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1 variance test is used to compare variance of sample to a known population, with the assumption that sample is drawn from a normally distributed population. This analyses the difference between an observed process std deviation and a specified std deviation. It determines whether the population variance differs from hypothesized value specified in the test Example 1: Let us take a Timber company ships its high quality products through sea to high priced markets of European countries. The quality analyst at the receiving country wants to determine whether the variance of the moisture content in the shipment Has increased from that of the specified value. If the variance is greater than maximum acceptable variance , analyst uses the confidence interval to determine how is the difference and whether this difference has practical significance. A test of single variance can be left tailed or Right tailed. Here in this case Let us assume if the moisture content variance specified is 0.1 gm/cc, it means our Null hypothesis is the population variance = specified variance and Alternate hypothesis is that population variance should be less than specified . Here the word “Less” indicates that it is left tailed test. Example 2: If the doctor wants to know the variation of Diastolic (low) Blood pressure levels in patients is more than 60mm /hg. In this case the word “more” indicates that it is right tailed test. The following Data considerations need to be satisfied when applying the chi – square test: 1> The sample must be randomly drawn from the population 2>The data needs to be ensured for normal distribution 3> Each observation should be independent from all other observations. How to derive the Test statistic for 1 variance? is given below:

Hello Pradeep 2 replicates are fine. It will take care of noise factors during the experimentations. Regards Mayank Gupta

Definition - Mixture design is a type of experimental design used when the factors being studied are proportions or compositions of multiple components that make it up. The sum of proportions of components is usually equal to 1 for each experimental run. This constraint makes the design and analysis of mixture experiments more challenging, but it also allows for more precise estimation of the effects of the different components on the response variable. The goal of mixture design is to determine the optimal combination of components leading to desired outcomes. It enables researchers to understand the effects of changing the proportions of components on the response variable and with these varying proportions, researchers can analyze the main effects, interactions, and non-linear effects of the components in the mixture. Different type of mixture designs exist – simplex, simplex centroid, extreme vertex. Key difference between mixture design and other broad type of DOEs are – In factorial design and RSM, factor levels are completely independent of each other. Ex. Temperature, speed, material type In mixture design, there are ingredients (or components) instead of factors. Equivalent of levels in fractional DOE is proportion of components in mixture design If the proportion of one ingredient is changed then the proportions of atleast one ingredients has to be changed to compensate (means there is interdependence of proportion of ingredients) Advantages of mixture design over traditional methods in DoE – It allows more efficient use of resources by considering the proportion of components rather than testing each component separately. It is effective at capturing interactions between components in a mixture as well as non-linear effects. Traditional designs often assume additivity (effect of different components are not confounded) which may not hold true for mixtures. Mixture designs provide a more accurate representation of complex relationships between components. Optimizing formulation by allowing exploration of component proportions to optimize the desired characteristics of the mixture. It can handle various constraints such as the requirement that the proportions sum to a constant value (e.g 1) Mixture designs can handle both continuous and categorical components in the mixture. It is statistically efficient in terms of estimating the effects of components and their interactions. The design can provide precise parameter estimates with fewer experimental runs compared to traditional designs. Examples - Mixture designs are commonly used in various fields such as product development, formulation optimization, industrial process optimization and market research. Examples where mixture designs are valuable – A cosmetic chemist while formulating a new skincare product, can use mixture design to determine optimal proportions of active ingredients, preservative, oil, surfactants, emulsifiers, fragrances, etc. can be determined to achieve desired texture, sensory attributes, efficacy. In developing a new recipe for a sauce, the ideal ratios of spices, herbs, etc. can be determined to achieve desired taste. In pharmaceutical manufacturing, mixture design can be used to determine optimal proportions of different raw materials APIs, excipients in drug formulation to maximize yield, control impurities. In formulating pesticides or fertilizers, the proportions of active ingredients carriers, surfactants, nitrogen, phosphorous, potassium, etc. can be optimized to control effective pest control, crop yield or desired nutrients level while minimizing environmental impact. Paint formulation optimization – the components pigments, binders, solvents, additives, etc. can be optimized to achieve desired properties such as color, gloss, durability, drying times. Determine the optimal proportions of various components to achieve desired concrete properties such as strength, workability, durability with the comprising components of cement, aggregates (sand and gravel), water and sometimes additives. In order to optimize the tensile strength of stainless steel, the factors would be proportion of iron, copper, nickel, chromium in the alloy. The mixture components are subject to the constraint that they must sum to one. In order to optimize the formulation of automotive clear coat paint. It’s a 3 component mixture of monomer, crosslinker, resin. 5 <= A – monomer <= 25, 25 <= B – crosslinker <= 40, 50 <= C – resin <= 70 with constraint that A + B + C = 100. In all the examples above, mixture design is ideal because it allows to study components simultaneously and study their combined effect. Researchers can use simplex centroid or simplex lattice design (type of factorial designs used for mixture designs) to identify specific combinations. Feature Simplex Centroid Simplex Lattice Location of design points Boundaries of simplex factor space Interior of simplex factor space Efficiency Less efficient More efficient Number of design points 2p - 1 3p - 3