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# Introduction to Statistical Methods STAT 301

UW

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This 6 page Class Notes was uploaded by Mrs. Triston Collier on Thursday September 17, 2015. The Class Notes belongs to STAT 301 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 34 views. For similar materials see /class/205083/stat-301-university-of-wisconsin-madison in Statistics at University of Wisconsin - Madison.

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Date Created: 09/17/15

STAT 301 TA Lisa Chung lchung statwiscedu DISCUSSION 5 Feb 21 2004 Properties for calculating probability PA m B PA 7 PA m B P Am B PB 7 PA B o PA O B PAPBA PBPAB o PBCA 17 PBA Bayes7 Formula For i12n the posterir probabilities are PB1A W W Bernoulli Trials Suppose we observe a sequence of 71 trials Let Xn denote the outcome of the nth trial The trials are called Bernoulli Trial if the following assumptions are satis ed H Each trial results in one of two possible outcomes which for convenience are labeled success and failure to The probability of obtaining a success remains constant from trial to trial This constant probability of success is denoted by the number p The probability of a failure is denoted by q OJ The trials are independent The Binomial Distribution lf random variables X1 X2 Xn are Bernoulli trials X 1 success in ith trial 1 T 0 failure in ith trial PXi1p PX0q17p 11727n Let X X1 1 X2 1 X which is the total number of successes The sampling distribution of X is given by PX77qu form701n Then we say that the random variable X has the binomial distribution denoted by X N Binnp 1 The mean of binomial distribution is M 71p 2 The standard deviation of binomial distribution is a 1Mpg where q 17p Of ce 1335 MSC 263 5948 10f ce Hour Wed100 200 and Thurs 1100 1200 STAT 301 TA Lisa Chung lchung statwiscedu Example 1 Calculate the posterior probabilities PMlAc and PFlAc using the prior probabilities and conditional probabilities given Prior probabilities PM O4PF 06 Conditional probabilities PAlM 08PAlF 03 Example 2 According to the Mendelian theory of inherited characteristics a cross fer tilization of related species of redand white owered plants produces a generation whose offspring contain 25 red owered plants Suppose that a hortriculturist wishes to cross 5 pairs of the crossfertilized species Of the 5 offspring what is the probability that a There will be one red owered plants b There will be 4 or more red owered plants Of ce 1335 MSC7 263 5948 2 Of ce Hour Wed100 200 and Thurs 1100 1200 STAT 301 TA Lisa Chung lchung statwiscedu DISCUSSION 10 Nov 15 2004 Sampling Distribution of a Statistic and the Central limit Theorem 0 Parameter A parameter is a numerical descriptive measure of the population It is calculatedestimated from observations in the population 0 Statistic A statistic is a numerical descriptive measure of a sample It is calculated from observations in the sample 0 Sampling Distribution The probability distribution of a statistic is called its sampling distribution 0 Mean and Standard Deviation of X Theidistribution of the sample mean based on a random sample size of n has mean EX u and sdX 0 Central Limit Theorem If the random sampling is from a normal population with mean 12 and std a then for any n the X 7 distribution of X is exactly normal with mean u std and x u Z 039 If the random sampling is from an arbitrary population with mean u and std a then when n is X i H x Z large n 2 30 the distribution of X is close to normal with mean u std and Example 1 A random sample of size 2 will be selected with replacement from the set of numbers 2 4 6 a List all possible samples and evaluate i and 52 for each b Determine the sampling distribution of X c Determine the sampling distribution of 52 Example 2 A population has mean 99 and standard deviatioin 7 Calculate EX and sdX for a random sample of size a 4 and b 25 Example 3 The heights of male students at a university have a nearly normal distribution with mean 70 inches and standard deviation 28 inches If 5 male students are randomly selected to make up an intramural basketball team what is the probability that the heights of the team will average over 720 inches Example 4 The weight of an almond is normally