8.3 critical region
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Testing a proportion using critical regions follows a similar process to testing a mean. The main difference is that a proportion represents a probability rather than a measurement. When using critical regions to test a proportion, if the sample test statistic falls within the critical region, the null hypothesis is rejected, and if it falls outside the critical region, the null hypothesis is not rejected. The test statistic is the z-score calculated from the sample proportion and null hypothesized proportion.
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7.3 daqy 2leblance
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Logarithmic functions are inverses of exponential functions. To graph a logarithmic function:
1. Identify the inverse exponential form.
2. Create a table of values for the exponential form.
3. Invert the ordered pairs.
4. Plot the points and sketch the graph of the logarithm.
Logarithmic functions can be transformed through stretching, compression, reflection, horizontal translation, and vertical translation compared to the parent logarithmic function. Examples are shown to demonstrate how transformations alter the graph.7.3
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This document introduces logarithmic functions as inverses of exponential functions. It defines logarithms as the inverse of an exponential function y = bx, such that y = bx is equivalent to logby = x. The document provides examples of writing exponential equations in logarithmic form and vice versa. It also demonstrates how to evaluate logarithms by using the definition to write them in exponential form and setting the exponents equal. Finally, it defines the common logarithm as a logarithm with base 10, which can be written as logx.7.1
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The document discusses exponential functions of the form f(x) = a*b^x, explaining that they always have a curved shape and asymptote at y=0. It distinguishes between exponential growth, where the value of y increases as x increases, and exponential decay, where the value of y decreases as x increases. Examples are provided to demonstrate how to determine if a function represents growth or decay and to find the y-intercept.7.2
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This document discusses properties and transformations of exponential functions, including stretch, compression, reflection, and horizontal and vertical translation. It also discusses the number e as the base for natural exponential functions and using the function A=Pe^rt to model continuously compounded interest. Examples are provided to demonstrate graphing transformations of exponential functions and using the continuously compounded interest formula.10.2
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This document discusses the chi-square goodness of fit test, which is used to check if observed data counts match the expected distribution of counts into categories. It examines whether a population follows a specified theoretical distribution.10.1 part2
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This document discusses testing for homogeneity among populations using chi-square tests. It defines homogeneity as populations having the same structure or composition. A test of homogeneity determines if different populations have the same proportions for various categories. It requires using a contingency table and chi-square distribution. An example tests if the same proportion of males and females prefer different pet types using survey data from college students.10.1 part 1
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The document provides an overview of the chi-square distribution and how it can be used for hypothesis testing. It discusses that the chi-square distribution is used to find critical values for determining the area under the curve for a given degrees of freedom. It also gives an example of how chi-square can be used to test if two variables such as keyboard type and time to learn typing are independent.5.4 synthetic division
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This document discusses synthetic division and the remainder theorem. Synthetic division is a process that simplifies long division when dividing a polynomial by a linear factor of the form x - a. It involves setting up the coefficients of the polynomial and multiplying/adding through the process. The remainder theorem states that if a polynomial P(x) is divided by x - a, then the remainder is equal to P(a). It provides a quick way to find the remainder of a polynomial division problem by evaluating the polynomial at the value of a. Examples are given to demonstrate evaluating polynomials using the remainder theorem.5.4 long division
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Long division can be used to divide polynomials in a similar way to dividing numbers. The key steps are to set up the division problem, divide the term of the dividend by the term of the divisor, multiply the divisor by the quotient term and subtract, then bring down the next term of the dividend and repeat. This polynomial long division allows polynomials to be factored by finding all divisor polynomials that give a remainder of zero. The factor theorem can also be used to check if a linear polynomial is a factor by setting it equal to zero and checking if it makes the other polynomial equal to zero.5.3
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This document discusses solving polynomial equations by factoring. It provides examples of factoring polynomials, including factoring the difference and sum of cubes. Factoring by substitution is also introduced as a method for factoring polynomials of degree 4 or higher. The document demonstrates solving polynomial equations by factoring the expressions and setting each factor equal to 0. Both real and imaginary solutions may be obtained depending on whether the factors are real or complex numbers. Graphing is presented as an alternative method to find real solutions of a polynomial equation.9.3 Part 1
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The document discusses inferences for correlation and regression. It provides an example of testing the correlation between percentage of population with a college degree (x) and percentage growth in income (y) for 6 Ohio communities. There is a positive correlation between x and y, but this does not necessarily mean higher education causes higher earnings. The document also discusses measuring the spread of data points around the least squares line, including the standard error of estimate, using an example of how much copper sulfate dissolves in water at different temperatures.5.2
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1. The document discusses writing polynomials in factored form and finding the zeros of polynomial functions. It defines linear factors, roots, zeros, and x-intercepts as equivalent terms.
