excursions in modern mathematics sixth edition

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Excursions in Modern Mathematics

Sixth Edition

Peter Tannenbaum

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Chapter 16Normal Distributions

Everything is Back to Normal (Almost)

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Normal Distributions

16.1 Approximately Normal Distributions of Data

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Normal Distributions

Approximately normal distributionApproximately normal distributionData sets that can be described as having bar graphs that roughly fit a bell-shaped pattern.

Normal distributionNormal distributionA distribution of data that has a perfect bell shape.

Normal curvesNormal curvesPerfect bell-shaped curves.

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Normal Distributions

16.2 Normal Curves and Normal Distributions

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Normal Distributions

SymmetrySymmetry

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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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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Normal Distributions

Median and Mean of a Normal DistributionMedian and Mean of a Normal DistributionIn 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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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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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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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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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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Standard Deviation of a Normal DistributionStandard 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 DistributionQuartiles of a Normal Distribution

In a normal distribution, Q3 + (0.675) and Q1 + (0.675) .

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Normal Distributions

16.3 Standardizing Normal Data

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Normal Distributions

StandardizingStandardizing

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-valueZ-value

A standardized data value.

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Normal Distributions

Standardizing RuleStandardizing Rule

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

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From From x x to to zz: Part 2: 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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Normal Distributions

16.4 The 68-95-99.7 Rule

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– The 68-95-99.7 RuleThe 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 = -1 and z = 1.

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Normal Distributions

– The 68-95-99.7 RuleThe 68-95-99.7 Rule

1 (cont). The remaining 32% are divided equally between data with standardized values z ≤ -1 and data with standardized values z ≥ 1 (see figure a).

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Normal Distributions

– The 68-95-99.7 RuleThe 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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– The 68-95-99.7 RuleThe 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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Normal Distributions

– The 68-95-99.7 RuleThe 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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16.5 Normal Curves as Models of Real-Life Data Sets

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The 68-95-99.7 Rule for Normal curves 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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16.6 Distributions of Random Events

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Coin-Tossing Coin-Tossing Experiments: Part 1Experiments: Part 1Distribution 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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16.7 Statistical Inference

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The Honest-Coin PrincipleThe Honest-Coin Principle

Suppose an honest coin is tossed n times (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 .( ) / 2n

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Coin-Tossing Experiments: Part 2Coin-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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Coin-Tossing Experiments: Part 2Coin-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!

( ) / 2n ( 256) / 2 8

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The Dishonest-Coin PrincipleThe Dishonest-Coin Principle

Suppose an arbitrary coin is tossed n times (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. .

1n p p

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Coin-Tossing Experiments: Part 3Coin-Tossing Experiments: Part 3

Let p = 0.20 n = 100

What can we say about X?

= n • p = 100 0.20 = 20

= 4 1n p p

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Coin-Tossing Experiments: Part 3Coin-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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Normal Distributions Conclusion

Bell-shaped (normal) curvesBell-shaped (normal) curvesStatistical InferenceStatistical InferenceLaws of probabilityLaws of probability