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### stat400lec37

Course: STAT 400, Fall 2008
School: University of Illinois,...
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400 Statistics Tests of Statistical Hypotheses Simple hypothesis Composite Hypothesis Types of Error Type I Error: If we reject Ho when Ho is true Type II Error: If we fail to reject Ho when Ho is false Ho True Ha True Reject Ho Type I error Correct Decision Fail to reject Ho Correct Decision Type II error = P(Type I error) = P(Type II error) Ping Ma STAT 400 -1- Example: You are inspecting light...

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400 Statistics Tests of Statistical Hypotheses Simple hypothesis Composite Hypothesis Types of Error Type I Error: If we reject Ho when Ho is true Type II Error: If we fail to reject Ho when Ho is false Ho True Ha True Reject Ho Type I error Correct Decision Fail to reject Ho Correct Decision Type II error = P(Type I error) = P(Type II error) Ping Ma STAT 400 -1- Example: You are inspecting light bulbs to determine whether they are defective or not. Ho: The bulbs are not defective Ha: The bulbs are defective Reject Ho if bulbs are defective Fail to reject Ho if bulbs are not defective Draw a table like above to characterize the Type I and Type II errors associated with this example. Ho: Bulbs are not defective Reject Bulbs Ha: Bulbs are defective Fail to reject Bulbs The significance level is the probability of a Type I error. It is the probability that the test will reject the null hypothesis when Ho is true. Ping Ma STAT 400 -2- Example An environmentalist collects a liter of water from 45 different locations along the banks of a stream. He measures the amount of dissolved oxygen in each specimen. The mean oxygen level is 4.62 mg, with a standard deviation of 0.92. A water purifying company claims that the mean level of oxygen in the water is 5 mg. Conduct a hypothesis test at a 99% confidence level to determine whether the mean oxygen level is less than 5 mg. Step 1. Null Hypothesis: Step 2: Alternative Hypothesis: Step 3: Test Statistic: Step 4: critical region Step 5: P-value/Probability Step 6: Decision Rule: Ping Ma STAT 400 -3- Step 7: Conclusion Confidence Intervals and Two-Sided Tests: When conducting a two-sided test it is possible to use a confidence interval to determine whether or not to reject the null hypothesis. Steps: Null Hypothesis: H o : = 0 Alternative Hypothesis: H a : 0 n Reject H o if 0 falls outside the confidence interval range Confidence Interval: z x * Example: For the previous problem, use a 95% confidence interval to determine if the mean oxygen level in the water is different from 5 mg. Step 1: Null Hypothesis Step 2: Alternative Hypothesis Step 3: Confidence Interval Step 4: Conclusions Ping Ma STAT 400 -4- Heads-Up about Confidence Intervals and Hypothesis Testing: Power and Inference In determining the usefulness of a confidence interval, we need to look at both the level of confidence and the margin of error. Level of confidence: how reliable is our estimate Margin of Error: how sensitive is our method; how closely we pin our estimate down High confidence is of little value if the interval is so wide that few values of the parameter are excluded A test with a small level of is of little value if it almost never rejects Ho even when the true value is far from the hypothesized value. Power: The probability that a fixed level significance test rejects Ho when Ha is true. This probability is called the power of the test against the alternative. **Don't worry about learning HOW to calculate power*** Instead focus on: How to increase Power: Increase , consequently d...

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University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Tests about Proportions Recall:^ Proportions: ( p = ) Y nCount the number of successes and take into account the sample Normal Approximation of proportions Draw a random sample of size n from a large population with p = P^ (Success
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Comparing Two Means and Two VariancesTwo Sample Problems: Compare the responses in two groups Each group is an individual sample from a distinct population The responses from either group are independent of each other When a two-s
University of Illinois, Urbana Champaign - STAT - 400
Final Review We consider random variables for which the function form of pdf is known, but of the parameter of the pdf, say , is unknown. Parameter space : all possible values of . Estimator: The function of X1, X2,., Xn used to estimate , say the st
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Empirical and Probability DistributionChapter One Pages 1-10 1.1 Basic concepts Discipline of Statistics: Collection and analysis of data. What is Data? A collection of outcomes from a series of random experiments What is random exper
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 2 Properties of Probability Chapter One Pages 13-20Review : An experiment with indeterministic outcome is called random experiment. The collection of all possible outcomes of a random experiment is called the Sample Space. G
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 3 Methods of EnumerationMultiplication Principle: Suppose that an experiment E1 has n1 outcomes and an experiment E2 has n2 possible outcomes. The composite experiment E1E2 has n1*n2 outcomes. Example: There are 5 seats in a c
