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Section 6.2

With a few little improvements and extras by D.R.S, University of Cordele. Section 6.2. Finding Area under a Normal Distribution IMPORTANT: “Area” is “_____________” IMPORTANT: “Probability” is “_________”.

raymond-sheppard
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Section 6.2

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  1. With a few little improvements and extras by D.R.S, University of Cordele. Section 6.2 Finding Area under a Normal Distribution IMPORTANT: “Area” is “_____________” IMPORTANT: “Probability” is “_________”.

  2. 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.

  3. 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 9 comma 1 . 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 has ShadeNorm(-1E99,1.37) 2ND DISTR right arrow to DRAW 1:ShadeNorm etc. 2ND DRAW (on the PRGM key) 1:ClrDraw gets rid of unwanted leftover drawings.

  4. 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.

  5. Example 6.3: Finding Area to the Left of a Negative z­Value Using a Table or a TI-83/84 Plus Calculator (cont.) Try TI-84: normalcdf(-1E99,-2.03) and optionally ShadeNorm(-1E99,-2.03) (remember 2ND DRAW 1:ClrDraw if you need to clear out previous drawing)

  6. 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

  7. 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 _______________

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

  9. 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 sameas the area to the _______ of z = ________

  10. 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)

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

  12. 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.

  13. 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 ( _____, _____) = _________________

  14. Example 6.6: Finding Area between Two z-Values Using Tables or a TI-83/84 Plus Calculator (cont.) Area to the left minus equals of the area to the left the area right endpoint of the between left endpoint the two endpoints

  15. 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.

  16. 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

  17. 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)

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

  19. 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.

  20. 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)

  21. Example 6.9: Finding Area in the Tails for Two z-Values Using a TI-83/84 Plus Calculator (cont.) Thus, (0.109349)(2)  0.2187. So, the total area in the two tails is approximately 0.2187.

  22. 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)

  23. 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. c. P(1.25 < z < 2.31) d.P(z < −2.5 or z > 2.5)

  24. 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. e. P(z < −4.01) f. P(z 3.98)

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