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1 Excursions in Modern Mathematics Sixth Edition Peter Tannenbaum

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2 Chapter 16 Normal Distributions Everything is Back to Normal (Almost)

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3 Normal Distributions Outline/learning Objectives To identify and describe an approximately normal distribution. To state properties of a normal distribution. To understand a data set in terms of standardized data values.

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4 Normal Distributions Outline/learning Objectives To state the 68-95-99.7 rule. To apply the honest and dishonest-coin principles to understand the concept of a confidence interval.

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5 Normal Distributions 16.1 Approximately Normal Distributions of Data

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6 Normal Distributions Approximately normal distribution Approximately normal distribution Data sets that can be described as having bar graphs that roughly fit a bell-shaped pattern. Normal distribution Normal distribution A distribution of data that has a perfect bell shape. Normal curves Normal curves Perfect bell-shaped curves.

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7 Normal Distributions 16.2 Normal Curves and Normal Distributions

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8 Normal Distributions Symmetry Symmetry Every normal curve has a vertical axis of symmetry, splitting the bell-shaped region outlined by the curve into two identical halves. We can refer to it as the line of symmetry.

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9 Normal Distributions Median/mean. Median/mean. We call the point of intersection of the horizontal axis and the line of symmetry of the curve the center of the distribution. The center is both the median and the mean (average) of the data. We use the Greek letter (mu) to denote this value.

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10 Normal Distributions Median and Mean of a Normal Distribution In a normal distribution, M = . (If the distribution is approximately normal, then M ). The fact that the median equals the mean implies that 50% of the data are less than or equal to the mean and 50% of the data are greater than or equal to the mean. For data fitting an approximately normal distribution, the median and the mean should be close to each other but we should not expect them to be equal.

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11 Normal Distributions Standard deviation. Standard deviation. The easiest way to describe the standard deviation of a normal distribution is to look at the normal curve. If you bend a piece of wire into a bell-shaped normal curve at the very top, you would be bending the wire downward (a),

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12 Normal Distributions Standard deviation. Standard deviation. but at the bottom you would be bending the wire upward (b). As you move your hands down the wire, the curvature gradually changes, and there is one point on each side of

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13 Normal Distributions Standard deviation. Standard deviation. the curve where the transition from being bent downward to being bent upward takes place. Such a point P (in figure c) is called a point of inflection of the curve.

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14 Normal Distributions The standard deviation of a normal distribution is the horizontal distance between the line of symmetry of the curve and one of the two points of inflection (P or P' )

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15 Normal Distributions Standard Deviation of a Normal Distribution Standard Deviation of a Normal Distribution In a normal distribution, the standard deviation equals the distance between a point of inflection and the line of symmetry of the curve. Quartiles of a Normal Distribution Quartiles of a Normal Distribution In a normal distribution, Q 3 + (0.675) and Q 1 + (0.675) .

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16 Normal Distributions 16.3 Standardizing Normal Data

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17 Normal Distributions Standardizing Standardizing To standardize a data value x, we measure how far x has strayed from the mean using the standard deviation as the unit of measurement. Z-value Z-value A standardized data value.

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18 Normal Distributions Standardizing Rule In a normal distribution with mean and standard deviation , the standardized value of a data point x is z = (x (x - )/ .

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19 Normal Distributions From x to z: z: Part 2 A normal distributed data set with mean = 63.18lb and standard deviation = 13.27lb. What is the standardized value of x = 91.54lb? z = (91.54 – 63.18)/13.27 = 2.13715… 2.14

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20 Normal Distributions 16.4 The 68-95- 99.7 Rule

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21 Normal Distributions – The – The 68-95-99.7 Rule 1. In every normal distribution, about 68% of all the data values fall within one standard deviation above and below the mean. In other words, 68% of all the data have standardized values between z = and z = 1.

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22 Normal Distributions – The – The 68-95-99.7 Rule 1 (cont). The remaining 32% are divided equally between data with standardized values z and data with standardized values z 1 (see figure a).

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23 Normal Distributions – The – The 68-95-99.7 Rule 2. In every normal distribution, about 95% of all the data values fall within two standard deviations above and below the mean. In other words, 95% of all the data have standardized values between z = -2 and z = 2.

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24 Normal Distributions – The – The 68-95-99.7 Rule 2 (cont). The remaining 5% of the data are divided equally between data with standardized values z -2 and data with standardized values z 2. (see figure b).

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25 Normal Distributions – The – The 68-95-99.7 Rule 3. In every normal distribution, about 99.7% (practically 100%) of all the data values fall within three standard deviations above and below the mean. In other words, 99.7% of all the data have standardized values between z = -3 and z = 3. There is a minuscule amount of data with standardized values outside the range (see figure b).

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26 Normal Distributions 16.5 Normal Curves as Models of Real- Life Data Sets

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27 Normal Distributions The 68-95-99.7 Rule for Normal curves 1. About 68% of the data values fall within (plus or minus) one standard deviation of the mean. 2. About 95% of the data values fall within (plus or minus) two standard deviations of the mean. 3. About 99.7%, or practically 100%, of the data values fall within (plus or minus) three standard deviations of the mean.

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28 Normal Distributions 16.6 Distributions of Random Events

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29 Normal Distributions Coin-Tossing Experiments: Part 1 Distribution of random variable X (number of Heads in 100 coin tosses) (a) 10 times, (b) 100 times, (c) 500 times, (d) 1000 times, (e) 5000 times, and (f) 10,000 times.

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30 Normal Distributions 16.7 Statistical Inference

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31 Normal Distributions The Honest-Coin Principle Suppose an honest coin is tossed n times (n (n 30), and let X denote the number of Heads that come up. The random variable X has an approximately normal distribution with mean = n/2 Heads and standard deviation heads.

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32 Normal Distributions Coin-Tossing Experiments: Part 2 An honest coin is going to be tossed 256 times. Let’s say that we can make a bet that if the number of Heads tossed falls somewhere between 120 and 136, we will win; otherwise we will lose. Should we make such a bet?

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33 Normal Distributions Coin-Tossing Experiments: Part 2 X = 256 = n/2 = 256/2 = 128 The values 120 to 136 are exactly one standard deviation below and above the mean of 128, which means that there is a 68% chance that the number of Heads will fall somewhere between 120 and 136. We should indeed make this bet!

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34 Normal Distributions The Dishonest-Coin Principle Suppose an arbitrary coin is tossed n times (n (n 30), and let X denote the number of Heads that come up. Suppose also that p is the probability of the coin landing heads, and (1 – p) is the probability of the coin landing tails. Then the random variable X has an approximately normal distribution with mean = n p Heads and standard deviation Heads..

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35 Normal Distributions Coin-Tossing Experiments: Part 3 Let p = 0.20n = 100 What can we say about X? = n p = 100 0.20 = 20 = 4

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36 Normal Distributions Coin-Tossing Experiments: Part 3 Applying the 68-95-99.7 rule gives the following: There is about a 68% chance that X will be somewhere between 16 and 24. There is about a 95% chance that X will be somewhere between 12 and 28. The number of Heads is almost guaranteed (about 99.7% chance) to fall somewhere between 8 and 32.

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37 Normal Distributions Conclusion Bell-shaped (normal) curves Bell-shaped (normal) curves Statistical Inference Statistical Inference Laws of probability Laws of probability

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