2. Examples are provided of writing polynomials in factored form using the factor theorem to find the zeros, and then graphing the polynomial function based on its zeros.
3. The factor theorem states that a linear expression x - a is a factor of a polynomial if and only if a is a zero of the related polynomial function. This allows writing a polynomial given its zeros.5.1 part 2
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This document discusses how to describe the shape of a cubic function by listing it in standard form, describing the end behavior of the graph, determining the possible number of turning points using a table of values, and determining the increasing and decreasing intervals. It explains that to describe the shape, you identify the sign of the leading coefficient to determine the end behavior and the number of turning points, which is one less than the possible degree. The document also discusses using differences of consecutive y-values in a table to determine the least degree of the polynomial function that could generate the data, with constant first differences indicating linear, constant second differences indicating quadratic, and constant third differences indicating cubic.5.1[1]
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This document defines key concepts related to polynomials and polynomial functions. It defines monomials as terms involving variables and exponents, and polynomials as sums of monomials. The degree of a polynomial is the highest exponent among its terms. Polynomial functions are polynomials written in terms of a single variable. Standard form arranges polynomial terms by descending degree. Polynomials are classified by degree and number of terms. Higher degree polynomials can have more turning points and their end behavior depends on the leading term. Examples show determining standard form, classifying polynomials, identifying end behavior and increasing/decreasing parts of graphs.9.1
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This document discusses scatter diagrams and linear correlation. It provides examples of scatter diagrams that do and do not show linear correlation. It defines the correlation coefficient r as a measure of linear correlation between two variables on a scatter plot, with values between -1 and 1. It presents formulas for calculating r and provides an example of computing r using wind velocity and sand drift rate data. It cautions that correlation does not necessarily imply causation and that lurking variables can influence the correlation between two variables.9.2 lin reg coeff of det
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1) Linear regression finds the "best-fitting" linear relationship between two variables by minimizing the vertical distances between the data points and the linear equation line.
2) The coefficient of determination, r^2, measures how well the linear relationship described by the regression line fits the actual data, with higher r^2 values indicating less unexplained variability.
3) r^2 has an interpretation as the percentage of the total variation in the response variable that is explained by the explanatory variable.Ad
8.3 critical region
- 1. AP/H Statistics Mrs. LeBlanc Perrone Name: ______________________________ 8.3: Critical RegionGuided Notes Date: ___________ 8.3 Testing a Proportion Using Critical Regions Testing a Proportion Using Critical Regions o To test a proportion using critical regions, we will use a similar process to those for testing the mean (8.1 & 8.2) o The main difference is that we are working with a _____________________________________ o When using the critical region method, we check if the sample test statistic falls in the critical region. If it _________, we ___________ If it _______________, then we ________________ Recall that when using critical regions, the ________________________ of the sketch is Since the distribution is _______________________________________ we find the critical values using invNorm 1
- 2. AP/H Statistics Mrs. LeBlanc Perrone How to Test 1. ________________________________________________________________________ Is this a binomial experiment with trials? Does represent the probability of success? Identify Is the sample large? (Is and ?) (Use p from ) If yes, then the distribution can be approximated by the normal distribution 2. ________________________________________________________________________ ________________________________________________________________________ 3. ________________________________________________________________________ Test Statistic: 4. ________________________________________________________________________ The area of the critical region = 5. Conclude the test __________________________________________________________________ __________________________________________________________________ 6. Interpret the results in the context of the application 2
- 3. AP/H Statistics Mrs. LeBlanc Perrone EXAMPLE: Testing , Critical Regions A botanist has produced a new variety of hybrid wheat that is better able to withstand drought than other varieties. The botanist knows that for the parent plants, the proportion of seeds germinating is 80%. The proportion of seeds germinating for the hybrid variety is unknown, but the botanist claims that it is 80%. To test this claim, 400 seeds from the hybrid plant are tested, and it is found that 312 germinate. Use a 5% level of significance to test the claim that the proportion germinating for the hybrid is 80%. 3
- 4. AP/H Statistics Mrs. LeBlanc Perrone Summary Questions 1. How is testing (using critical values) similar to testing (using critical values)? ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ 2. When testing using critical values, what is the test statistic for? ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ HOT Question: __________________________________________________________________________________________ __________________________________________________________________________________________ 4