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 4 ProbabilityLabor Day Break is coming, let's playhttp:/math.ucsd.edu/~crypto/Monty/monty.htmlSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. Y
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 5 Section 1.5 Independence Events Independence: Definition: Outcome of one event has on effect on the outcome of another event Tossing a coin twiceDefinition: Event A and event B are independent if P(A B)=P(A and B) = P (A)
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 6 Bayes' Theorem Review: Conditional Probability: The goal is to find the probability of a certain event given that a specific condition is satisfied. Assume P (A) &gt; 0 P(B|A) = P(A and B) P (A) P (A and B) = P(B|A) * P (A) =P(A
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 7 2.1 Random variables of the discrete type Review: Random Variable: (R.V.) is a variable whose value is a numerical outcome of a random phenomenon. Notation: X, Y, Z Discrete Random Variable: 1 A discrete R.V. has a countable
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 8 Mathematical Expectation Random Variable: (R.V.) is a variable whose value is a numerical outcome of a random phenomenon. Notation: X, Y, Z Note: When a random variable X describes a random phenomenon, the sample space S just
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 9 Section 2.4 Bernoulli Trials Bernoulli Experiment: A Bernoulli experiment is a random experiment, the outcome of which can be classified as success and failure. Let X be random variable of Bernoulli distribution. Outcome Prob
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 10 Section 2.5 The Moment-generating function Moment-generating function (Laplace Transform)tx M(t)=E[etX]= e f (x )M'(t)= M&quot;(t)= xetxf (x ) f ( x)M'(0)= M&quot;(0)= x f ( x ) =E[X] x2x e2 txf ( x ) =E[X2]Thus
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 11 Section 2.6 The Poisson DistributionPoisson distributionf ( x) =x e - , x=0,1,2,. x!E[X] = Example:Var[X]= (1) Let X have a Poisson distribution with a variance of 3, Find P(X=2), P(X 6)(2) If X has a Poisson
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 12 Continuous Type Data Random Variable: (R.V.) is a variable whose value is a numerical outcome of a random phenomenon. Notation: X, Y, Z Note: When a random variable X describes a random phenomenon, the sample space S just li
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 13 Continuous Distributions Random Variable: (R.V.) is a variable whose value is a numerical outcome of a random phenomenon. Notation: X, Y, Z Note: When a random variable X describes a random phenomenon, the sample space S jus
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 14 Waiting Time Review: Continuous distribution Percentile : The (100p)th percentile p is a number such that the area under f(x) to the left of p is p. That is:first quartile: 25th percentile median: 50th percentile third qua
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 15 Waiting Time Review: Example: Customers arrive at Staples according a Poisson distribution with mean rate 1/3 per minute. On Thanksgiving morning, Staples opens 5:00am in the Morning. The first 10 customers will get a free f
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 16 Review Random Variable: (R.V.) is a variable whose value is a numerical outcome of a random phenomenon. Notation: X, Y, Z Probability Theorem 2.1-1: Complementary Rule: P (A') = 1-P (A) Theorem 2.1-2: P( )=0 Theorem 2.1-3:
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 17 Functions of a Random Variable X is a continuous random variable Y is a continuous random variable defined as Y=u(X). What is the distribution of Y CDF of Y is G(y)=P(Y&lt;y)=P[u(X)&lt;y] pdf of Y is g(y)=G'(y) Example: X has a ga
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Correlation CoefficientDiscrete random variables (X,Y)Joint probability mass function of (X,Y) x1 y1 y2 y3 f(x1, y1) f(x1, y2) f(x1, y3) . . ym f(x1, ym) x2 f(x2, y1) f(x2, y2) f(x2, y3) . f(x2, ym) x3 f(x3, y1) f(x3, y2) f(x3, y3)
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Conditional DistributionDiscrete random variables (X,Y)Joint probability mass function of (X,Y) x1 y1 y2 y3 f(x1, y1) f(x1, y2) f(x1, y3) . . ym f(x1, ym) x2 f(x2, y1) f(x2, y2) f(x2, y3) . f(x2, ym) x3 f(x3, y1) f(x3, y2) f(x3, y3)
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Transformation of Random Variables Too complicated to discuss in this class One important distribution to remember Let U and V are two independent chi-square random variables with r1 and r2 degrees of freedom, respectively.F= U / r1 V
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 4.5 Random Sample If the distributions of the independent random variables X1 and X2 are the same, the collection of the two random variables X1 and X2 is called a random sample of size 2 from the distribution.If the distribu
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 4.6 Distribution of sums of the independent random variables Theorem 4.6-1 If X1, X2, ., Xn are n independent random variables with respective 2 2 2 means 1, 2, ., n and variances 1 , 2 , ., n , then the mean and n varianc
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 The Normal Distribution Normal Curves: Describe the Normal Distribution All normal curves have the same shape: Probability density function1 ( x - )2 f ( x) = exp[ - ] 2 2 2Graph: bell shaped-&lt; x &lt; Mean: : at the center of
