Statistics Unit 2
Statistics Unit 2 Math 141-08
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This 5 page Class Notes was uploaded by Kasandra Angermeier on Tuesday September 20, 2016. The Class Notes belongs to Math 141-08 at Lincoln Land Community College taught by Dr. Richard Monke in Fall 2016. Since its upload, it has received 3 views. For similar materials see Statistics in Mathmatics at Lincoln Land Community College.
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Date Created: 09/20/16
Probability, Random Variables and Sampling Distributions Probability Terminology Experiment – a process with observable outcomes Sample Space – the set of all possible outcomes for an experiment Event – Any subset of the sample space Example – A Dice Game has an observable outcome of sums. *Above is a Sample Space* ~The Number of Elements in event E is denoted by n(E). In the example the n(S) = 36. ~Assuming each outcome is likely, then the probability of event E is expressed as follows: Example – Another Dice Game with winning and nonwinning probabilities *The Highlighted sets are the winning sets* ~The probability of losing is higher than the probability of winning, making the game unfair. ~Probabilities range in value from zero to one. An event is certain if the probability is one. A probability of zero is an impossible event. ~Theoretical Probability – deals with expected probabilities based on a model. ~Experimental Probability – deals with actual outcomes of an experiment. Example – Tossing a fair coin 100 times under the equally likely assumption, we would expect 50 heads and 50 tails to occur. This results in a theoretical probability of 0.5. ~Events are mutually exclusive if the sets have no common observations. ~Events are said to be nonmutually exclusive if there is at least one common observation. ~Events are independent if the occurrence of one event does not affect the probability of the other event occurring. ~Conditional Probability – the probability of event B occurs given event A has occurred. It’s denoted by P(B|A). Use the following formula to find conditional probabilities: Example – At LLCC, 326 students in a group are taking English or Math. Of those students, 192 are enrolled in an English course while 292 are enrolled in a Math course. What is the probability that a student is enrolled in Math given the student is taking English? ~Consider an experiment of flipping a fair coin three times and recording the number of heads that occur. The possible outcomes are 0,1,2, or 3. Assigning a quantitative variable to the outcomes is termed a random variable since the outcomes depend on chance. Specifically, a discrete random variable since the outcomes can be listed. The mean of a discrete random variable is given by m=S x P(X=x), where x is a discrete value and P(X=x) is the probability the discrete random variable takes on the value of x. This is termed expected value or expectation. ~Binomial Distributions – when something is either a success or failure, it becomes a special class of probability distribution. ~A binomial distribution has the following characteristics: n identical trails are performed only success or failure are possible for each trial the trials are independent the probability of success remains the same from trial to trial ~Binomial Probability Formula: ~ Normal distribution shape is characterized by the bellshaped curve. ~Percentiles: the data value dividing the data set is the percentile value. Example For boys of age 30 months, 34.5, 36, and 37 are the percentile values for the 90th, 95th, and 97th percentiles, respectively. Thus, a boy age 30 months weighing 34.75 pounds would weigh more than 90% of boys his age. ~ When the data set is large, a histogram reflecting the bellshaped curve indicates the data set is normal. The following histogram represents a sample of size 1000 taken from a standard normal distribution, m=0 and s=1. A sample size of 15 is taken from the same distributions as shown in the following: *It is not clear with the previous that the data set is normal. ~Plotting an ordered data set against typical values found in a standard normal distribution will produce a linear graph if the data set is normal. Example – The following is an example of a data set that is not normal: ~ In most instances, it is not practical to conduct a census to gather information regarding a population under study. For this reason, representative samples of the population are gathered to infer information about the population. When this is done, the sample will not exactly reflect the population under study. This imprecise characteristic is referred to as sampling error. *Is it possible that a sample size of 3 exhibits a greater sample error than a sample size of 1? *YES* *The distribution of the sample mean can also be approximated as normal if the sample size is relatively large. Typcially, if n is greater than 30 the distribution of the sample mean will be normal regardless of the distribution of the population.
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