Making Statistical Inferences from Samples

Abstract

This chapter covers various concepts and methods dealing with statistical inference, namely point estimation, interval or confidence interval estimation, hypothesis testing and significance testing. These methods are used to infer point and interval estimates about a population from sample data using knowledge of probability and probability distributions. Classical univariate and multivariate techniques as well as non-parametric and Bayesian methods are presented. Further, various types of sampling methods are also described, which is followed by a discussion on estimators and their desirable properties. Finally, resampling methods are treated which, though computer intensive, are conceptually simple, versatile, and allow robust point and interval estimation.

Problems

Pr. 4.1

The specification to which solar thermal collectors are being manufactured requires that their lengths be between 8.45–8.65 feet and their width between 1.55–1.60 ft. The modules produced by a certain assembly line have lengths that are normally distributed about a mean of 8.56 ft with standard deviation 0.05 ft, and widths also normally distributed with a mean of 1.58 ft with standard deviation 0.01 ft. For the modules produced by this assembly line, find:

• (a) the % that will not be within the specified limits for length; state the implicit assumption in this approach

• (b) the % that will not be within the specified limits for width; state the implicit assumption in this approach

• (c) the % that will not meet the specifications; state the implicit assumption in this approach.

Pr. 4.2

The pH of a large lake is to be determined for which purpose 9 test specimens were collected and tested to give: {6.0, 5.7, 5.8, 6.5, 7.0, 6.3, 5.6, 6.1, 5.0}.

• (a) Calculate the mean pH for the 9 specimens

• (b) Find an unbiased estimate of the standard deviation of the population of all pH samples

• (c) Find the 95% CL interval for the mean of this population if it is known from past experience that the pH values have a standard deviation of 0.6. State the implicit assumption in this approach

• (d) Find the 95% CL interval for the mean of this population if no previous information about pH value is available. State the implicit assumption in this approach.

Pr. 4.3

Two types of cement brands A and B were evaluated by testing the compressive strength of concrete blocks made from them. The results for 7 test pieces for A and 5 for B are shown in Table 4.14.

Table 4.14 Data table for Problem 4.3

• (a) Estimate the mean difference between both types of concrete

• (b) Estimate the 95% confidence limits of the difference of both types of concrete

• (c) Repeat the above using the bootstrap method with 1000 samples and compare results.

Pr. 4.4

Using classical and Bayesian approaches to verify claimed benefit of gasoline additive

An inventor claims to have developed a gasoline additive which increases gas mileage of cars. He specifically states that tests using a specific model and make of car resulting in an increase of 50 miles per filling. An independent testing group repeated the tests on six identical cars and obtained the results shown in Table 4.15.

Table 4.15 Data table for Problem 4.4

• (a) Assuming normal distribution, perform parametric tests at the 95% CL to verify the inventor’s claim

• (b) Repeat the problem using the bootstrap method with 1000 samples and compare results.

Pr. 4.5

Analyzing distribution of radon concentration in homes

The Environmental Protection Agency (EPA) determined that an indoor radon concentration in homes of 4 pCi/L was acceptable though there is an increased cancer risk level for humans of 10^–6. Indoor radon concentrations in 43 residences were measured randomly, as shown in Table 4.16.

Table 4.16 Data table for Problem 4.5

The Binomial distribution is frequently used in risk assessment since only two states or outcomes can exist: either one has cancer or one does not.

• (a) Determine whether the normal or the t-distributions better represent the data

• (b) Compute the standard deviation

• (c) Use the one-tailed test to evaluate whether the mean value is less than the threshold value of 4 pCi/L at the 90% CL

• (d) At what confidence level can one state that the true mean is less than 6 pCi/L ?

• (e) Compute the range for the 90% confidence intervals.

Pr. 4.6

Table 4.17 assembles radon concentrations in pCi/L for U.S. homes. Clearly it is not a normal distribution. Researchers have suggested using the lognormal distribution or the power law distribution with exponent of 0.25. Evaluate which of these two functions is more appropriate:

Table 4.17 Data table for Problem 4.6

• (a) graphically using quantile plots

• (b) using appropriate statistical tests for distributions.

Pr. 4.7

Using survey sample to determine proportion of population in favor of off-shore wind farms

A study is initiated to estimate the proportion of residents in a certain coastal region who do not favor the construction of an off-shore wind farm. The state government decides that if the fraction of those against wind farms at the 95% CL drops to less than 0.50, then the wind farm permit will likely be granted.

• (a) A survey of 200 residents at random is taken from which 90 state that they are not in favor. Based on this data, would the permit be granted or not?

