[PDF] 85&86: The Normal Distribution and Applications - TAMU Math

40944_6141LN_8_5_8_6.pdf c

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8.5&8.6: The Normal Distribution and Applications

RECALL:To nd a probability distribution with adiscrete niterandom variable can be represented graphically

by a histogram. Characteristics of a histogram: Each rectangle is centered aroundx. Each rectangle has a base of width 1. The height of each rectangle represents the probability for that particular value ofX,P(X=x).

The area of any given rectangle (width x height) is equal to the probability of that random variable.

If you want to nd the probability of a range ofXvalues, we would add up the ares over the range ofX

values. Continuous Probability Distributionsare probability distributions associated with continuous random variables. For a continuous random variable, the graph of the probability distribution becomes a smooth curve. To nd probability with acontinuousrandom variable,X, we use a probability density function,f(x). This function has the properties:

1.f(x)0 for all values ofX;

2. The area under the graph off(x) is equal to 1 (on the values of X).P(a < x < b) = area under the probability distribution fromx=atox=b:REMARK1.Because the area under one point of the graph offis equal to zero, we see that:P(a < x < b) =P(a < xb) =P(ax < b) =P(axb)

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We concentrate on a special class of continuous probability distributions known asnormal distributions. Each normal distribution is dened by(mean) and(standard deviation;determines the sharpness or atness of the curve).

Characteristics of the Normal Distribution Curve:

1. The curve has a peak atx=.

2. The curve is symmetric about a vertical line passing through the mean,x=.

3. The curve always lies above thex-axis but approaches thex-axis asxextends indenitely

in either direction. (The curve never crosses thex-axis.)

4. The area under the curve is always 1.

5. Regardless of the value ofand:

68:27% of the area lies within 1from the mean. 95:45% of the area lies within 2from the mean.

99:73% of the area lies within 3from the mean.EXAMPLE2.Shade the area under the normal curve that represents these probabilities:

(a)P(Xa) c

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(b)P(c < Xd)(c)P(Xb)(d)P(X=a) c

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Tond the probability of a normal distribution, we will use the built-in feature of your calculator:

Go to DIST which is found by pressing2ndVARS.

normalcdf(lower,upper,;) computes the probability that a continuous random variableXis between the lower bound and the upper bound,P(lower< X P(X < a)-1E99anormalcdf(-1EE99,a,;)P(a < X < b)abnormalcdf(a,b,;)P(X > b)b1E99normalcdf(b,1EE99,;)REMARK3.To enter 1E99 press1EE99. The EE represents scientic notation and theEEis found

by pressing2nd,. EXAMPLE4.SupposeXis a normal random variable with= 70and= 4. each of the following probabilities: (a)P(66X <74) (b)P(X >70) (c)P(X72) DEFINITION5.Thestandard normal curveis the normal curve with= 0and= 1. The random variable for the standard normal curve isZ. EXAMPLE6.Let Z be the standard normal variable. Make a sketch of the appropriate region under the standard normal curve, and then nd the values of (a)P(1:5Z <2)(b)P(Z >0:5) c

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EXAMPLE7.Find the area under the standard normal curve to the left of z = 1.27.To nd the cuto,when given a probability:

Go to DIST which is found by pressing2ndVARS. Your 3rd choice isinvnorm": invnorm(probability,;). Here the probability is given and is sometimes referred to as "area". REMARK8.In order for this feature on your calculator to work, you must always enter in the

probability LESS THAN the cuto you are looking for, i.e.we useinvnorm(d;;) to ndcinP(X < c) =d, givend:EXAMPLE9.LetXbe a normally distributed random variable with= 53and= 4.

(a)Findcsuch thatP(X < c) = 0:325 (b)Findbsuch thatP(Xb) = 0:525 c

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EXAMPLE10.Find the value ofcsuch that

(a)P(Z < c) = 0:14 (b)P(Z > c) = 0:65 (c)P(c < z < c) = 0:84 EXAMPLE11.A certain company manufactures articial starter logs for replaces. These logs are accepted by the buyer if they fall within the tolerance limits of0:695inches and0:780inches in length. Assuming that the length of the logs is normally distributed with a mean of0:72inches and a standard deviation of0:03inches, estimate the percentage of logs that will be rejected by the buyer. c

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EXAMPLE12.At a certain hospital, the weight of infants is normally distributed with a mean of7:5lbs and a standard deviation of1:1lbs. (a)What is the probability that a randomly selected infant at this hospital weighs more than8 lbs? (b)What is the probability that a randomly selected infant at this hospital weighs exactly7:5lbs? (c)Only1%of all infants at this hospital weigh less thanlbs. (d)25%of all infants at this hospital weigh more thanlbs. (e)If you randomly access records of1000infants at this hospital, how many of those infants would you expect to weigh more than9lbs? c

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EXAMPLE13.An instructor gave an exam to his class that had an average of65and standard deviation of13. He decided to assign grades as follows: the top6%and the bottom6%will receive A's and F's, respectively. The next16%in either direction will be given B's and D's, and the remaining students will receive C's. Assuming that the grades on the exam are normally distributed, nd the cutos for each grade level.

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