Module01 random variable
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A random variable is a variable that can take on a set of possible values based on the outcome of a random experiment. There are two types of random variables: discrete and continuous. Discrete random variables take on countable whole number values, like the number of children in a family. Continuous random variables take on any real number value within an interval, like time or distance. A discrete probability distribution lists the probability of each possible outcome for a discrete random variable. It assigns a probability value to each outcome. Examples are given of finding the probability of drawing red balls from a box with red and blue balls.
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- If the derivative of f(x) is g(x), then an antiderivative of g(x) is f(x)
- Several integration formulas and examples are provided to demonstrate how to find antiderivatives of common functions like polynomials, trigonometric, exponential, and logarithmic functions
Module 7 the antiderivative
This document discusses differentiation and integration (antidifferentiation). It provides:
1) An example of differentiating and then integrating (antidifferentiating) a function to reverse the process.
2) The definition that integration is the reversing of differentiation, known as antidifferentiation or indefinite integration.
3) Several formulas for integrating common functions like polynomials, trigonometric, exponential, and logarithmic functions.
The algebraic techniques module4
This document outlines the algebraic techniques of differentiation, including the derivatives of constants, power functions, sums and differences of functions, products of functions, and quotients of functions. It provides examples of applying the differentiation rules to find the derivatives of various functions.
Module 6 the chain rule
The chain rule provides a formula for finding the derivative of a composite function h(x) = f(g(x)). The formula is: h'(x) = f'(g(x))g'(x). This takes the derivative of the outer function f with respect to the inner function g, and multiplies it by the derivative of the inner function g. An example applies the chain rule to find the derivative of h(x) = (x3 + 4x)3/2 by identifying the inner and outer functions, taking their individual derivatives, and plugging into the chain rule formula.
Module 6 the extreme value theorem
A continuous function on a closed interval will always have both a maximum and minimum value. The maximum value is the largest value the function takes on within the interval, while the minimum value is the smallest value. To find the extreme values, take the derivative to find any critical values, then evaluate the original function at the critical values and endpoints to determine which yields the maximum and minimum.
The algebraic techniques module4
This document outlines the algebraic techniques of differentiation, including the derivatives of constants, power functions, sums and differences of functions, products of functions, and quotients of functions. It provides examples of applying the differentiation rules to find the derivatives of various functions.
Module10 the regression analysis
This document discusses regression lines and linear regression. It defines independent and dependent variables in scatter plots. The best-fit line, or line of best fit, is the straight line that best illustrates the trend of data points in a scatter plot. The regression equation for a best-fit line is y=mx + b, where m is the slope and b is the y-intercept. Formulas are provided to calculate the slope and y-intercept from sample data. Two examples are worked through to find the regression equation for sets of (x, y) data points and estimate y values.
Module9 the pearson correlation
The Pearson Product Moment Correlation Coefficient (r) measures the strength of the linear relationship between two variables. The r value is calculated using a formula involving the sum of the products of paired values and sum of squares for the two variables. r values range from +1 to -1, with values closer to these extremes indicating a stronger correlation and values closer to 0 indicating a weaker or no correlation. A positive r represents a positive correlation and a negative r represents a negative correlation. The example calculates r = 0.962 for time spent studying and test scores, indicating a strong positive correlation.
Module9 the pearson correlation
The Pearson Product Moment Correlation Coefficient (r) measures the strength of the linear relationship between two variables. The r value is calculated using a formula involving the sum of the products of paired values and sum of squares for the two variables. r values range from +1 to -1, with values closer to these extremes indicating a stronger correlation and values closer to 0 indicating weaker or no correlation. A positive r represents a positive correlation and a negative r represents a negative correlation. The example calculates r = 0.962 for time spent studying and test scores, indicating a strong positive correlation.
Review on module 9
This document discusses bivariate data and scatter plots. Bivariate data involves collecting two variables in a research study: an independent variable that can cause changes in the dependent variable, which is influenced by the independent variable. Scatter plots are diagrams used to show the relationship between two sets of bivariate data by plotting them on an x-y coordinate plane. They can reveal positive or negative correlations depending on whether the points form an upward or downward sloping line. Two examples are given of scatter plots analyzing students' study/game time versus test scores.
Module08 hypotheses testing proportions
The Central Limit Theorem states that the mean of a sufficiently large random sample from any population will be approximately normally distributed, even if the population is not normally distributed. The mean of the sampling distribution equals the population mean, and its standard deviation equals the population standard deviation divided by the square root of the sample size.
To test hypotheses about a population proportion using a sample proportion, we 1) state the null and alternative hypotheses, 2) choose a significance level, 3) calculate the test statistic as the sample proportion minus the hypothesized proportion divided by the standard error, 4) determine the critical value, and 5) make a decision about whether to reject the null hypothesis based on where the test statistic falls relative to the critical value.