• (b) If the survey size is increased, what factors could intervene which could result in a reversal of the above course of action?

• (c) A major public relation campaign is initiated by the wind farm company in an effort to sway public opinion in their favor. After the campaign, a new sample of 100 residents at random was taken, and now only 30 stated that they were not in favor. Did the fraction of residents change significantly at the 95% CL from the previous fraction?

• (d) Would a permit likely to be granted in this case?

Pr. 4.8

Using ANOVA to evaluate pollutant concentration levels at different times of day

The transportation department of a major city is concerned with elevated air pollution levels during certain times of the day at some key intersections. Samples of SO₂ in ($\mu g/{m^3}$) are taken at three locations during three different times of the day as shown in Table 4.18.

Table 4.18 Data table for Problem 4.8

• (a) Conduct an ANOVA test to determine whether the mean concentrations of SO₂ differ during the three collection periods at $\alpha$ = 0.05

• (b) Create an effects plot of the data

• (c) Use Tukey’s multiple comparison procedure to determine which collection periods differ from one another.

Pr. 4.9

Using non-parametric tests to identify the better of two fan models

The facility manager of a large campus wishes to replace the fans in the HVAC system of his buildings. He narrows down the possibilities to two manufacturers and wishes to use Wilcoxon Rank sum at significance level $\alpha$ = 0.05 to identify the better fan manufacturer based on the number of hours of operation prior to servicing. Table 4.19 assembles such data (in hundreds of hours) generated by an independent testing agency:

Table 4.19 Data table for Problem 4.9

Pr. 4.10

Parametric test to evaluate relative performance of two PV systems from sample data

A consumer advocate group wishes to evaluate the performance of two different types of photovoltaic (PV) panels which are very close in terms of rated performance and cost. They convince a builder of new homes to install 2 panels of each brand on two homes in the same locality with care taken that their tilt and orientation towards the sun are identical. The test protocol involves monitoring these two PV panels for 15 weeks and evaluating the performance of the two brands based on their weekly total electrical output. The weekly total electrical output in kWh is listed in Table 4.20. The monitoring equipment used is identical in both locations and has an absolute error of 3 kWh/week at 95% uncertainty level. Evaluate using parametric tests whether the two brands are different at a significance level of $\alpha$ = 0.05 with measurement errors being explicitly considered.

Table 4.20 Weekly electrical output in kWh (Problem 4.10)

Pr. 4.11

Comparing two instruments using parametric, nonparametric and bootstrap methods

A pyranometer meant to measure global solar radiation is being cross-compared with a primary reference instrument. Several simultaneous observations in (kW/m²) were taken with both instruments deployed side by side as shown in Table 4.21. Determine, at a significance level $\alpha$ =0.05, whether the secondary field instrument differs from the primary based on:

Table 4.21 Data table for Problem 4.11

• (a) Parameteric tests

• (b) Non-parametric tests

• (c) The bootstrap method with a sample size of 1000.

Pr. 4.12

Repeat Example 4.2.1 using the Bayesian approach assuming:

• (a) the sample of 36 items tested have a mean of 15 years and a standard deviation of 2.5 years

• (b) the same mean and standard deviation but the sample consists of 9 items only.

Pr. 4.13

An electrical motor company states that one of their product lines of motors has a mean life 8100 h with a standard deviation of 200 h. A wholesale dealer purchases a consignment and tests 10 of the motors. The sample mean and standard deviation are found to be 7800 h with a standard deviation of 100 h. Assume normal distribution. Compute:

• (a) The 95 % confidence interval based on the classical approach

• (b) The 95 % confidence interval based on the Bayesian approach

• (c) The probability that the consignment has a mean value less than 4000 h.

Pr. 4.14

The average cost of electricity to residential customers during the three summer months is to be determined. A sample of electric cost in 25 residences is collected as shown in Table 4.22. Assume a normal distribution with standard deviation of 80.

Table 4.22 Data table for Problem 4.14

• (a) If the prior value is a Gaussian with N(325, 80), find the posterior distribution for the mean $\mu$

• (b) Find a 95% Bayesian credible interval for $\mu$

• (c) Compare the interval with that from the traditional method

• (d) Perform a traditional test for: ${H_0}:\mu = 350\;{\rm{ versus }}\;{H_1}^{\prime}:\mu\ne 350$at the 0.05 significance level

• (e) Perform a Bayesian test of the hypothesis: ${H_0}:\mu\le 350\;{\rm{ versus }}\;{H_1}^{\prime}:\mu> 350$ at the 0.05 significance level.

