Completely determined by its mean and standard

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Nature of Mathematics
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Chapter 14 / Exercise 56
Nature of Mathematics
Smith
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completely determined by its mean and standard deviation and, as a result, the standard deviation is a natural "yardstick"jor a normal distribution In any nonnal distribution, the same percentage of items will fall within some specified number of standard deviations of the mean. We can find "what% of items fall where" in a normal distribution simply by expressing our variable X's value in terms of"how many standard deviations away from the mean it is X-f.l 7'-~ i.e., the "standard normal variable" Z = (J ~ - f C 2
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Nature of Mathematics
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Chapter 14 / Exercise 56
Nature of Mathematics
Smith
Expert Verified
j Exercise: Normal Distribution Respond to the following questions: 1. I'd speculate that the "height of adult males in Minnesota" could be modeled accurately by a normal distribution with a mean of 70 inches, and a standard deviation of 3 inches. Based on these assumptions, what % of males are taller than 76 inches? Only 10% of males are taller than ? 2. Suppose packages of cream cheese coming from an automated processor have weights that are normally distributed. As the process is presently operating, the mean package weight is 8.2 ounces and the standard deviation of package weights is .1 ounce. a. If the packages of cream cheese are labeled "8 ounces", what proportion of the packages weigh less than the labeled amount? b. What would the standard deviation need to be reduced to, so that only .5% of the packages weigh less than the labeled amount? c. Suppose they are unable to reduce the variability in this process, but they still want only .5% of the packages to weigh less than the labeled amount. To what level do they need to increase the mean amount per package in order to accomplish this? 3
SAMPLING DISTRIBUTIONS Consider the very simplistic population made up of only the 5 numbers [1, 2, 3, 4, 5] Suppose we select a simple random sample of 2 items from this population: Possible Outcomes 1, 2 1, 3 1, 4 1, 5 2, 3 2, 4 2, 5 3, 4 31 5 4, 5 Possible sample Probability 1.0 means Probability 1.5 2.0 2.5 3.0 3.5 4 . 0 4.5 Possible sample means 1.5 2.0 2.5 3.0 2.5 3.0 3.5 3.5 4.0 4.5 What we've developed here is a list of all the possible values for a variable, and the probabilities associated with each of those possible values; i.e. this is a probability distribution. But unlike the probability distributions we've seen thus far, the "variable" of interest here is not an individual outcome, but rather is a sample statistic. The probability distribution for a sample statistic is called a "Sampling Distribution" 1
CHARACTERISTICS OF THE "SAMPLING DISTRIBUTION OF THE MEAN" Has a mean of Has a standard deviation of Has a shape/pattern, according to the Central Limit Theorem NOTE : wh en a population is described by a probability distribution, the mean of the population is also referred to as the "expected value", defined as I I I I 2:: X P(X) So the "expected value" of our derived population of sample mean s here is found by: E(Xbar)=l.5(.1)+2.0(.1)+2.5(.2)+3.0(.2)+3.5(.2)+4.0(.1)+4.5(.1) i . e. E(Xbar) = p(Xbar) = 3.0 And the mean of our original population here was: p = 1 + 2 + 3 + 4 + 5 p = 3 5 And this equality isn't coinc iden tal here, nor a quirk of the small population or sample size .

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