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Probability And Statistics For Engineering And The Sciences 9th Edition Jay L. Devore - Solutions

By how much must the sample size n be increased if the width of the CI (7.5) is to be halved? If the sample size is increased by a factor of 25, what effect will this have on the width of the interval? Justify your assertions.
Let Î±1 > 0, Î±2 > 0, with Î±1 + Î±2 = Î±. Thena. Use this equation to derive a more general expression for a 100(1 - Î±)% CI for m of which the interval (7.5) is a special case. b. Let Î± = .05 and
a. Under the same conditions as those leading to the interval (7.5), P[(x̅ = µ)/(σ/√n) < 1.645] = .95. Use this to derive a one-sided interval for m that has infinite width and provides a lower confidence bound on µ. What is this interval for the data in Exercise 5(a)? b. Generalize the
For each of the following assertions, state whether it is a legitimate statistical hypothesis and why: a. H: σ > 100 b. H: = 45 c. H: s ≤ .20 d. H: σ1/σ2 < 1 e. H: - = 5 f. H: λ ≤ .01, where l is the parameter of an exponential distribution used to model component lifetime.
A regular type of laminate is currently being used by a manufacturer of circuit boards. A special laminate has been developed to reduce warpage. The regular laminate will be used on one sample of specimens and the special laminate on another sample, and the amount of warpage will then be determined
Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second. Consider testing H0: p = .5 versus Ha: p ≠ .5 based on a random sample of 25 individuals. Let the
A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than 1300 KN/m2. The mixture will not be used unless experimental evidence indicates conclusively that the strength specification has been met. Suppose compressive strength for
The calibration of a scale is to be checked by weighing a 10-kg test specimen 25 times. Suppose that the results of different weighings are independent of one another and that the weight on each trial is normally distributed with σ = .200 kg. Let m denote the true average weight reading on the
A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be 120 ft. It is proposed that the new design be implemented only if sample data strongly indicates a reduction
Let μ denote the true average reaction time to a certain stimulus. For a z test of H0: μ = 5 versus Ha: μ > 5, determine the P-value for each of the following values of the z test statistic. a. 1.42 b. .90 c. 1.96 d. 2.48 e. -.11
Newly purchased tires of a particular type are supposed to be filled to a pressure of 30 psi. Let μ denote the true average pressure. A test is to be carried out to decide whether μ differs from the target value. Determine the P-value for each of the following z test statistic values. a. 2.10 b.
Answer the following questions for the tire problem in Example 8.7. a. If = 30,960 and a level α = .01 test is used, what is the decision? b. If a level .01 test is used, what is β(30,500)? c. If a level .01 test is used and it is also required that β(30,500) = .05, what sample size n is
Reconsider the paint-drying situation of Example 8.5, in which drying time for a test specimen is normally distributed with σ = 9. The hypotheses H0: μ = 75 versus Ha: μ < 75 are to be tested using a random sample of n = 25 observations. a. How many standard deviations (of ) below the null
The melting point of each of 16 samples of a certain brand of hydrogenated vegetable oil was determined, resulting in = 94.32. Assume that the distribution of the melting point is normal with σ = 1.20. a. Test H0: μ = 95 versus Ha: μ ≠ 95 using a two-tailed level .01 test. b. If a level .01
For the following pairs of assertions, indicate which do not comply with our rules for setting up hypotheses and why (the subscripts 1 and 2 differentiate between quantities for two different populations or samples): a. H0: μ = 100, Ha: μ > 100 b. H0: σ = 20, Ha: σ ≤ 20 c. H0: p ≠ .25, Ha:
Lightbulbs of a certain type are advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless it can be conclusively demonstrated that the true average lifetime is smaller than
The desired percentage of SiO2 in a certain type of aluminous cement is 5.5. To test whether the true average percentage is 5.5 for a particular production facility, 16 independently obtained samples are analyzed. Suppose that the percentage of SiO2 in a sample is normally distributed with σ = .3
To obtain information on the corrosion-resistance properties of a certain type of steel conduit, 45 specimens are buried in soil for a 2-year period. The maximum penetration (in mils) for each specimen is then measured, yielding a sample average penetration of = 52.7 and a sample standard
Automatic identification of the boundaries of significant structures within a medical image is an area of ongoing research. The paper "Automatic Segmentation of Medical Images Using Image Registration: Diagnostic and Simulation Applications" (J. of Medical Engr. And Tech., 2005: 53-63) discussed a
Unlike most packaged food products, alcohol beverage container labels are not required to show calorie or nutrient content. The article "What Am I Drinking? The Effects of Serving Facts Information on Alcohol Beverage Containers" (J. of Consumer Affairs, 2008: 81-99) reported on a pilot study in
Body armor provides critical protection for law enforcement personnel, but it does affect balance and mobility. The article "Impact of Police Body Armour and Equipment on Mobility" (Applied Ergonomics, 2013: 957-961) reported that for a sample of 52 male enforcement officers who underwent an
The recommended daily dietary allowance for zinc among males older than age 50 years is 15 mg/day. The article "Nutrient Intakes and Dietary Patterns of Older Americans: A National Study" (J. of Gerontology, 1992: M145-150) reports the following summary data on intake for a sample of males age
Show that for any Δ > 0, when the population distribution is normal and σ is known, the two-tailed test satisfies β(μ0 - Δ) = β(μ0 - Δ), so that β(μʹ) is symmetric about μ0.
