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Stat 281: Intro to Statistics F2003, Dr. Galster Test 2 Practice Name__________________ 1. Matching: Fill in the appropriate letters in the blanks below. A. B. C. D. E. F. G. H. I. J. K. L. M. Binomial Distribution Combination Complementary Cumulative Distribution Function Discrete Distribution Empirical Probability Event Exhaustive Independent Mean of a Distribution Mutually Exclusive n-Factorial Outcome N. O. P. Q. R. S. T. U. V. W. X. Y. Z. Poisson Distribution Probability (in general) Probability Density Function Probability Experiment Random Variable Sample Space Subjective Probability Theoretical Probability Trial Standard Normal Uniform Distribution Variance of a Distribution z ( ) _O_ The expected relative frequency of an event in the long run _F _ Probability is determined by counting frequency of actual occurances _U _ Probability determined by analyzing the known characteristics of an experiment _T _ Probability determined on the basis of personal opinion or experience _Q_ A repeatable process that yields a recordable result or observation _V_ The smallest unit of an experiment, or one repetition _M_ The immediate or physical result of an experiment _S _ The collection of all outcomes _G _ A subset of the sample space _K _ Two or more events that cannot occur at the same time _H _ Two or more events that, together, make up the whole sample space _C _ Each event contains all the outcomes that are not in the other _I__ The occurrence of one event does not change the probability of another _R _ A function that assigns a numerical value to an outcome of an experiment _J__ Expected Value of X _Y _ E( X )2 _L _ n(n 1)(n 2) (2)(1) _B _ The number of ways to choose x objects out of a group of n objects _E _ Gives the probabilities of all values of a discrete random variable _A _ Fixed number of trials, independent trials, binary response, constant P(success). _N _ Fixed mean number of occurrences per unit time or space, infinite discrete values _P _ Gives the height of the graph we use to calculate continuous probabilities _D_ Gives P( X x) for a continuous distribution _X_ Same probability for any sub-intervals of the same size within some closed interval _W Also known as N (0,1) _Z _ z-value for a particular probability to the right of z. 2. Circle the letters of any statements that are true of discrete distributions: a. The purpose of a probability function is to give the probability for each value of X. 3. Circle B if the statement applies to the binomial distribution, P if it applies to Poisson. (you may circle one, both, or neither). B There is a fixed number of trials involved. P The possible values of this random variable are infinite. B P This is a discrete distribution. P We record the number of successes or occurrences in a unit of time or space. B P Zero is a possible value of the random variable. P The mean is equal to the variance. The mean, median, and mode are always the same. B There are independent trials with a constant probability of success. 4. Circle C if the statement is true for all continuous distributions, U if it is true for Uniform distributions, and N if it is true for all Normal distributions (If C is marked, there is no need to mark U and N, you may have U and/or N without C, or none). C Half of the probability will always be below the median. U Within an interval that has positive probability, any sub-intervals that are the same width have the same probability. C P(X=1)=0 U N The distribution is symmetric about the mean. N The pdf is a “bell-shaped curve.” C The probability of a measurement must be based on an interval that accounts for the precision of the measurement. N The mean, median, and mode are all the same. U If a<p<q<b, P(p<X<q)=(q-p)/(b-a) C The area under the pdf for any interval is always less than or equal to one. C There are infinitely many values of the random variable. P(X<-1) always equals P(X>1). N The probabilities get smaller as you go further away from the mean. 5. The purpose of the pdf (probability density function) is a. To define an upper boundary for the area that represents probability 6. The purpose of the discrete probability function is b. To give the probability for any real number value of X 7. The following table shows the values for a random variable, X, and the values of its probability function. Complete the table. x 0 2 3 5 P( x) .1 .2 .3 .4 μ μ2 x P( x) 0.0 0.4 0.9 2.0 3.3 10.89 x2 0 4 9 25 Sum Variance Standard Deviation x 2 P( x) 0 .8 2.7 10 13.5 2.61 1.62 8. The pdf of a continuous random variable is given below. Find its cdf. if x 0 0 1 if 0 x 2 4 1 x if 0 x 2 4 1 F ( x) f ( x) if 2 x 3 1 x 1 if 2 x 3 2 2 2 0 otherwise 1 if x 3 9. Sally has some unusual ideas about color coordination. She has a pair of yellow pants that she will wear with probability 1/3, otherwise she will wear a red pair. She has three shirts to choose from: yellow, red, and blue. She will not wear the same color shirt as pants, and she is twice as likely to choose blue as the other available color of shirt. Make a tree diagram to illustrate the sample space and calculate the probability she will wear each color of shirt. 10. Suppose P(A)=P(B)=P(C)=.4, P(AUB)=.6, P(AUC)=.8, and P(B∩C)=.16. a) What is P(A∩B)? .2 b) What is P(A∩C)? 0 c) What is P(A|B)? .5 d) What is P(B|C)? .4 e) Are A and B mutually exclusive? No f) Are A and C mutually exclusive? Yes g) Are A and B independent? No h) Are A and C independent? No i) Are B and C independent? Yes 11. Suppose X is a binomial random variable with n=10 and p=.3. Calculate the probability that X=0, 1, 2, and more than two. Also give the mean and variance. P(0)=.0282, P(1)=.1211, P(2)=.2335, P(X>2)=.6172, mean is 3, variance is 2.1. 12. Suppose Y is a Poisson random variable with μ=5. Calculate the probability that Y=0, 1, 2, and more than two. P(0)=.0067, P(1)=.0337, P(2)=.0842, P(X>2)=.8753 13. Suppose X is a Uniform continuous random variable with positive probability on [50,100]. Find P(80<X<95). P(80<X<95)=15/50=.3 14. Find the following probabilities from a standard normal distribution. a) P(Z>2.08)=.0188 b) P(Z<1.05)=.8531 c) P(-2.06<Z<1.06)=.8357 d) P(0<Z<3)=.4987 15. Find the following probabilities from a normal distribution with μ=100 and σ=5. a) P(X>112.5)=.0062 b) P(X<87.4)=.0059 c) P(97.1<X<105.4)=.5790 d) P(X<112.5)=.9938 16. Find a z-score associated with each probability. a) P(0<Z<z)=.4382 z=1.54 b) P(Z<z)=.0011 z= -3.06 c) P(-z<Z<z)=.4108 z=.54 d) z(.0055)=2.54