Review on module8
Hypotheses testing is used by statisticians to determine whether to reject statements about populations. There are 5 steps: 1) State the null and alternative hypotheses. 2) Choose the significance level. 3) Compute the test statistic. 4) Determine the critical value or p-value. 5) Draw a conclusion. An example tests if the average daily pay of jeepney drivers is different than claimed, using a test statistic, critical value, and concluding the average is significantly different at the 0.05 level.
Module 7 hypothesis testing
Hypotheses testing is used by statisticians to determine whether to reject statements about populations. It involves stating the null and alternative hypotheses, choosing a significance level, computing the test statistic, determining critical values or p-values, and drawing a conclusion. Two examples are provided to demonstrate the process of hypotheses testing for means when variance and distribution are known.
Review on module 7
The null hypothesis (Ho) states that there is no difference between a population parameter and a claimed value, and uses an equal sign. The alternative hypothesis (Ha) states there is a significant difference, and uses inequality symbols. Examples show identifying the null and alternative hypotheses for claims about mean heights and defective bulb percentages. After identifying the hypotheses, researchers calculate a test statistic from the sample and establish a rejection region and critical value based on the level of significance to determine whether to reject the null hypothesis.
Statisitcal hypotheses module06
The null hypothesis (Ho) represents no difference between a population parameter and a claimed value, using an equal sign. The alternative hypothesis (Ha) represents a significant difference, using inequality symbols. Examples show identifying the null (Ho) and alternative (Ha) hypotheses for statistical tests on mean heights and defective bulb percentages. The rejection region, critical value, and level of significance are used to determine whether to reject the null hypothesis or accept the alternative. Type I error rejects a true null hypothesis, while Type II error fails to reject a false null hypothesis.
Derivatives of trigo and exponential functions module5
The document discusses rules for taking derivatives of trigonometric and exponential functions. It provides the derivatives of sine, cosine, and exponential functions, as well as examples of using these rules to find the derivatives of more complex functions that include trigonometric and exponential terms. Specifically, it states that the derivative of sine is cosine, the derivative of cosine is negative sine, and the derivative of an exponential function is itself. It then shows examples of finding the derivatives of f(x) = sin2x + cos 3x and f(x) = sin 2x/ex.
The algebraic techniques module4
This document outlines the algebraic techniques of differentiation, including the derivatives of constants, power functions, sums and differences of functions, products of functions, and quotients of functions. It provides examples of applying the differentiation rules to find the derivatives of various functions.
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- 1. A random variable is a quantitative variable that is derived from the outcomes of a random experiment RANDOM VARIABLE DISCRETE RANDOM VARIABLE CONTINUOUS RANDOM VARIABLE - is a random variable that can take only whole number values/outcomes that are countable. It is associated with experiments for which there are finite number of possible outcomes - is a random variable that can take on non- integers as they take on values contained in an interval. It is associated for experiments with infinitely many possible outcomes, and is commonly used for measurements such as lengths, weight and time. EXAMPLES the number of children in the family five cards are drawn from the deck, one at a time, without replacement The effect on the number of students in a class in getting scores in the class the time it takes for a child to complete a lesson module the distance traveled between classes the effect on the number of hours studying in getting high grades in the subject of statistics and probability THE DISCRETE PROBABILITY DISTRIBUTION A discrete probability distribution is a list of probabilities for each of the distinct outcomes of a discrete random variable. ( ) ( ) ( ) The probability ( ) for each possible value of is: ( ) EXAMPLE #1: In a box are 2 balls – one red and one blue. Two balls picked one at a time with replacement. Find the probability of the number of red balls is drawn. 𝐿𝑒𝑡 𝑅 𝑟𝑒𝑑 𝑏𝑎𝑙𝑙 𝐵 𝑏𝑙𝑢𝑒 𝑏𝑎𝑙𝑙 𝑆 *𝐵𝐵, 𝐵𝑅, 𝑅𝐵, 𝑅𝑅+ 𝐸1: 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑒𝑑 𝑏𝑎𝑙𝑙𝑠 𝑏𝑒𝑖𝑛𝑔 𝑑𝑟𝑎𝑤𝑛 𝑃(𝑋) 1 4 + 2 4 + 1 4 1 Outcomes 𝐸1: 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑒𝑑 𝑏𝑎𝑙𝑙𝑠 𝑏𝑒𝑖𝑛𝑔 𝑑𝑟𝑎𝑤𝑛 Probability 𝑃(𝑋) BB 0 𝑃(𝑋) 1 4 BR, RB 1 𝑃(𝑋) 2 4 RR 2 𝑃(𝑋) 1 4
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