For a fixed alternative value μʹ, show that β(μʹ) → 0 as n → ∞ for either a one-tailed or a two-tailed z test in the case of a normal population distribution with known σ.
A sample of n sludge specimens is selected and the pH of each one is determined. The one-sample t test will then be used to see if there is compelling evidence for concluding that true average pH is less than 7.0. What conclusion is appropriate in each of the following situations? a. n = 6, t =
The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let μ denote the true average reflectometer reading for a new type of paint under consideration. A test of H0: μ = 20 versus Ha: μ > 20 will be based on a random sample of size n from a normal
The relative conductivity of a semiconductor device is determined by the amount of impurity "doped" into the device during its manufacture. A silicon diode to be used for a specific purpose requires an average cut-on voltage of .60 V, and if this is not achieved, the amount of impurity must be
The article "The Foreman's View of Quality Control" (Quality Engr., 1990: 257-280) described an investigation into the coating weights for large pipes resulting from a galvanized coating process. Production standards call for a true average weight of 200 lb per pipe. The accompanying descriptive
The following observations are on stopping distance (ft) of a particular truck at 20 mph under specified experimental conditions ("Experimental Measurement of the Stopping Performance of a Tractor-Semitrailer from Multiple Speeds," NHTSA, DOT HS 811 488, June
The article "Uncertainty Estimation in Railway Track Life-Cycle Cost" (J. of Rail and Rapid Transit, 2009) presented the following data on time to repair (min) a rail break in the high rail on a curved track of a certain railway line. 159 120 480 149 270 547 340 43 228 202 240 218 A
Have you ever been frustrated because you could not get a container of some sort to release the last bit of its contents? The article "Shake, Rattle, and Squeeze: How Much Is Left in That Container?" (Consumer Reports, May 2009: 8) reported on an investigation of this issue for various consumer
The accompanying data on cube compressive strength (MPa) of concrete specimens appeared in the article "Experimental Study of Recycled Rubber-Filled High-Strength Concrete" (Magazine of Concrete Res., 2009: 549-556):a. Is it plausible that the compressive strength for this type of concrete is
A random sample of soil specimens was obtained, and the amount of organic matter (%) in the soil was determined for each specimen, resulting in the accompanying data (from "Engineering Properties of Soil," Soil Science, 1998: 93-102).The values of the sample mean, sample standard deviation, and
Reconsider the accompanying sample data on expense ratio (%) for large-cap growth mutual funds first introduced in Exercise 1.53.A normal probability plot shows a reasonably linear pattern. a. Is there compelling evidence for concluding that the population mean expense ratio exceeds 1%? Carry out
Polymer composite materials have gained popularity because they have high strength to weight ratios and are relatively easy and inexpensive to manufacture. However, their nondegradable nature has prompted development of environmentally friendly composites using natural materials. The article
A spectrophotometer used for measuring CO concentration [ppm (parts per million) by volume] is checked for accuracy by taking readings on a manufactured gas (called span gas) in which the CO concentration is very precisely controlled at 70 ppm. If the readings suggest that the spectrophotometer is
Consider using a z test to test H0: p = .6. Determine the P-value in each of the following situations. a. Ha: p > .6, z = 1.47 b. Ha: p < .6, z = -2.70 c. Ha: p ≠ .6, z = -2.70 d. Ha: p < .6, z = .25
A common characterization of obese individuals is that their body mass index is at least 30 [BMI = weight/(height)2, where height is in meters and weight is in kilograms]. The article "The Impact of Obesity on Illness Absence and Productivity in an Industrial Population of Petrochemical Workers"
A manufacturer of nickel-hydrogen batteries randomly selects 100 nickel plates for test cells, cycles them a specified number of times, and determines that 14 of the plates have blistered. a. Does this provide compelling evidence for concluding that more than 10% of all plates blister under such
