chpt6ReviewExercise - 202 Chapter 6 Sampling Distributions...

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Unformatted text preview: 202 Chapter 6 Sampling Distributions 3. When the underlying distribution is normal with mean ,u and variance 0 2 ’ calculate exact probabilities for X using the normal distribution with mean 0.2 pt and variance —— n 4. Apply the central limit theorem, wlgn the sample size is large, to approx- imate the sampling distribution of X by a normal distribution with mean 02 u. and variance —. n b the standard normal probability P (Z < | . Don’t confuse the population distribution, which describes the variation for P(X§b)=P<Z§ b—u) 6J5 — 03%)- Don’ts a single random variable, with the sampling distribution of a statistic. 2. When sampling from a _finite population of size N, don’t use U/fi as the ; standard deviation of X unless the finite population correction factor is i nearly 1. 3. When the population distribution is noticgbly nonnormal, don’t try to' conclude that the sampling distribution of X is normal unless the sample size is at least moderately large, 30 or more. Review Exercises 6.51 6.52 6.53 6.54 The panel for a national science fair wishes to select 10 states from which a student representative will be chosen at random from the students participating in the state science fair. (a) Use Table 7 to select the 10 states. (b) Does the total selection process give each student who participates in some state science fair an equal chance of being selected to be a representative at the national science fair? How many different samples of size n = 2 can be cho- sen from a finite population of size (a) N = 7; (b) N = 18? With reference to Exercise 6.52, what is the probability of choosing each sample in part (a) and the probability of choosing each sample in part (b), if the samples are to be random? Referring to Exercise 6.52, find the value of the finite population correction factor in the formula for 0% for part (a) and part (b). 6.55 The time to check out and process payment info m tion at an office supplies Web site can be modeled a: random variable with mean [L = 63 seconds and a ance 02 = 81. If the sample mean Y will be has w a random sample of n = 36 times, what can we . about the probability of getting a sample mean :4 ., than 66.75, if we use (a) Chebyshev’s theorem; (b) the central limit theorem? 6.56 The number of pieces of mail that a departrnen ceives each day can be modeled by a distribution 1} ing mean 44 and standard deviation 8. For a m” sample of 35 days, what can be said about the prob If; ity that the sample mean will be less than 40 or y than 48 using (a) Chebyshev’s theorem; (b) the central limit theorem? 6.57 If measurements of the specific gravity of a me : E be looked upon as a sample from a normal w; tion having a standard deviation of 0.04, what '3 i The probability P( X 5 b) is approximately equal to 3 probability that the mean of a random sample of size 25 will be “off” by at most 0.02? ‘8 Adding graphite to iron can improve its ductile qual- ities. If measurements of the diameter of graphite spheres within an iron matrix can be modeled as a nor- mal distribution having standard deviation 0.16, what is the probability that the mean of a sample of size 36 will differ from the population mean by more than 0.06? ".59 If 2 independent random samples of size 111 = 9 and 6.62 M = 16 are taken from a normal population, what is the probability that the variance of the first sample will ~ be at least 4 times as large as the variance of the second sample? If 2 independent sample of sizes 111 = 26 and 112 = 8 are taken from a normal population, what is the prob- ability that the variance of the second sample will be at least 2.4 times the variance of the first sample? When we sample from an infinite population, what happens to the standard error of the mean if the sample size is (a) increased from 100 to 200; (b) increased from 200 to 300; (c) decreased from 360 to 90? A traffic engineer collects data on traffic flow at a busy intersection during the rush hour by recording the number of westbound cars that are waiting for a green light. The observations are made for each light change. 6.63 6.64 Key Terms 203 Explain why this sampling technique will not lead to a random sample. Explain why the following may not lead to random samples from the desired populations: (a) To determine the smoothness of shafts, a manu- facturer measures the roughness of the first piece made each morning. (b) To determine the mix of cars, trucks, and buses in the rush hour, an engineer records the type of vehicle passing a fixed point at 1-minute intervals. Several pickers are each asked to gather 30 ripe apples and put them in a bag. (a) Would you expect all of the bags to weigh the same? For one bag, let X1 be the weight of the first apple, X 2 the weight of the second apple, and so on. Relate the weight of this bag, 30 2x:- i=1 to the approximate sampling distribution of Y. (b) Explain how your answer to part (a) leads to the sampling distribution for the variation in bag weights. (C) If the weight of an individual apple has mean ,u = 02 pound and standard deviation 0 = 0.03 pound, find the probability that the total weight of the bag will exceed 6.2 pounds. Central limit theorem 184 Infinite population 175 Chi square distributlon 189 Law of large numbers 182 Convolution formula 199 Moment generating function Degrees of freedom 188 method 195 Discrete uniform distribution 179 Parameter 178 Distribution function method 197 : F distribution 190 Population 175 ' Finite population 175 Finite population correction Random sample factor 1 8 1 Representation 192 Probability integral transformation 197 Probable error of the mean 176 Sample 175 Sampling distribution 180 Standard error of the mean 183 Standardized sample mean 183 Statistic 178 tdistribution 187 Transformation method 198 1 86 ...
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