section 6.2

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Section 6.2

Finding Area under a Normal DistributionIMPORTANT: “Area” is “_____________”IMPORTANT: “Probability” is “_________”.

With a few little improvements and extras by D.R.S, University of Cordele.






Example 6.2: Finding Area to the Left of a Positive z-Value Using a Cumulative Normal Table

Find the area under the standard normal curve to the left of z = 1.37. Use the printed table.

z 0.05 0.06 0.07 0.08 0.091.0 0.8531 0.8554 0.8577 0.8599 0.86211.1 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8944 0.8962 0.8980 0.8997 0.90151.3 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9265 0.9279 0.9292 0.9306 0.93191.5 0.9394 0.9406 0.9418 0.9429 0.9441






Example 6.2: Finding Area to the Left of a Positive z-Value Using a Cumulative Normal Table (cont.)

TI-84: normalcdf(-1E99,1.37)2ND DISTR (on the VARS key)2:normalcdf(negative 1 2ND comma 9 9comma1 . 3 7 ) right paren ENTER

-1E99 is calculator language for which is a huge negative number that we use to represent , “negative __________”

TI-84 also hasShadeNorm(-1E99,1.37)2ND DISTRright arrow to DRAW1:ShadeNorm etc.

2ND DRAW (on the PRGM key)1:ClrDraw gets rid of unwanted leftover drawings.






Example 6.3: Finding Area to the Left of a Negative z-Value Using a Table or a TI-83/84 Plus Calculator

Find the area under the standard normal curve to the left of z = −2.03.

z 0.04 0.03 0.02 0.01 0-2.2 0.0125 0.0129 0.0132 0.0136 0.0139-2.1 0.0162 0.0166 0.0170 0.0174 0.0179-2.0 0.0207 0.0212 0.0217 0.0222 0.0228-1.9 0.0262 0.0268 0.0274 0.0281 0.0287-1.8 0.0329 0.0336 0.0344 0.0351 0.0359-1.7 0.0409 0.0418 0.0427 0.0436 0.0446






Example 6.3: Finding Area to the Left of a Negative z Value Using a Table or a TI-83/84 Plus Calculator (cont.)

Try TI-84: normalcdf(-1E99,-2.03)and optionallyShadeNorm(-1E99,-2.03)

(remember 2ND DRAW 1:ClrDraw if you need to clear out previous drawing)

Excel: Area to the left of z = -2.03

=NORM.S.DIST(z value, TRUE)The “TRUE” tells it to give you Cumulative, from -∞ to z






Example 6.4 : Finding Area to the Right of a Positive z-Value Using a Cumulative Normal Table

Find the area under the standard normal curve to the right of z = 1.37. Table Method: Total area under curve is _______, Use Subtraction: Total area ________ Minus area to the left of z = 1.37, which is ________ Equals area to the right of z = 1.37, which is ________

TI-84 Method: normalcdf(left endpoint, right endpoint)normalcdf(1.37, 1E99); the result is _______________






Example 6.4 : Finding Area to the Right of a Positive z-Value Using a Cumulative Normal Table (cont.)






Example 6.4 : Finding Area to the Right of a Positive z-Value Using a Cumulative Normal Table (cont.)

• Alternative Table Method: Because of symmetry, the area to the right of z = 1.37 is the same asthe area to the _______ of z = ________z 0.09 0.08 0.07 0.06 0.05

-1.6 0.0455 0.0465 0.0475 0.0485 0.0495-1.5 0.0559 0.0571 0.0582 0.0594 0.0606-1.4 0.0681 0.0694 0.0708 0.0721 0.0735-1.3 0.0823 0.0838 0.0853 0.0869 0.0885-1.2 0.0985 0.1003 0.1020 0.1038 0.1056-1.1 0.1170 0.1190 0.1210 0.1230 0.1251






Example 6.5: Finding Area to the Right of a Negative z-Value Using a Table or a TI-83/84 Plus Calculator

Find the area under the standard normal curve to the right of z = -0.90.

Table Method: TI-84 method:(show details) (show the command and the result)






Example 6.5: Finding Area to the Right of a Negative z Value Using a Table or a TI-83/84 Plus Calculator (cont.)

Excel: Area to the right of z = -0.90

=1-NORM.S.DIST(z value, TRUE)Like the printed table, NORM.S.DIST only gives you area to the left. It doesn’t do area “between” two z values like the TI-84’s normalcdf() does.So the “1 minus area to the left” technique is needed.






Example 6.6: Finding Area between Two z-Values Using Tables or a TI-83/84 Plus Calculator

Find the area under the standard normal curve between z1 = -1.68 and z2 = 2.00.

Table Method:Area to the left of z = _______ is ________Area to the left of z = _______ is ________Subtract: _______ - _______ = ________TI-84 Method:normalcdf ( _____, _____) = _________________






Example 6.6: Finding Area between Two z-Values Using Tables or a TI-83/84 Plus Calculator (cont.)

Area to the left minus equalsof the area to the left the arearight endpoint of the between left endpoint the two endpoints

Excel: Area between z = -1.68 and z=2.00

=NORM.S.DIST(high z,TRUE)-NORM.S.DIST(low z, TRUE)Like the printed table, NORM.S.DIST only gives you area to the left. It doesn’t do area “between” two z values like the TI-84’s normalcdf() does.So the subtraction of two areas technique is needed.






Example 6.7: Finding Area between Two z-Values Using a TI-83/84 Plus Calculator

Find the area under the standard normal curve between z1 = 1.50 and z2 = 2.75.

Solution – show your table and/or TI-84 details






Example 6.8: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator

Find the total of the areas under the standard normal curve to the left of z1 = −2.50 and to the right of z2 = 3.00.

Solution There are two areas that we must find. (Show details here)






Example 6.8: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator (cont.)







Note an alternative method for finding this area that is particularly clever. By definition, we know that the total area under the curve equals 1. Using this fact, the area in the tails can be obtained by finding the area between z1 = −2.50 and z2 = 3.00 and then subtracting that area from 1.






Example 6.9: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator

Find the total of the areas under the standard normal curve to the left of z1 = -1.23 and to the right of z2 = 1.23. Use SYMMETRY.

Solution Area to the left of z = -1.23 times 2.(Show details)







Thus, (0.109349)(2) 0.2187. So, the total area in the two tails is approximately 0.2187.






Example 6.11 : Finding Probabilities for the Standard Normal Distribution Using Tables or a TI 83/84 Plus Calculator ‑

Find the following probabilities using the cumulative normal distribution tables or a TI-83/84 Plus calculator.Write details and draw sketches. a. P(z < 1.45) b. P(z −1.37)







Find the following probabilities using the cumulative normal distribution tables or a TI-83/84 Plus calculator.Write details and draw sketches. c. P(1.25 < z < 2.31) d. P(z < −2.5 or z > 2.5)







Find the following probabilities using the cumulative normal distribution tables or a TI-83/84 Plus calculator.Write details and draw sketches.

e. P(z < −4.01) f. P(z 3.98)

section 6.2

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