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VII Asignación de Estadística A 2018 APELLIDO PATERNO ALLCCA CONTRERAS DE LA TORRE GENEBROZO HUAMANI LINARES PEREZ SOLÍS VALENCIA

APELLIDO MATERNO FLORES RAMOS CÓRDOVA GUERRA ROQUE CABALLERO VITE VEGA CHINARRO

NOMBRES DIEGO ALONSO DIANA ELTON MAX ELIZABETH CRISTINA ADA MADISON LISET WINNIE EVELYN LUZ MARÍA FRANCO NICOLAS

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Enunciado Problema Se afirma que la dureza, en grados Shore, de un determinado caucho, debe ser 65. Se probaron 14 especímenes y se obtuvo una medida promedio de dureza igual a 63.1 y una desviación estándar de 1,4. ¿Hay evidencia 1 suficiente como para rechazar la afirmación anterior al nivel del 5% de significación? ¿Qué hipótesis se necesita para que la respuesta sea válida? Acidity of mouthwash. Acid has been found to be a primary cause of dental caries (cavities). It is theorized that oral mouthwashes contribute to the development of caries due to the antiseptic agent oxidizing into acid over time. This theory was tested in the Journal of Dentistry, Oral Medicine and Dental Education (Vol. 3, 2009). Three bottles of mouthwash, each of a different brand, were randomly selected from a drug store. The pH level (where lower pH levels indicate higher acidity) of each bottle was measured on the date of purchase and after 30 days. The data are shown in the table. Use a 95% confidence interval to determine if the mean initial pH level of mouthwash differs significantly from the mean pH level after 30 days. Mouthwash Brand LMW SMW RMW

Initial pH 4.56 6.71 5.65

Final pH 4.27 6.51 5.58

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Source: Chunhye, K.L. & Schmitz, B.C. “Determination of pH, total acid, and total ethanol in oral health products: Oxidation of ethanol and recommendations to mitigate its association with dental caries”, Journal of Dentistry, Oral Medicine and Dental Education, Vol. 3, No. 1, 2009 (Table 1) A chemical products manufacturer must identify a new supplier for a raw material that is an essential component of a particular product. The previous supplier was able to deliver material with a mean molecular weight of 3500. The new supplier must show equivalence to this value of molecular weight. If the new supplier can deliver material that has a molecular weight that is within 50 units of this value, it will be considered equivalent. A random sample of 10 lots of product is available, and the sample mean and standard deviation of molecular weight are 3550 and 25, respectively.

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(a) State the appropriate hypotheses that must be tested to demonstrate equivalence. (b) What are your conclusions using α = 0 05. The mean bond strength of a cement product must be at least 1000 psi. The process by which this material is manufactured must show equivalence to this standard. If the process can manufacture cement for which the mean bond strength is at least 9750 psi, it will be considered equivalent to the standard. A random sample of six observations is available, and the sample mean and standard deviation of bond strength are 9360 psi and 42.6 psi, respectively. (a) State the appropriate hypotheses that must be tested to demonstrate equivalence. (b) What are your conclusions using α = 0 05. An article in Biological Trace Element Research [“Interaction of Dietary Calcium, Manganese, and Manganese Source (Mn Oxide or Mn Methionine Complex) or Chick Performance and Manganese Utilization” (1991, Vol. 29(3), pp. 217–228)] showed the following results of tissue assay for liver manganese (ppm) in chicks fed high Ca diets. 6.02 6.08 7.11 5.73 5.32 7.10 5.29 5.84 6.03 5.99 4.53 6.81 (a) Test the hypothesis H0: σ2 = 0,6 versus H1: σ2 ≠ 0 6 using α = 0,01. (b) What is the P-value for this test? (c) Discuss how part (a) could be answered by constructing a 99% two-sided confidence interval for σ. Alcoholic fermentation in wines. Determining alcoholic fermentation in wine is critical to the wine-making process. Must/wine density is a good indicator of the fermentation point since the density value decreases as sugars are converted into alcohol. For decades, winemakers have measured must/wine density with a hydrometer. Although accurate, the hydrometer employs a manual process that is very time-consuming. Consequently, large wineries are searching for more rapid measures of density measurement. An alternative method utilizes the hydrostatic balance instrument (similar to the hydrometer, but digital). A winery in Portugal collected the must/wine density measurements for white wine samples randomly selected from the fermentation process for a recent harvest. For each sample, the density of the wine at 20°C was measured with both the hydrometer and the hydrostatic balance. The densities for 40 wine samples are saved in the WINE40 file. The winery will use the alternative method of measuring wine density only if it can be demonstrated that the mean difference between the density measurements of the two methods does not exceed .002. Perform the analysis for the winery and give your recommendation. A polymer is manufactured in a batch chemical process. Viscosity measurements are normally made on each batch, and long experience with the process has indicated that the variability in the process is fairly stable with σ = 20. Fifteen batch viscosity measurements are given as follows: 724, 718, 776, 760, 745, 759, 795, 756, 742, 740, 761, 749, 739, 747, 742

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A process change that involves switching the type of catalyst used in the process is made. Following the process change, eight batch viscosity measurements are taken: 735, 775, 729, 755, 783, 760, 738, 780 Assume that process variability is unaffected by the catalyst change. If the difference in mean batch viscosity is 10 or less, the manufacturer would like to detect it with a high probability. (a) Formulate and test an appropriate hypothesis using α = 0.10. What are your conclusions? Find the P-value. (b) Find a 90% confidence interval on the difference in mean batch viscosity resulting from the process change. (c) Compare the results of parts (a) and (b) and discuss your findings. An article in Nature (2003, Vol. 48, p. 1013) described an experiment in which subjects consumed different types of chocolate to determine the effect of eating chocolate on a measure of cardiovascular health. We will consider the results for only dark chocolate and milk chocolate. In the experiment, 12 subjects consumed 100 grams of dark chocolate and 200 grams of milk chocolate, one type of chocolate per day, and after one hour, the total antioxidant capacity of their blood plasma was measures in an assay. The subjects consisted of seven women and five men with an average age range of 32.2 ±1 years, an average weight of 65.8 ± 3.1 kg, and average body mass index of 21.9 ± 0.4 kg/m2. Data similar to that reported in the article follows. Dark Chocolate Milk Chocolate 118.8, 122.6, 115.6, 113.6, 119.5, 115.9,

102.1, 105.8, 99.6, 102.7, 98.8, 100.9,

115.8, 115.1, 116.9, 115.4, 115.6, 107.9

102.8, 98.7, 94.7, 97.8, 99.7, 98.6

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Is there evidence to support the claim that consuming dark chocolate produces a higher mean level of total blood plasma antioxidant capacity than consuming milk chocolate? An article in the journal Hazardous Waste and Hazardous Materials (1989, Vol. 6) reported the results of an analysis of the weight of calcium in standard cement and cement doped with lead. Reduced levels of calcium would indicate that the hydration mechanism in the cement is blocked and would allow water to attack various locations in the cement structure. Ten samples of standard cement had an average weight percent calcium of 𝑥̅1 = 90.0 with a sample standard deviation of s1 = 5,0, and 15 samples of the lead-doped cement had an average weight percent calcium of 𝑥̅2 = 87,0 with a sample standard deviation of s2 = 4,0. .

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