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8 Pages

### Midterm 1

Course: IEE 380, Spring 2008
School: ASU
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Word Count: 1544

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method: I+.05 for kth %ile. Returns item place in array of ordered set. K as integer. N+1 method: . K in decimal form. For interpolators (e.g. 3.5), ipart + fpart(larger-smaller). Sample mean is just average Sample variance PDFs: P(a&lt;x&lt;b) = 1) F(x) &gt;= 0 i.e. 2) CDF: F(x) = P(X&lt;x) = f(x) = some PDF =E[x]= . Shows amount of probability to that point. Normal Distribution: X:n(, 2)...

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method: I+.05 for kth %ile. Returns item place in array of ordered set. K as integer. N+1 method: . K in decimal form. For interpolators (e.g. 3.5), ipart + fpart(larger-smaller). Sample mean is just average Sample variance PDFs: P(a<x<b) = 1) F(x) >= 0 i.e. 2) CDF: F(x) = P(X<x) = f(x) = some PDF =E[x]= . Shows amount of probability to that point. Normal Distribution: X:n(, 2) -> P(Z>z)=P(Z<-z) PMFs: 1) F(x) >= 0 for all x 2) =1 Binomial distribution (discrete): Deals with some # of trials with one of two outcomes (success or failure in general) E[x]=n-p V(x)=np(1-p) CDF of: Poisson Distribution (discrete): P(X=x)= X:p(x, ), and x is # of some event in some time/distance Poisson is for some defined interval with events occurring at random Assure events are random through the interval, and can be split into subintervals so that: 1) Probability of more than one count in a subinterval is 0 2) Prob. Of one count of subinterval is same for all, and proportional to length 3) Count in each sub. Is independent of other subintervals. Exponential Distribution (continuous): Timed, so not binomial/poisson Time between successive events of a Poisson process PDF: for Good for slow wear-out Has lack of memory E(X) = , V(x)= stretch e.g. X=distance until first crack, and need to find probability of no cracks in 10-mile P(X>10)= If there are a # of cracks in 10 miles, use Poisson Joint PDFs: F(x,y)->P(a<x<b, c<Y<d) = Let Y = x+c E[Y]=E(X+c), E[x]= =E[x]+E[c]->+c V(Y)=V(X)+V(c)=V(X)=2 Y=cx1, E[Y]=cE[x]=c V(Y)=c2V(X) = c22 Y=c1x1+c2x2+cnxn E(Y)=c11+c22++cnn V(Y)=c1212+c2222 Normal Prob. Plots 1) Rank data in ascending order 2) Plot each pt with i-.5 method (P=(i-0.5)/n E(A-B) = E(A)-E(B) V(A+-B) = V(A)+V(B) e.g. A:n(stuff) B:n(stuff) C:n(stuff) D=A-B-C E(D) = A+B+C V(D) = A2+B2+C2 Central Limit Theorem: If underlying population is unknown, X1,x2xn R.S. (mu, sigma squared), and if x-bar is sampling mean, Then is n(0, 1) as n-> infinity Type I error: Rejecting H0 when it is true. Type II error: Failing to reject H0 when it is false. =P(type I error) = P(reject H0 when H0 is true) P-value is smallest level of significance that would lead to rejection of H0. 1) Parameter of interest 2) Null Hypothesis 3) Alternative Hypothesis 4) Test statistic 5) Reject H0 if 6) Computations 7) Conclusions Inference on the mean of a population, variance known Random sample size n, normally distributed. P-value: for two-sided. 1-sided = 1-(Z0) when > o, or (Z0) when < 0. Reject if z0 is > z/2 or z0 is < z/2 and fail to reject if z2 <= z0 <= z/2 (two-sided). } Critical For one-sided, reject if z0 > za ( > 0), reject if z0 < -za ( < 0). } regions! Probability of Type II Error for Two-Sided Alternative Hypothesis on the Mean, Variance Known For One-sided: Sample Size for Two-Sided Alternative Hypothesis on the Mean, Variance Known , where Z = For One-sided: and Confidence Interval on the Mean, Variance Known Sample Size for a Specified E on the Mean, Variance Known We can be (100-)% confident that the error will not exceed a specified amount E when the sample size is: One-sided Confidence Bounds on Mean, Variance Known Upper-bound: . Lower bound: Upper-bound is for < 0. Lower bound is for > 0. Hypothesis Testing on the Mean with Unknown Variance Test statistic: , S = sample standard deviation, t distribution with n-1 degrees of freedom t-distribution table provides percentage points of t distribution. Let t,k be the value of the random variable T with k degrees of freedom above which we find an area (or probability) . Thus, t,k is an upper-tail 100 percentage point of the t distribution with k degrees of freedom. E.g. t.05,10 = 1.812. That is: P(T10 > t0.05,10) = P(T10 > 1.812) = 0.05. Because t distribution is symmetric about 0, we have t1- = -t. P-value: 2-sided: P=2P(Tn-1 > |t0|). Small p-value is evidence against H0. So if P-value is sufficiently small (typically 0.05), reject H0. For > 0, P = P(Tn-1 > t0), and for < 0, P = P(Tn-1 < t0) Rejection regions: 2-sidedReject H0 if t0 > ta/2,n-1 or t0 < -ta/2,n-1 If alt is > 0, reject