A random sample of 150 recent donations at a certain blood bank reveals that 82 were type A blood. Does this suggest that the actual percentage of type A donations differs from 40%, the percentage of the population having type A blood? Carry out a test of the appropriate hypotheses using a
It is known that roughly 2/3 of all human beings have a dominant right foot or eye. Is there also right-sided dominance in kissing behavior? The article "Human Behavior: Adult Persistence of Head-Turning Asymmetry" (Nature, 2003: 771) reported that in a random sample of 124 kissing couples, both
The article "Effects of Bottle Closure Type on Consumer Perception of Wine Quality" (Amer. J. of Enology and Viticulture, 2007: 182-191) reported that in a sample of 106 wine consumers, 22 (20.8%) thought that screw tops were an acceptable substitute for natural corks. Suppose a particular winery
With domestic sources of building supplies running low several years ago, roughly 60,000 homes were built with imported Chinese drywall. According to the article "Report Links Chinese Drywall to Home Problems" (New York Times, Nov. 24, 2009), federal investigators identified a strong association
A plan for an executive travelers' club has been developed by an airline on the premise that 5% of its current customers would qualify for membership. A random sample of 500 customers yielded 40 who would qualify. a. Using this data, test at level .01 the null hypothesis that the company's premise
Each of a group of 20 intermediate tennis players is given two rackets, one having nylon strings and the other synthetic gut strings. After several weeks of playing with the two rackets, each player will be asked to state a preference for one of the two types of strings. Let p denote the proportion
A manufacturer of plumbing fixtures has developed a new type of washerless faucet. Let p = P(a randomly selected faucet of this type will develop a leak within 2 years under normal use). The manufacturer has decided to proceed with production unless it can be determined that p is too large; the
In a sample of 171 students at an Australian university that introduced the use of plagiarism-detection software in a number of courses, 58 students indicated a belief that such software unfairly targets students ("Student and Staff Perceptions of the Effectiveness of Plagiarism Detection
Reconsider the paint-drying problem discussed in Example 8.5. The hypotheses were H0: μ = 75 versus Ha: μ < 75, with s assumed to have value 9.0. Consider the alternative value μ = 74, which in the context of the problem would presumably not be a practically significant departure from H0. a. For
Consider the large-sample level .01 test in Section 8.4 for testing H0: p = .2 against Ha: p > .2. a. For the alternative value p = .21, compute β(.21) for sample sizes n = 100, 2500, 10,000, 40,000, and 90,000. b. For = x/n = .21, compute the P-value when n = 100, 2500, 10,000, and 40,000. c.
Consider carrying out m tests of hypotheses based on independent samples, each at significance level (exactly) .01. a. What is the probability of committing at least one type I error when m = 5? When m = 10? b. How many such tests would it take for the probability of committing at least one type I
A 95% CI for true average amount of warpage (mm) of laminate sheets under specified conditions was calculated as (1.81, 1.95), based on a sample size of n = 15 and the assumption that amount of warpage is normally distributed. a. Suppose you want to test H0: μ = 2 versus Ha: μ ≠ 2 using α =
A sample of 50 lenses used in eyeglasses yields a sample mean thickness of 3.05 mm and a sample standard deviation of .34 mm. The desired true average thickness of such lenses is 3.20 mm. Does the data strongly suggest that the true average thickness of such lenses is something other than what is
In Exercise 57, suppose the experimenter had believed before collecting the data that the value of s was approximately .30. If the experimenter wished the probability of a type II error to be .05 when μ = 3.00, was a sample size 50 unnecessarily large? In Exercise 57 A sample of 50 lenses used in
It is specified that a certain type of iron should contain .85 g of silicon per 100 g of iron (.85%). The silicon content of each of 25 randomly selected iron specimens was determined, and the accompanying Minitab output resulted from a test of the appropriate hypotheses.a. What hypotheses were
Let m denote the true average radioactivity level (picocuries per liter). The value 5 pCi/L is considered the dividing line between safe and unsafe water. Would you recommend testing H0: μ = 5 versus Ha: μ < 5 or H0: μ = 5 versus Ha: μ < 5? Explain your reasoning.