Pr. 4.15

Comparison of human comfort correlations between Caucasian and Chinese subjects

Human indoor comfort can be characterized by to the occupants’ feeling of well-being in the indoor environment. It depends on several interrelated and complex phenomena involving subjective as well as objective criteria. Research initiated over 50 years back and subsequent chamber studies have helped define acceptable thermal comfort ranges for indoor occupants. Perhaps the most widely used standard is ASHRAE Standard 55-2004 (ASHRAE 2004). The basis of the standard is the thermal sensation scale determined by the votes of the occupants following the scale in Table 4.23.

Table 4.23 ASHRAE thermal sensation classes

The individual votes of all the occupants are then averaged to yield the predicted mean vote (PMV). This is one of the two indices relevant to define acceptability of a large population of people exposed to a certain indoor environment. PMV = 0 is defined as the neutral state (neither cool nor warm), while positive values indicate that occupants feel warm, and vice versa. The mean scores from the chamber studies are then regressed against the influential environmental parameters so as to yield an empirical correlation which can be used as a means of prediction:

$$ PMV = a^*{T_{db}} + b^*{P_v} + c^* $$

(4.49)

where ${T_{db}}$ is the indoor dry-bulb temperature (degrees C), ${P_v}$ is the partial pressure of water vapor (kPa), and the numerical values of the coefficients a*, b* and c* are dependent on such factors as sex, age, hours of exposure, clothing levels, type of activity, …. The values relevant to healthy adults in an office setting for a 3 h exposure period are given in Table 4.24.

Table 4.24 Regression parameters of the ASHRAE PMV model (Eq. 4.49) for 3 h exposure

In general, the distribution of votes will always show considerable scatter. The second index is the percentage of people dissatisfied (PPD), defined as people voting outside the range of − 1 to + 1 for a given value of PMV. When the PPD is plotted against the mean vote of a large group characterized by the PMV, one typically finds a distribution such as that shown in Fig. 4.21. This graph shows that even under optimal conditions (i.e., a mean vote of zero), at least 5% are dissatisfied with the thermal comfort. Hence, because of individual differences, it is impossible to specify a thermal environment that will satisfy everyone. A correlation between PPD and PMV has also been suggested:

Fig. 4.21

Predicted percentage of dissatisfied (PPD) as function of predicted mean vote (PMV) following Eq. 4.50

$$\begin{aligned} PPD = & 100 - 95 \cdot \\&\exp \left[ -0.03353.PM{V^4} + 0.2179.PM{V^2})\right]\end{aligned} $$

(4.50)

Note that the overall approach is consistent with the statistical approach of approximating distributions by the two primary measures, the mean and the standard deviation. However, in this instance, the standard deviation (characterized by PPD) has been empirically found to be related to the mean value, namely PMV (Eq. 4.50).

A research study was conducted in China by Jiang (2001) in order to evaluate whether the above types of correlations, developed using American and European subjects, are applicable to Chinese subjects as well. The environmental chamber test protocol was generally consistent with previous Western studies. The total number of Chinese subjects in the pool was about 200, and several tests were done with smaller batches (about 10–12 subjects per batch evenly split between males and females). Each batch of subjects first spent some time in a pre-conditioning chamber after which they were moved to the main chamber. The environmental conditions (dry-bulb temperature T _db , relative humidity RH and air velocity) of the main chamber were controlled such that: ${T_{db}}( \pm {0.3^0}C),RH( \pm 5 \% )$ and air velocity < 0.15 m/s. The subjects were asked to vote about every ½ hr over 2½ h in accordance with the 7-point thermal sensation scale. However, in this problem we consider only the data relating to averages of the two last votes corresponding to 2 and 2½ h since only then was it found that the voting had stabilized (this feature of the length of exposure is also consistent with American/European tests).

Three separate sets each consisting of 18 tests were performed; one for females only, one for males only, and one for combined⁵. The chamber ${T_{db}}{\rm{, }}RH$ and the associated partial pressure of water needed in Eq. 4.49 (which can be determined from psychrometric relations) along with the PMV and PPD measures are tabulated as shown in Table P4.15 (see Appendix B). The conditions under which these were done is better visualized if plotted on a psychrometric chart shown in Fig. 4.22. Based on this data, one would like to determine whether the psychological responses of Chinese people are different from those of American/European people.

Fig. 4.22

Chamber test conditions plotted on a psychrometric chart for Chinese subjects

Hint:

One of the data points is suspect. Also use Eqs. 4.49 and 4.50 to generate the values pertinent to Western subjects prior to making comparative evaluations.

• (a) Formulate the various different types of tests one would perform stating the intent of each test

• (b) Perform some or all of these tests and draw relevant conclusions

• (c) Prepare a short report describing your entire analysis.