Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the alcohol content of each bottle is determined. Let u denote the average alcohol content for the population of all bottles of the brand under study. Suppose that resulting 95% confidence interval is (7.8, 9.6). (a) Would a 90% confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning. O The 90% would be narrower since the z critical value for 90% is smaller than the z critical value for 95%. O The 90% would be the same since the z critical value for 90% is the same as the z critical value for 95%. O The 90% would be wider since the z critical value for 90% is larger than the z critical value for 95%. O The 90% would be wider since the z critical value for 90% is smaller than the z critical value for 95%. O The 90% would be narrower since the z critical value for 90% is larger than the z critical value for 95%. (b) Consider the following statement: There is a 95% chance that u is between 7.8 and 9.6. Is this statement correct? Why or why not? O It is not a correct statement. We are 95% confident in the general procedure for creating the interval, but the mean may or may not be enclosed in this interval. O It is not a correct statement. There is a 5% chance that the mean is between these values. O It is not a correct statement. Each interval contains the mean by definition. O It is a correct statement. Each interval contains the mean by definition. O It is a correct statement. There is only a 5% chance that the mean is not between these values. (c) Consider the following statement: We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 7.8 and 9.6. Is this statement correct? Why or why not? O It is not a correct statement. The interval is an estimate for the sample mean, not a boundary for sample values. O It is a correct statement. This is the definition of a confidence interval. O It is not a correct statement. The interval is an estimate for the population mean, not a boundary for population values. O It is a correct statement. This interval is a great estimate of boundaries for population values. O t is not a correct statement. The interval is an estimate for the sample mean, not a boundary for population values. (d) Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% interval is repeated 100 times, 95 of the resulting intervals will include u. Is this statement correct? Why or why not? O It is not a correct statement. We expect 95 out of 100 intervals will contain the mean, but we don't know this to be true.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
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