One method for straightening wire before coiling it to make a spring is called "roller straightening." The article "The Effect of Roller and Spinner Wire Straightening on Coiling Performance and Wire Properties" (Springs, 1987: 27-28) reports on the tensile properties of wire. Suppose a sample of
Contamination of mine soils in China is a serious environmental problem. The article "Heavy Metal Contamination in Soils and Phytoaccumulation in a Manganese Mine Wasteland, South China" (Air, Soil, and Water Res., 2008: 31-41) reported that, for a sample of 3 soil specimens from a certain restored
The article "Orchard Floor Management Utilizing Soil-Applied Coal Dust for Frost Protection" (Agri. and Forest Meteorology, 1988: 71-82) reports the following values for soil heat flux of eight plots covered with coal dust. 34.7 35.4 34.7 37.7 32.5 28.0 18.4 24.9 The mean soil heat flux for
The article "Caffeine Knowledge, Attitudes, and Consumption in Adult Women" (J. of Nutrition Educ., 1992: 179-184) reports the following summary data on daily caffeine consumption for a sample of adult women: n = 47, = 215 mg, s = 235 mg, and range = 5 - 1176. a. Does it appear plausible that the
Annual holdings turnover for a mutual fund is the percentage of a fund's assets that are sold during a particular year. Generally speaking, a fund with a low value of turnover is more stable and risk averse, whereas a high value of turnover indicates a substantial amount of buying and selling in an
The true average breaking strength of ceramic insulators of a certain type is supposed to be at least 10 psi. They will be used for a particular application unless sample data indicates conclusively that this specification has not been met. A test of hypotheses using α = .01 is to be based on a
The accompanying observations on residual flame time (sec) for strips of treated children's nightwear were given in the article "An Introduction to Some Precision and Accuracy of Measurement Problems" (J. of Testing and Eval., 1982: 132-140). Suppose a true average flame time of at most 9.75 had
The incidence of a certain type of chromosome defect in the U.S. adult male population is believed to be 1 in 75. A random sample of 800 individuals in U.S. penal institutions reveals 16 who have such defects. Can it be concluded that the incidence rate of this defect among prisoners differs from
In an investigation of the toxin produced by a certain poisonous snake, a researcher prepared 26 different vials, each containing 1 g of the toxin, and then determined the amount of antitoxin needed to neutralize the toxin. The sample average amount of antitoxin necessary was found to be 1.89 mg,
The sample average unrestrained compressive strength for 45 specimens of a particular type of brick was computed to be 3107 psi, and the sample standard deviation was 188. The distribution of unrestrained compressive strength may be somewhat skewed. Does the data strongly indicate that the true
Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high-pressure oil-filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of sheath thickness is less than .05 mm. What hypotheses should be tested, and why?
The Dec. 30, 2009, the NewYork Times reported that in a survey of 948 American adults who said they were at least somewhat interested in college football, 597 said the current Bowl Championship System should be replace by a playoff similar to that used in college basketball. Does this provide
When X1, X2,..., Xn are independent Poisson variables, each with parameter μ, and n is large, the sample mean has approximately a normal distribution with μ = E() and V() = μ/n. This implies that Z = - μ / √μ/n has approximately a standard normal distribution. For testing H0: μ = μ0,
An article in the Nov. 11, 2005, issue of the San Luis Obispo Tribune reported that researchers making random purchases at California Wal-Mart stores found scanners coming up with the wrong price 8.3% of the time. Suppose this was based on 200 purchases. The National Institute for Standards and
The article "Heavy Drinking and Polydrug Use Among College Students" (J. of Drug Issues, 2008: 445-466) stated that 51 of the 462 college students in a sample had a lifetime abstinence from alcohol. Does this provide strong evidence for concluding that more than 10% of the population sampled had
The article "Analysis of Reserve and Regular Bottlings: Why Pay for a Difference Only the Critics Claim to Notice?" (Chance, Summer 2005, pp. 9-15) reported on an experiment to investigate whether wine tasters could distinguish between more expensive reserve wines and their regular counterparts.
