Intro to Stats 5-7
Intro to Stats 5-7 MATH-10041-002
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This 5 page Bundle was uploaded by Amy Turk on Saturday April 2, 2016. The Bundle belongs to MATH-10041-002 at Kent State University taught by Dr. Joseph Minerovic in Spring 2016. Since its upload, it has received 19 views. For similar materials see Introductory Statistics in Math at Kent State University.
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Date Created: 04/02/16
Math Chapter 5 ● randomness is hard to achieve without help from a computer or some other randomizing device ● computer-generated random numbers are sometimes called pseudo-random numbers ○ if a researcher inputs the same seed number, the researcher will always see the same sequence of pseudo-random numbers ● empirical probabilities are based on short-run relative frequencies ● simulations = experiments used to produce empirical probabilities ● theoretical probability = the relative frequency at which an event happens after infinitely many repetitions ○ the value never changes ○ not based on experiment ● empirical probabilities are based on an experiment ● for any event to be truly random, the underlying set of outcomes should be equally likely ● random means that no predictable pattern occurs and that no digit is more likely to appear than any other ● to generate random numbers, one can use the internet, use a random number table, or use a computer program ● probabilities are always numbers between 0 and 1 ○ if the probability of an event happening is 0, the event never happens ○ if the probability of an event is 1, then the event always happens ● the sample of a random experiment is the set of all possible and equally likely outcomes of the experiment ○ often represented with the letter S ● the probability that an event does not occur is 1 minus the probability that the event will occur ● because it contains all possible outcomes, the probability that something in the sample space will occur is 1 ● when two events are mutually exclusive, they have no outcomes in common ● probabilities are always numbers between 0 and 1 ● empirical numbers are based on short-term relative frequencies ○ ex. throwing a thumbtack on a table to see how it lands ○ sometimes the only way to do an experiment ● simulations = experiments used to produce empirical probabilities ● A^c = compliment of A ○ have to give all other probabilities in the sample space ● tree diagram = useful technique for finding sample spaces ● mutually exclusive events = two events having no common outcome ○ no overlap ● “or” rule ○ use if the two events are mutually exclusive ○ p(A or B) = p(A) + p(B) ○ p(AUB) = p(A OR B) ○ U = union ● if the two events are not mutually exclusive… ○ p(A or B) = p(A) + p(B) - p(A and B) ● probability is the long-run relative frequency with which we expect a process that has chance behavior to occur ○ 3 kinds ■ subjective = info you gain from an expert ■ empirical = short-term relative frequency ■ theoretical ● probability rules ○ the probability of an event must always be between zero and one ■ if zero, then the event is impossible ■ if one, the event will always occur ■ you can’t have a negative probability ○ every event has a complementary event ■ opposites ■ when you take the probability of an event and add its complementary event, the sum is 1 ● the compliment of A = all the outcomes that aren’t in A ● “And” problems ○ when events are independent, the probability of the first and second are occurring, you multiply the probability of the first and second ○ the outcome of one does not influence the outcome of another ● replacement causes two events to be independent ● if the events are mutually exclusive, they can’t be independent or associated ● each probability must be between 0 and 1 ● the sum of probabilities must equal 1 ● mean = expected value ● normal distribution curves ○ area under curve = 1 ○ highest point is the mean ○ symmetric and unimodal ○ tails approach but do not touch the x-axis ○ 50% of the area lies to the left of the mean and 50% to the right ○ 2 infections points = occur one SD from the mean ● a smaller standard deviation results in a higher z-score Binomial Model ● exactly two outcomes (successes and failures) ● fixed number of trials ● trials are independent ● the probability of success must remain fixed at each trial ● the outcome of one does not affect the probability of the 2nd ● as N grows in size, it becomes bell shaped ● when N is small, it can take on many shapes ● the shape of the binomial distribution depends on N ● “pdf” stands for probability distribution function ● areas under curves can only represent continuous probability distributions ● the total area under a probability density curve equals 1 ○ it represents the probability that the outcome will be somewhere on the x-axis ● a random variable is a numerical measure of the outcome of a probability experiment ○ its value is determined by chance ● the standard deviation of the binomial = square root of (n * p)(1-p) ● the binomial probability model is useful in situations with discrete - valued numerical variables ● a binomial model ○ the probability of success is the same at each trial ○ the trials are independent ○ there are only two outcomes ○ there is a fixed number of trials ● the shape of a binomials distribution is determined by the number of trials, n, and the probability of success, p. ● the expected value of a probability distribution is another name for the mean LECTURE/CHAPTER 7 ● statistics are sometimes called estimates because they are representative of the true parameter ● measurement bias = asking questions that don’t produce results ● statisticians evaluate the method used for a survey, not the outcome of a single survey ● accuracy = bias ● standard error =precision ● we try to make the bias zero ● p = entire population’s proportion ● p hat = sample proportion ● one population, but many sample proportions ● as you increase the sample size, the standard error decreases ● the mean of all sample proportions always equals the population proportion ● the standard error will be smaller for larger sample sizes ● the size of the population has no effect on the distribution ● p hat - p = bias ○ goal is to get the bias down to zero ● the precision is better for larger sample sizes ● convenience sample = when only the people easily available are surveyed ● a voluntary-response bias = when only people who have strong feelings on the subject respond ● nonresponse bias = when people fail to answer a question or respond to a survey ○ the people who do not respond might have different opinions than those who do respond ● population = a group of objects or people to be studied ● a statistic is sometimes called an estimator ● parameter = numerical value that characterizes some aspect of the population ● measurement bias = occurs when the method of data collection does not produce valid results ● the quality of the survey depends on the method used for the survey ● when taking samples from a population and computing the proportion of each sample, the population proportion is always the same ○ the population proportion = the true proportion and is assumed to be independent of measurement ● standard error = the standard deviation of the sampling distribution ● the bias of a sampling distribution is measured by computing the distance between the center of the sampling distribution and the population parameter ● the precision of an estimator does not depend on the size of the population ● the precision of an estimate depends on the size of the sample ● surveys based on larger sample sizes have smaller standard errors
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