distributed with mean 05 ounce and standard deviation 015 ounce Find the probability that a package of 100 almonds will weigh between 48 and 53 ounces That is nd the probability that X will be between 048 and 053 ounce Off Hour W 100 300 pm 1 1275A MSC 262 1577 STAT 301 TA Lisa Chung lchungstatwiscedu DISCUSSION 3 Sep 27 2004 Announcement 0 Please submit your homework by putting in the folder which will be circulating in class 0 Prepare a formula sheet for quiz 0 Show me all your work to get to answer speci cally Combination Number of ways choosing r distinct items from a group of n distinct objects lt71nn71n72nir1 r 777121 7 Choose r from n or n choose r Example 1 Evaluate 20 a 30 30 b lt26 gt7 lt 4 gt Example 2 Of 9 available candidates for membership in a university committee 5 are men and 4 women The committee is to consist of 4 persons a How many different selections of the committee are possible b How many selectioins are possible if the committee must have 2 men and 2 women Example 3 An instructor will choose 3 problems from a set from a set of 6 containing 3 hard and 3 easy problems If the selection is made at random what is the probability that at least one hard problem is chosen Example 4 Harder ln STAT301 course there are 9 students Adam Bruce Charley Douglas Emilie Fisher Gauss Hotelling lngi Julia Among them 2 are in Dis 331 3 are in Dis 332 4 are in Dis 333 4 students are randomly chosen a What is the probability that one was from Dis 331 and one was from Dis 333 b What is the probability that all 4 are from Dis 333 c What is the probability that all 4 are from Dis 332 d What is the probability that Fisher will be chosen Random Variable A random variable X is a rule or function that assigns one and only one numerical value to each simple event of an experiment 1 Discrete Random Variable it has a nite or in nite many values which can be arranged in a list 2 Continuous Random Variable it has all possible values in an interval Example 5 Identify the variable as a discrete or continuous random variable a The loss of weight following a diet program Off Hour W 100 300 pm 1 1275A MSC 262 1577 STAT 301 TA Lisa Chung lchungstatwiscedu b The seating capacity of an airplane c The number of cars sold at a dealership on one day d The percentage of fruit juice in a drink mix Probability Distribution The probability distribution of a discrete random variable X is a list of the distinct numerical values of X along with their associated probabilities Example 6 Let the random varible X represent the sum of the points in two tosses of a die a List the possible values of X b Obtain the probability distribution of X Example 7 The probability distribution of z is given by the function 1 4 x 7 forxi 1234 Find a P X 2 b P X is odd Off Hour W 100 300 pm 2 1275A MSC 262 1577 STAT 301 TA Lane Burgette burgette statwiscedu DISCUSSION 6 March 8 9 1 Continuous Random Variables Probability Density Function The probability density function pdf 1 describes the distribution of probability for a continu ous random variable It has the properties 1 The total area under the probability density curve is 1 2 Pa X S b is the area under the probability density curve between a and b 3 fz 2 0 for all z Remark With a continuous random variable7 the probability that X z is always 0 1 PX a 0 for all a 2 Pa X bPaltX bPa XltbPaltXltb Median The median is a point with 50 percent of the distribution on either side of it This does not have to be the mean 100 p th Percentile 100 p th percentile is a point that has area p to the left and 1 p to its right p is between 0 and 1 Quantiles are the 25th7 50th and 75 percentiles The most common continuous random variable for our purposes will be the normal random variable It is denoted N01 0 To use the standard normal table to nd probabilities7 we almost always must standardize We do this by subtracting off the mean7 and dividing by the standard deviation That is ZX is a standard normal random variable N07 17 assuming that X N Nu7 0 Example 1 Determine the following probabilities for a standard normal random variable P5 lt X lt 1 cP1lt X g 15 wwunggm Example 2 Determine the following for X N N27 3 aP0 X lt 5 PGltX 1 c Find the median Off Hours R 230 430 pm 1 1245F MSC

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