Bhisham C. Gupta - Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP

Figure 2.5.4Distribution of amounts of soft drink contained in bottles.

Example 2.5.12(Applying the empirical rule) At the end of each fiscal year, a manufacturer writes off or adjusts its financial records to show the number of units of bad production occurring over all lots of production during the year. Suppose that the dollar values associated with the various units of bad production form a bell‐shaped distribution with mean and standard deviation = $2500. Find the percentage of units of bad production that has a dollar value between $28,200 and $43,200 .

Solution:From the information provided, we have and = $2500. Since the limits $28,200 and $43,200 are three standard deviations away from the mean, applying the empirical rule shows that approximately 99.7% units of the bad production has dollar value between $28,200 and $43,200.

Figure 2.5.5 Dollar value of units of bad production.

If the population data have a distribution that is not bell‐shaped, then we use another result, called Chebyshev's inequality, which states:

Chebyshev's inequality:For any , at least of the data values fall within standard deviations of the mean.

Figure 2.5.6a,b illustrates the basic concept of Chebyshev's inequality. Chebyshev's inequality is further discussed in Chapter 5.

The shaded area in Figure 2.5.6a contains at least of the data values. The shaded area in Figure 2.5.6b contains at least of the data values. Note that Chebyshev's inequality is also valid for sample data.

Example 2.5.13(Using Chebyshev's inequality) Sodium is an important component of the metabolic panel. The average sodium level for 1000 American male adults who were tested for low sodium was found to be 132 mEq/L with a standard deviation of 3 mEq/L. Using Chebyshev's inequality, determine at least how many of the adults tested have a sodium level between 124.5 and 139.5 mEq/L.

Figure 2.5.6Shaded area lies within the intervals: (a) and (b) .

Solution:From the given information, we have that the mean and the standard deviation of sodium level for these adults are

To find how many of 1000 adults have their sodium level between 124.5 and 139.5 mEq/L, we need to determine the value of . Since each of these values is 7.5 points away from the mean, then using Chebyshev's inequality, the value of is such that , so that

Hence, the number of adults in the sample who have their sodium level between 124.5 and 139.5 mEq/L is at least

Numerical measures can easily be determined by using any one of the statistical packages discussed in this book. We illustrate the use of MINITAB and R with the following example. The use of JMP is discussed in Section 2.11, which is available on the book website: www.wiley.com/college/gupta/statistics2e.

Example 2.5.14(Using MINITAB and R) Calculate numerical measures for the following sample data:

6, 8, 12, 9, 14, 18, 17, 23, 21, 23

1 Enter the data in column C1.

2 From the Menu bar, select Stat Basic Statistics Display Descriptive Statistics. This prompts the following dialog box to appear on the screen:

3 In this dialog box, enter C1 in the box under variables and click at the box Statistics. Then, the following dialog box appears:In this dialog box, check the statistics you would like to determine (for instance, we checked Mean, Mode, Median, Variance, Standard Deviation, Minimum, Maximum, and Range) and then click OK, again, click OK. The numerical measures shown below appear in the Session window:

USING R:

We can use the built in ‘summary()’ function in R to get basic summary statistics. However, the extra functions ‘mean()’, ‘sd()’, ‘var()’, and ‘median()’ are used to calculate the sample mean, standard deviation, variance, and median, respectively. The mode can be obtained using the manual calculation specify in the following R code. The task can be completed by running the following R code in the R Console window.