Probability Statistics for Engineers Scientists

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1.2 Sampling Procedures; Collection of Data

In sections that follow, we introduce and discuss speciﬁc quantities that can be computed in samples; the quantities give a sense of the nature of the sample withrespect to center of location of the data and variability in the data. While wemight draw conclusions about the role of humidity and the impact of coating the specimens from the ﬁgure, we cannot truly evaluate the results from an analyti-cal point of view without taking into account the variability around the average.

2 As an example, suppose the data set is the following: 1.7, ., 3.9, 3.11, and

˜Clearly, the mean is inﬂuenced considerably by the presence of the extreme obser- vation, 14.7, whereas the median places emphasis on the true “center” of the dataset. Note that the sample median is, indeed, a special case ofthe trimmed mean in which all of the sample data are eliminated apart from the / / 5.6 3.0 5.2 4.0 4.4 2.8 3.7 3.3 (c) Calculate the sample mean tensile strength of the two samples.(d) Calculate the median for both.

4.8 Assume that the measurements are a simple random sample

While the details of the analysis of this type of data set are deferred to Chap- ter 9, it should be clear from Figure 1.1 that variability among the no-nitrogenobservations and variability among the nitrogen observations are certainly of some consequence. Perhaps there is something about the inclusion of nitrogen that not only increases the stem height (¯ x of 0.565 gramcompared to an ¯ x of 0.399 gram for the no-nitrogen sample) but also increases the variability in stem height (i.e., renders the stem height more inconsistent).

1.4 Measures of Variability

2 Definition 1.3: The sample variance, denoted by s , is given by n 2 (x x) i 2 − ¯ s = .n − 1 i=1 The sample standard deviation, denoted by s, is the positive square root of 2 s , that is,√ 2 s = s . In fact, since thelast value of x − ¯x is determined by the initial n − 1 of them, we say that these 2 are n − 1 “pieces of information” that produce s .

2 The sample variance s is given by

Units for Standard Deviation and Variance It should be apparent from Deﬁnition 1.3 that the variance is a measure of the average squared deviation from the mean ¯ x. As a result, the sample variance possesses units that are the square of the units in the observed data whereas the sample standard deviationis found in linear units.

1.1. Which Variability Measure Is More Important?

Compute the sample variance and the sample standard 1.9 Exercise 1.3 on page 13 showed tensile strength deviation for both control and treatment groups.data for two samples, one in which specimens were ex- posed to an aging process and one in which there was 1.12 For Exercise 1.6 on page 13, compute the sample no aging of the specimens.standard deviation in tensile strength for the samples separately for the two temperatures. The question arises, “Is the decrease in the sample proportion 5000 from 0.02 to 0.018 substantial enough to suggest a real improvement in the pop- ulation proportion?” Both of these illustrations require the use of the statisticalproperties of sample averages—one from samples from a continuous population, and the other from samples from a discrete (binary) population.

15 Tensile Strength

Separationis made into four parts by quartiles, with the third quartile separating the upper quarter of the data from the rest, the second quartile being the median, and the ﬁrstquartile separating the lower quarter of the data from the rest. They are (i) What is the nature of the impact of relative humidity on the corrosion of the aluminum alloy within the range of relative humidity in this experiment?(ii) Does the chemical corrosion coating reduce corrosion levels and can the eﬀect be quantiﬁed in some fashion?

13.8 Nonsmokers:

, 9.(b) Construct a relative frequency histogram, draw an estimate of the graph of the distribution, and dis-cuss the skewness of the distribution.(c) Compute the sample mean, sample median, and sample standard deviation. Indicate on the plot both shrinkage means,that for low injection velocity and high injection velocity.(b) Based on the graphical results in (a), using the lo- cation of the two means and your sense of variabil-ity, what do you conclude regarding the effect of injection velocity on shrinkage at low mold tem-perature?

1.31 Consider the situation of Exercise 1.28. But now

Could the type of information found in Exer- ature is raised to a high level and held constant.cises 1.28 and 1.31 have been found in an observational study in which there was no control on injection veloc- Shrinkage values at low injection velocity: ity and mold temperature by the analyst? Use the sample means and variances 92.98 93.47 93.75 93.89 91.62and the types of plots presented in this chapter to sum- (a) As in Exercise 1.28, construct a dot plot with bothmarize any features that draw a distinction between the data sets on the same graph and identify bothdistributions of shoe sizes for males and females.

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