The American Academy of Pediatrics recommends a vitamin D level of at least 20 ng/ml for infants. The article "Vitamin D and Parathormone Levels of Late-Preterm Formula Fed Infants During the First Year of Life" (European J. of Clinical Nutr., 2012: 224-230) reported that for a sample of 102
Chapter 7 presented a CI for the variance σ2 of a normal population distribution. The key result there was that the rv χ2 = (n - 1)S2/σ2 has a chi-squared distribution with n - 1 df. Consider the null hypothesis H0: σ2 = σ02 (equivalently, σ = σ0). Then when H0 is true, the test statistic
Referring to Exercise 76, suppose an investigator wishes to test H0: σ2 = .04 versus Ha: σ2 < .04 based on a sample of 21 observations. The computed value of 20s2/.04 is 8.58. Place bounds on the P-value and then reach a conclusion at level .01.
When the population distribution is normal and n is large, the sample standard deviation S has approximately a normal distribution with E(S) ≈ σ and V(S) ≈ σ2/(2n). We already know that in this case, for any n, is normal with E() = μ and V() = σ2/n. a. Assuming that the underlying
Let X1, X2,..., Xn be a random sample from an exponential distribution with parameter λ. Then it can be shown that 2λ∑Xi has a chi-squared distribution with v = 2n (by first showing that 2λXi has a chi-squared distribution with v = 2). a. Use this fact to obtain a test statistic for testing
Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require
Because of variability in the manufacturing process, the actual yielding point of a sample of mild steel subjected to increasing stress will usually differ from the theoretical yielding point. Let p denote the true proportion of samples that yield before their theoretical yielding point. If on the
An article in the November 1983 Consumer Reports compared various types of batteries. The average lifetimes of Duracell Alkaline AA batteries and Eveready Energizer Alkaline AA batteries were given as 4.1 hours and 4.5 hours, respectively. Suppose these are the population average lifetimes. a. Let
An experiment was performed to compare the fracture toughness of high-purity 18 Ni maraging steel with commercial-purity steel of the same type (Corrosion Science, 1971: 723-736). For m = 32 specimens, the sample average toughness was = 65.6 for the high purity steel, whereas for n = 38 specimens
The level of lead in the blood was determined for a sample of 152 male hazardous-waste workers ages 20-30 and also for a sample of 86 female workers, resulting in a mean 6 standard error of 5.5 ± 0.3 for the men and 3.8 ± 0.2 for the women ("Temporal Changes in Blood Lead Levels of Hazardous
The accompanying summary data on total cholesterol level (mmol/l) was obtained from a sample of Asian postmenopausal women who were vegans and another sample of such women who were omnivores ("Vegetarianism, Bone Loss, and Vitamin D: A Longitudinal Study in Asian Vegans and Non-Vegans," European J.
