Lesson 7 - QR

Quiz Review

Objectives

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Review for the chapter 7 quiz on sections 7-1 through 7-3

Vocabulary

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Continuous random variable – has infinitely many values

•

Uniform probability distribution – probability distribution where the probability of occurrence is equally likely for any equal length intervals of the random variable X

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Normal curve – bell shaped curve

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Normal distributed random variable – has a PDF or relative frequency histogram shaped like a normal curve

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Standard normal – normal PDF with mean of 0 and standard deviation of 1 (a z statistic!!)

Continuous Uniform PDF

1 0.75

0.5

0.25

P(x=1) = 0 P(x ≤ 1) = 0.33

P(x ≤ 2) = 0.66

P(x ≤ 3) = 1.00

0 0 1 2 3 Since the area under curve must equal one. The height or P(x) will always be equal to 1/(b-a), where b is the upper limit and a the lower limit. Probabilities are just the area of the appropriate rectangle.

Properties of the Normal Density Curve

• • • • • • •

It is symmetric about its mean, μ Because mean = median = mode , the highest point occurs at x = μ It has inflection points at μ – σ and μ + σ Area under the curve = 1 Area under the curve to the right of μ equals the area under the curve to the left of μ, which equals ½ As x increases or decreases without bound (gets farther away from μ), the graph approaches, but never reaches the horizontal axis (like approaching an asymptote) The Empirical Rule applies

Empirical Rule

μ ± 3σ μ ± 2σ μ ± σ 99.7

% 95% 68% 0.15% 2.35% 13.5% 34% 34% 13.5% 2.35% 0.15% μ - 3σ μ - 2σ μ - σ μ μ + σ μ + 2σ μ + 3σ

Normal Curves

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Two normal curves with different means (but the same standard deviation) [on left]

–

The curves are shifted left and right

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Two normal curves with

different standard deviations

(but the same mean) [on right]

–

The curves are shifted up and down

Area under a Normal Curve

The area under the normal curve for any interval of values of the random variable X represents either

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The proportion of the population with the characteristic described by the interval of values or

•

The probability that a randomly selected individual from the population will have the characteristic described by the interval of values [the area under the curve is either a proportion or the probability]

Standardizing a Normal Random Variable

our Z statistic from before X μ Z = ---------- σ where μ is the mean and σ is the standard deviation of the random variable X Z is normally distributed with mean of 0 and standard deviation of 1 Z measures the number of standard deviations away from the mean a value of X is

Normal Distributions on TI-83

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normalcdf

cdf = Cumulative Distribution Function

This function returns the cumulative probability from zero up to some input value of the random variable x. Technically, it returns the percentage of area under a continuous distribution curve from negative infinity to the x. You can, however, set a different lower bound.

•

Syntax:

normalcdf (lower bound, upper bound, mean, standard deviation)

(note: we use -E99 for negative infinity and E99 for positive infinity)

Normal Distributions on TI-83

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invNorm

inv = Inverse Normal PDF

This function returns the x-value given the probability region to the left of the x-value. (0 < area < 1 must be true.) The inverse normal probability distribution function will find the precise value at a given percent based upon the mean and standard deviation.

•

Syntax:

invNorm (probability, mean, standard deviation)

Obtaining Area under Standard Normal Curve

Approach Find the area to the left of z

a P(Z < a)

Graphically Shade the area to the left of z

a

Solution Use Table IV to find the row and column that correspond to z

a

. The area is the value where the row and column intersect.

Normcdf(-E99,a,0,1) a Find the area to the right of z

a

Shade the area to the right of z

a

Use Table IV to find the area to the left of z

a

. The area to the right of z

a

is 1 – area to the left of z

a . P(Z > a) or 1 – P(Z < a)

Find the area between z

a

and z

b

Normcdf(a,E99,0,1) or 1 – Normcdf(-E99,a,0,1) a Shade the area between z

a and z b

Use Table IV to find the area to the left of z

a

and to the left of z

a

. The area between is area zb – area za .

P(a < Z < b)

Normcdf(a,b,0,1) a b

Problems

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Standard Normal Random Variable P(Z < 1.96) = 0.975 normalcdf(-E99,1.96) P(Z > 0.57) = 0.284 normalcdf(0.57,E99) P(-2.71 < Z < 1.09) = 0.859 normalcdf(-2.71,1.09)

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Regular Normal Random Variable P(x<4) = 0.965 with μ=2 σ=1.1

normalcdf(-E99,4,2,1.1) P(x>16) = 0.965 with μ=10 σ=3.84

normalcdf(16,E99,10,3.84)

Problems

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Standard Normal Random Variable What is the Z value associate with 91 st percentile?

Z = invNorm(0.91) =

•

Regular Normal Random Variable What is the X value associated with 57% to the right with μ = 11 and σ = 3?

X = invNorm(1-0.57,11,3) = 10.47

invNorm uses area to the left!