Binomial Experiment Binomial Experiment A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties: The experiment consists of n repeated trials. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. The probability of success, denoted by p, is the same on every trial. The trials are independent; that is, the outcome on on trial does not affect the outcome on other trials. Here is an example of a binomial experiment. You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because: The experiment consists of repeated trials. We flip a coin 2 times. Each trial can result in just two possible outcomes - heads or tails. The probability of success is constant - 0.5 on every trial. Know More About Algebra 2 Help
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The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials. Example 1 What is the probability of obtaining 45 or fewer heads in 100 tosses of a coin? Solution: To solve this problem, we compute 46 individual probabilities, using the binomial formula. The sum of all these probabilities is the answer we seek. Thus, b(x < 45; 100, 0.5) = b(x = 0; 100, 0.5) + b(x = 1; 100, 0.5) + . . . + b(x = 45; 100, 0.5) b(x < 45; 100, 0.5) = 0.184 Example 2 The probability that a student is accepted to a prestigious college is 0.3. If 5 students from the same school apply, what is the probability that at most 2 are accepted? Solution: To solve this problem, we compute 3 individual probabilities, using the binomial formula. The sum of all these probabilities is the answer we seek. Thus, b(x < 2; 5, 0.3) = b(x = 0; 5, 0.3) + b(x = 1; 5, 0.3) + b(x = 2; 5, 0.3) b(x < 2; 5, 0.3) = 0.1681 + 0.3601 + 0.3087 b(x < 2; 5, 0.3) = 0.8369
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Correlation Coefficients Correlation Coefficients The common usage of the word correlation refers to a relationship between two or more objects (ideas, variables...). In statistics, the word correlation refers to the relationship between two variables. We wish to be able to quantify this relationship, measure its strength, develop an equation for predicting scores, and ultimately draw testable conclusion about the parent population. This lesson focuses on measuring its strength, with the equation coming in the next lesson, and testing conclusions much later. Examples: one variable might be the number of hunters in a region and the other variable could be the deer population. Perhaps as the number of hunters increases, the deer population decreases. This is an example of a negative correlation: as one variable increases, the other decreases. A positive correlation is where the two variables react in the same way, increasing or decreasing together. Temperature in Celsius and Fahrenheit have a positive correlation.
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Pearson Product Moment :- How can you tell if there is a correlation? By observing the graphs, a person can tell if there is a correlation by how closely the data resemble a line. If the points are scattered about then there may be no correlation. If the points would closely fit a quadratic or exponential equation, etc., then they have a nonlinear correlation. In this course we will restrict ourselves to linear correlations and hence linear regression. Since the data are almost linear, the data can be enclosed by an ellipse. The major axis (length) of the ellipse relative to the minor axis (width) of the ellipse, are an indication of the degree of correlation. How can you tell by inspection the type of correlation? If the graph of the variables represent a line with positive slope, then there is a positive correlation (x increases as y increases). If the slope of the line is negative, then there is a negative correlation (as x increases y decreases). An important aspects of correlation is how strong it is. The strength of a correlation is measured by the correlation coefficient r. Another name for r is the Pearson product moment correlation coefficient in honor of Karl Pearson who developed it about 1900. There are at least three different formulae in common used to calculate this number and these different formulae somewhat represent different approaches to the problem. However, the same value for r is obtained by any one of the different procedures. First we give the raw score formula. n has the usual meaning of how many ordered pairs are in our sample. It is also important to recognize the difference between the sum of the squares and the squares of the sums!
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