The level of monoamine oxidase (MAO) activity in blood platelets (nm/mg protein/h) was determined for each individual in a sample of 43 chronic schizophrenics, resulting in x̅ = 2.69 and s1 = 2.30, as well as for 45 normal subjects, resulting in y̅ = 6.35 and s2 = 4.03. Does this data strongly
a. Show for the upper-tailed test with σ1 and σ2 known that as either m or n increases, b decreases when µ1 = µ2. Δ0. b. For the case of equal sample sizes (m = n) and fixed a, what happens to the necessary sample size n as β is decreased, where β is the desired type II error probability at
To decide whether two different types of steel have the same true average fracture toughness values, n specimens of each type are tested, yielding the following results: Type.............. Sample Average............. Sample SD 1 ................................60.1........................ 1.0 2
Determine the number of degrees of freedom for the two-sample t test or CI in each of the following situations: a. m = 10, n = 10, s1 = 5.0, s2 = 6.0 b. m = 10, n = 15, s1 = 5.0, s2 = 6.0 c. m = 10, n =15, s1 = 2.0, s2 = 6.0 d. m = 12, n = 24, s1 = 5.0, s2 = 6.0
Which way of dispensing champagne, the traditional vertical method or a tilted beer-like pour, preserves more of the tiny gas bubbles that improve flavor and aroma? The following data was reported in the article "On the Losses of Dissolved CO2 during Champagne Serving" (J. Agr. Food Chem., 2010:
Suppose µ1 and µ2 are true mean stopping distances at 50 mph for cars of a certain type equipped with two different types of braking systems. Use the two-sample t test at significance level .01 to test H0: µ1 - µ2 < -10 versus Ha: µ1 - µ2 < -10 for the following data: M = 6 , = 115.7, s1 =
The National Health Statistics Reports dated Oct. 22, 2008, included the following information on the heights (in.) for non-Hispanic white females:a. Calculate and interpret a confidence interval at confidence level approximately 95% for the difference between population mean height for the younger
Use the data of Exercise 19 to calculate a 95% CI for the difference between true average stopping distance for cars equipped with system 1 and cars equipped with system 2. Does the interval suggest that precise information about the value of this difference is available? Data From Exercise
Quantitative noninvasive techniques are needed for routinely assessing symptoms of peripheral neuropathies, such as carpal tunnel syndrome (CTS). The article "A Gap Detection Tactility Test for Sensory Deficits Associated with Carpal Tunnel Syndrome" (Ergonomics, 1995: 2588-2601) reported on a test
According to the article "Modeling and Predicting the Effects of Submerged Arc Weldment Process Parameters on Weldment Characteristics and Shape Profiles" (J. of Engr. Manuf., 2012: 1230-1240), the submerged arc welding (SAW) process is commonly used for joining thick plates and pipes. The heat
Fusible interlinings are being used with increasing frequency to support outer fabrics and improve the shape and drape of various pieces of clothing. The article "Compatibility of Outer and Fusible Interlining Fabrics in Tailored Garments" (Textile Res. J., 1997: 137-142) gave the accompanying data
Damage to grapes from bird predation is a serious problem for grape growers. The article "Experimental Method to Investigate and Monitor Bird Behavior and Damage to Vineyards" (Amer. J. of Enology and Viticulture, 2004: 288-291) reported on an experiment involving a bird-feeder table, time-lapse
The accompanying data consists of prices ($) for one sample of California cabernet sauvignon wines that received ratings of 93 or higher in the May 2013 issue of Wine Spectator and another sample of California cabernets that received ratings of 89 or lower in the same issue.Assume that these are
The article "The Influence of Corrosion Inhibitor and Surface Abrasion on the Failure of Aluminum-Wired Twist-On Connections" (IEEE Trans. on Components, Hybrids, and Manuf. Tech., 1984: 20-25) reported data on potential drop measurements for one sample of connectors wired with alloy aluminum and
Anorexia Nervosa (AN) is a psychiatric condition leading to substantial weight loss among women who are fearful of becoming fat. The article "Adipose Tissue Distribution After Weight Restoration and Weight Maintenance in Women with Anorexia Nervosa" (Amer. J. of ClinicalNutr., 2009: 1132-1137) used
As the population ages, there is increasing concern about accident-related injuries to the elderly. The article "Age and Gender Differences in Single-Step Recovery from a Forward Fall" (J. of Gerontology, 1999: M44-M50) reported on an experiment in which the maximum lean angle-the farthest a
The article "Effect of Internal Gas Pressure on the Compression Strength of Beverage Cans and Plastic Bottles" (J. of Testing and Evaluation, 1993: 129-131) includes the accompanying data on compression strength (lb) for a sample of 12-oz aluminum cans filled with strawberry drink and another
Pilates is a popular set of exercises for the treatment of individuals with lower back pain. The method has six basic principles: centering, concentration, control, precision, flow, and breathing. The article "Efficacy of the Addition of Modified Pilates Exercises to a Minimal Intervention in

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