histograms based on the five sets of data given in Table 8.13 are presented in Figs. 8.10, 8.11, 8.12, 8.13, and 8.14, respectively. The means associated with Figs. 8.10, 8.11, 8.12, 8.13, and 8.14 are .4458, .4857, .4776, .48688, and .49650, respectively; the standard deviations associated with Figs. 8.10, 8.11, 8.12, 8.13, and 8.14 are .1927, .1300, .0890, .06235, and .04414. By comparing these five figures, we can draw two important conclusions. First, when sample size increases from 2 to 50, the shape of the histogram becomes more similar to the bell-shaped normal distribution. Second, as the sample size increases, the standard deviation of the sample mean falls drastically. In sum, this data simulation reinforces the central limit theorem discussed in Sect. 8.6.
Histogram of 40 sample means (n = 2)
Histogram of 40 sample means (n = 5)
Histogram of 40 sample means (n = 10). Histogram of 40 sample means (n = 5)
Histogram of 40 sample means (n = 25). Histogram of 40 sample means (n = 5)
Histogram of 40 sample means (n = 50)
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Lee, CF., Lee, J.C., Lee, A.C. (2013). Sampling and Sampling Distributions. In: Statistics for Business and Financial Economics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5897-5_8
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