H0 if t0 > ta,n-1. If < 0, t0 < -ta,n-1. When true value of mean is = 0 + delta, distribution for T0 is called the noncentral t distribution with n-1 DoF and noncentrality parameter (delta*sqrt(n))/sigma. If delta = 0, noncentral t distribution reduces to usual central t distribution. Thus, type 2 error of 2-sided T0 alternative: where denotes noncentral t random variable. Use operating Curves (OC). . For one-sided, use - 0 for > 0, and 0- for < 0. Confidence Interval on the Mean with Unknown Variance P(-ta/2,n-1 <= T <= ta/2,n-1) = 1-a Or To find a 100(1-a)% lower bound on with unknown variance, replace ta/2,n-1 with ta,n-1 in lower bound of above eqn and set upper bound to INF. For upper bound, set lower to INF and replace ta/2,n-1 with ta,n-1. TESTING HYPOTHESES ON A BINOMIAL POPULATION p represents portion of defective items produced (or similar), X is number of observations in an R.S. of size n that belongs to class of interest. H0: p = p0, H1: p != p0. Test statistic: P-value: for 2-sided: P=2[1-(Z0)]. P>p0, P=1-(Z0). P<p0, P=(Z0). P-hat = X/n is sample proportion for the class of interest. New z-stat has P-hat p0 over . Above is for 2-sided. For p<p0, 1- . P>p0, . Sample size for a 2-sided Hyp. Test on a Binomial Proportion For one-sided, replace Za/2 with Za. Confidence Interval on a Binomial Proportion Sample Size for a Specified E on a Binomial Proportion 100(1- )% confident error |P-hat p| will not exceed specified E when sample is n from above equation Upper bound: Prediction Interval for a single future observation from a normal distn Tolerance Interval To contain at least % of the values in a normal population with confidence level 100(1- )%: . Values of k are given for 1- = 0.90, 0.95, and 0.99 confidence level and for gamma = 90, 95, 99% Sample variance Central Limit Theorem: If underlying population is unknown, X1,X2Xn R.S. (, 2), and if x-bar is sampling mean, Then is n(0, 1) as n-> infinity Decision Making for Two Samples Inference on the Means of 2 Populations, Variances Known Assumptions: Both samples are random samples, independent, and are normal (if not normal, C.L.T. apply) E(Xbar1 Xbar2) = E(Xbar1) E(Xbar2) = 1-2 V(Xbar1-Xbar2)=V(Xbar1)+V(Xbar2)= Hypothesis Testing on the Difference in Means, Variances Known Null hypothesis: 1-2 = delta_0 Test stat: Alternative Hyp: != delta_0P-value: P above Z_0 and P below Z_0, P = 2[1-phi(|Z_0|)] Rejection: Z_0 > Z_a/2 or Z_0 < -Z_a/2 Hyp > Delta_0P-value: P above Z_0, P = 1-phi(Z_0), reject Z_0 > Z_a Hyp < Delta_0P-value: P below Z_0, P = phi(Z_0), reject Z_0 < Z_a Type II Error and Choice of Sample Size Sample Size for 2-sided Alt Hyp. On Difference in Means, Variances known, n1=n2 For one-sided: Confidence Intervals on the Difference in Means, Variances Known Sample Size for a specified E on the Difference in Means, and variances known when n1=n2 Inference on the Means of Two Populations, Variances Unknown Hypothesis Testing on the Difference in Means Pooled estimator of 2 = Sp2: Alt. Hyp: != delta_0: P-value: Sum of P above t_0 and below t_0. Reject if t_0>ta/2,n1+n2-2 or t_0<-ta/2,n1+n2-2 Dif-means > delta_0: P-value is P above t_0, reject if t_0 > ta,n1+n2-2 Dif-means < delta_0: P-value is P below t_0, reject if t_0 < -ta,n1+n2-2 Test Statistic for Difference in Means of Two Normal Distributions, Variances Unknown, not necessarily Equal Degrees of freedom: Confidence Interval on Difference in Means Case 1: CI on DiM of2 Norm Distns, Variances Unknown and Equal Where Sp= Case 2: CI on DiM of 2 Norm Distns, Variances Unknown and Unequal V as defined above For one-sided bounds, replace /2 with just for proper bound. Lower confidence = -t, upper = t. Paired t-Test When observations on two populations are collected in pairs D=E(X1-X2) = E(X1)-E(X2)=1-2 Null hyp: D=delta_0 Test stat: Alt. hyp: != delta_0, P-value is sum of P above t_0 and below t_0, reject for t_0 > ta/2,n-1 or t_0 < -ta/2,n-1 Hyp > delta_0, p-value is P above t_0, reject if t_0 > ta,n-1 Hyp < delta_0, p-value is P below t_0, reject if t_0 < -ta,n-1 D-bar is sample average of the n differences D1, D2,Dn. SD is sample standard dev. Of the diff. CI on D for paired observations Inferences on the Ratio of Variances of Two Normal Populations F distribution: DoF. F has pdf: where W and Y are independent chi-square random vars with u and v Fu,v, u in numerator, v in denom. Mean for F distribution: =v/(v-2) for v > 2 and p. 245!..f1-a,u,v=1/fa,v,u Test stat.: Testing Hypotheses on the Equality of Variances of Two Normal Distributions Null hyp: 12=22, test stat Alt. hyp. 12!=22, reject if 12>22, reject if 12<22, reject if C.I. on the Interval of the Ratio of Variances of 2 Normal Distributions
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