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 5.3 Distribution of sums of the independent random variables Review If X1, X2, ., Xn are n independent random variables with respective 2 2 2 means 1, 2, ., n and variances 1 , 2 , ., n , then the mean and n variance of Y=
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 5.4 Central limit theorem Central Limit Theorem (CLT) If X1, X2, ., Xn are observations of a random sample of size n from a distribution with mean and variance 2 , Then we have W=X - X i - n = N(0,1) as / n nnIn anoth
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 5.5 Central limit theorem Central Limit Theorem (CLT) If X1, X2, ., Xn are observations of a random sample of size n from a distribution with mean and variance 2 , Then we have W=X - X i - n = N(0,1) as / n nnIn anoth
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 5.6 Bivariate normal distribution Two random variables X and Y have a bivariate normal distribution with 2 means X and Y , variances X and Y2 , correlation coefficient .2 X X X ~ , N Y Y X Y X Y
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 29 Review Continuous distribution: Probability density function Properties of p.d.f f(x): (a) f(x)&gt;0; (b) f ( x )dx = 1 ; (c) P(a&lt;X&lt;b)= a f ( x )dx Cumulative distribution function (c.d.f) F(x)= P( X x ) = F'(x)=f(x) Expected
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Chapter 6 Estimation Maximum likelihood estimates Example: If X1, X2, ., X16 are observations of a random sample of size 16 from a normal distribution N(50,100), Find Blah blah blah How do we know that the mean is 50 and the variance i
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 6.2 Method of momentsReview:We consider random variables for which the function form of pdf is known, but of the parameter of the pdf, say , is unknown. Parameter space : all possible values of . Estimator: The function of X
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Confidence intervals for means The Statistical End Goal: draw conclusions about the data we have collected and analyzed. Every time we estimate using statistic X , we will never get the same answer. Due to sampling variability Need t
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 6.5 Confidence Intervals for Proportions Binomial distribution Let Y ~ b(n,p) Objective: construct confidence interval for pY - np Y /n- pW= np(1 - p ) = p(1 - p ) / n N(0,1) &quot;sufficiently large&quot;: np 5 and n(1-p) 5Ping
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 6.7 Sample Size Calculation Review: Confidence Interval Form: estimate margin of error n s x t / 2 * nx z / 2 *Y Y / n (1 - Y / n ) z / 2 n nThe length of the interval = 2*margin of error Tow factors associated with w
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Tests of Statistical Hypothesis Confidence Intervals: Creates an interval where we think the true parameter we are estimating will fall with a certain probability or level of confidence. Serve the purpose when the goal is to estimate
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Tests of Statistical Hypotheses Simple hypothesis Composite HypothesisTypes of Error Type I Error: If we reject Ho when Ho is true Type II Error: If we fail to reject Ho when Ho is false Ho True Reject Ho Type I error Ha True Correct
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Tests about Proportions Recall:^ Proportions: ( p = ) Y nCount the number of successes and take into account the sample Normal Approximation of proportions Draw a random sample of size n from a large population with p = P ^ (Success
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Comparing Two Means and Two Variances Two Sample Problems: Compare the responses in two groups Each group is an individual sample from a distinct population The responses from either group are independent of each other When a two-sa
University of Illinois, Urbana Champaign - STAT - 400
Final Review We consider random variables for which the function form of pdf is known, but of the parameter of the pdf, say , is unknown. Parameter space : all possible values of . Estimator: The function of X1, X2,., Xn used to estimate , say the st
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Virginia Tech - CS - 6104
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Virginia Tech - CS - 6104
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
CHM101- B Quiz #1Name _KEY_Question #1 (4 points) Classify each of the following as a pure substance, homogeneous mixture or heterogeneous mixture. (a) air : homogeneous mixture (c) sugar : pure (b) propane gas : pure (d) tomato juice : heterogen
UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
CHM101 Quiz #2 (B)Name_Key_Section#_1. One of the steps for converting ammonia to nitric acid is the conversion of NH3 to NO: 4NH3(g) +MW=175O2(g)MW=324NO(g) + 6H2OMW=30 MW=18(a) What is the limiting reactant when 26.5g of ammonia reac
UNC Wilmington - CHM - 101
CHM101 Quiz #2 (C)Name_ Section#_1. The complete combustion of acetylene (C2H2) proceeds as follow. 2C2H2(g) + 5O2(g)MW=26 MW=324CO2(g) + 2H2O(g)MW=44 MW=18(a) What is the limiting reactant when 26.5g of acetylene reacts with 26.5g of oxyge
UNC Wilmington - CHM - 101
CHM101 Quiz #3 (A)Name_KEY_ Section#_1. Calculate Go for the following reaction. Is the reaction spontaneous at 25oC? [5 points] CH4 (g) + 2O2 (g) CO2(g) + 2H2O (g) Hf (kJ/mol) So (J/mol-K)oO2(g) 0 205.0CH4(g) -74.80 186.3CO2(g) -393.5 21