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Inrto Stat Week 6 Notes

by: Qihua Wu

Inrto Stat Week 6 Notes TMATH 110 C

Qihua Wu
University of Washington Tacoma
Intro Stat Applications

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About this Document

This week includes finding the probability of a variable with z and individual distributions.
Intro Stat Applications
Class Notes
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This 0 page Class Notes was uploaded by Qihua Wu on Monday November 9, 2015. The Class Notes belongs to TMATH 110 C at University of Washington Tacoma taught by KENNEDY,MAUREEN C. in Fall 2015. Since its upload, it has received 8 views. For similar materials see Intro Stat Applications in Math at University of Washington Tacoma.

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Date Created: 11/09/15
Standard normal distribution a combination of curves mainly de nes by the mean and the standard deviation and used to calculate normal random variable39s probability As a data39s standard deviation away from the mean increases the density of that data decreases and vice versa you can see standard deviation as the distance Since normal distribution are symmetric within the same deviation for the mean the density is the same like the area between the left 2 standard deviation from the mean and the area between the right 2 standard deviation from the mean are the same Any normal distribution can be changed into standard normal distribution To nd the probability of normal random variable 1 Start by drawing a standard normal curve to identify the range of variables you are looking for 2 2 normal random variable mean standard deviation of normal random variable Z represents the number of standard deviations the normal random variable is away from the mean thus Z is called the standard normal deviate Z has a normal distribution with a mean of 0 and standard deviation of 1 When rounding standard deviation it has same number of decimals as the mean 3 Find the probability that is associated with Z with a Ztable that is the probability of the normal random variable Tips for the reading the table and nding the probability of 2 Always check whether the area under the curve is to the left or to the right of 2 most of the time is to the left of z The probabilities for positive 2 value and the corresponding negative 2 value always add up to 1 just in case if you are only given one page of zvalue table After you identify the range you are looking for if it is left of 2 you can just take the probability of the 2 value if you are looking for the area right of 2 you can use 1 to subtract the probability on the left of the 2 value If you are looking for a range between 2 values rst nd the probability of each values then use the larger probability to subtract the smaller probability to nd the probability of the range between two values Let s say you are looking for the probability of 13 lt 2 lt15 then what you want to do is nd the probability of zlt15 and zlt13 then use the probability of zlt15 zlt13 This is easier to understand when you have drawn a normal distribution what you are looking for in this kind of problem is the area between 13 and 15 in this particular case therefore the area greater than 15 is not considered which means we can directly take the probability associated with a 2 value of 15 and the probability associated with a zvalue of 15 includes all the probability that is less than 15 however any area less than 13 is not needed therefore by subtracting the probability that is associated with a 2 value of 13 we exclude all the probability that is less than 13 You can also approach this kind of problem by using 1 as a whole then subtract the probability that is greater than 15 which means you have to use 1 to subtract the 2 value associated with 15 to nd the probability of z greater than 15 then subtract the resulted probability from 1 and then subtract the probability that is less than 13 If you want to nd the absolute value of 2 between a range let s say absolute value of z is lt12 what you are really getting is 12ltzlt12 You can approach it with the rst method where you use the zlt12 subtract zlt12 or you can approach it by 12the probability of zlt15 The second part works because normal distribution has the property of being symmetrical and when you are nding this kind of range what you do is cutting the same amount of area from both ends We use negative 2 value instead of the positive 2 is because the area is cumulative from the left of 2 if we use a positive 2 we would be cutting off not the edge of the end but from one end all the way to the positive 2 You can check if you have cut off the right area by drawing a normal distribution and see if the area you are looking for matched the probability you get at the end for example if the area you are looking for in your curve is more than half of the curve you should not end up with a probability smaller than 05 For sketching distributions we should include labels and mean numbers of these values one standard deviation within mean and two standard deviations within mean all includes positive and negative and corresponding zscores When we are given the probability and nd 2 we go to the probability is given on the table and nd the corresponding 2 score If you are given a normal distribution graph and the probability you are looking for is 2 from the right end the probability you got is 102 which is 98 Discrete variable distribution is rightskewed which means it would not be symmetrical and is de ned by mean and variance If the population parameter is not given for the distribution we use sample statistics to estimate the values of the parameters after taking a random sample Rounded probability number of times an event is observed number of trials Sampling distribution under repeated sampling of a given sample size we nd the distribution of the statistics Simple random sampling equal opportunity for all possible values being selected which means it is with replacement that is after each drawn the subject is being returned to the whole population However in the real world replacement only matters when you have small sample size since it would not affect the variability of the big sample size with just small samples Central limit theory regardless of the distribution as the sample size increases the distribution of the sample mean approaches to a normal distribution since discrete samples are always nite so it is only approaching this is very reliable once your sample size is greater than 30


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