Calculating test statistics for means and proportions for one- and two-tailed tests

With any hypothesis test, we need to state the null and alternative hypotheses, then determine the level of significance. We’ve already covered these first two steps, and now we want to learn how to calculate the test statistic, which will depend on whether we’re running a two-tail test or a one-tail test.

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Using laws of logarithms (laws of logs) to solve log problems

Laws of logarithms (or laws of logs) include product, quotient, and power rules for logarithms, as well as the general rule for logs (and the change of base formula we’ll cover in the next lesson), can all be used together, in any combination, in order to solve log problems.

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U-substitution to solve definite integrals

U-substitution in definite integrals is just like substitution in indefinite integrals except that, since the variable is changed, the limits of integration must be changed as well. If you don’t change the limits of integration, then you’ll need to back-substitute for the original variable at the end.

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Using the comparison test to determine convergence or divergence

The comparison test for convergence lets us determine the convergence or divergence of the given series by comparing it to a similar, but simpler comparison series. We’re usually trying to find a comparison series that’s a geometric or p-series, since it’s very easy to determine the convergence of a geometric or p-series.

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Modeling sales decline with exponential equations

In order to model sales decline with the exponential decay equation, the decline must have a constantly and exponentially rate of decline. If it does, we can use our standard exponential change equation.

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Every fraction has three signs (signs of fractions)

There are three signs associated with every fraction, one with the numerator, one with the denominator, and one with the fraction in general. But this can be hard to remember, because not all of the signs are always visible.

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Polar coordinates vs. rectangular coordinates

Any point in the coordinate plane can be expressed in both rectangular coordinates and polar coordinates. Instead of moving out from the origin using horizontal and vertical lines, like we would with rectangular coordinates, in polar coordinates we instead pick the angle, which is the direction, and then move out from the origin a certain distance.

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Vertical angles as congruent angles

Vertical angles are angles in opposite corners of intersecting lines. So vertical angles always share the same vertex, or corner point of the angle. They’re a special angle pair because their measures are always equal to one another, which means that vertical angles are congruent angles.

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How to get the domain and range from the graph of a function

The domain is all x-values or inputs of a function and the range is all y-values or outputs of a function. When looking at a graph, the domain is all the values of the graph from left to right. The range is all the values of the graph from down to up.

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Theorem of Pappus to find volume using the centroid

The Theorem of Pappus tells us that the volume of a three-dimensional solid object that’s created by rotating a two-dimensional shape around an axis is given by V=Ad. V is the volume of the three-dimensional object, A is the area of the two-dimensional figure being revolved, and d is the distance traveled by the centroid of the two-dimensional figure.

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Calculating absolute values

The absolute value operation turns any value inside it into its distance from the origin, essentially turning both positive and negative numbers into only positive numbers. Always calculate the value inside the absolute value first, then apply the absolute value last.

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Number sets in the real number system

The vast majority of the numbers you’ll use in most math classes are called real numbers, and the whole universe of real numbers is what makes up the Real Number System. Let’s start with a diagram.

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The alternating series estimation theorem to estimate the value of the series and state the error

The alternating series estimation theorem gives us a way to approximate the sum of an alternating series with a remainder or error that we can calculate. To use the theorem, the alternating series must follow two rules.

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Probability density functions and probability of X in an interval

Probability density refers to the probability that a continuous random variable X will exist within a set of conditions. It follows that using the probability density equations will tell us the likelihood of an X existing in the interval [a,b].

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Solving systems of three linear equations

Remember that a solution to a system of equations is the set of numbers that makes all of the equations true. If a three variable system has a solution, it’ll have a solution for each of the three variables.

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Intersecting tangents and secants of circles, intersecting inside the circle and outside the circle

There’s a special relationship between two secants that intersect outside of a circle. The length outside the circle, multiplied by the length of the whole secant is equal to the outside length of the other secant multiplied by the whole length of the other secant.

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