normal distribution - home - math - the university of utahlzhang/teaching/3070spring2009... ·...
TRANSCRIPT
![Page 1: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/1.jpg)
Normal Distribution
Table A.3 Standard Normal Curve Areas
z .00 .01 .02 .03 .04 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · ·-1.2 0.1151 0.1131 0.1112 0.1094 0.1075 · · · 0.0985-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 · · · 0.1170· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
![Page 2: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/2.jpg)
Normal Distribution
Table A.3 Standard Normal Curve Areas
z .00 .01 .02 .03 .04 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · ·-1.2 0.1151 0.1131 0.1112 0.1094 0.1075 · · · 0.0985-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 · · · 0.1170· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
![Page 3: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/3.jpg)
Normal Distribution
Z ∼ N(0, 1), calculate (a)P(Z ≤ 1.61); (b)P(Z > −1.12); and(c)P(−1.12 < Z ≤ 1.61).
z .00 .01 .02 .03 .04 · · · .09· · · · · · · · · · · · · · · · · · · · · · · ·-1.2 0.1151 0.1131 0.1112 0.1094 0.1075 · · · 0.0985-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 · · · 0.1170· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
P(Z ≤ 1.61) = 0.9463;P(Z > −1.12) = 1− P(Z ≤ −1.12) = 1− 0.1314 = 0.8686;P(−1.12 < Z ≤ 1.61) = P(Z ≤ 1.61)− P(Z ≤ −1.12) =0.9463− 0.1314 = 0.8149.
![Page 4: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/4.jpg)
Normal Distribution
Z ∼ N(0, 1), calculate (a)P(Z ≤ 1.61); (b)P(Z > −1.12); and(c)P(−1.12 < Z ≤ 1.61).
z .00 .01 .02 .03 .04 · · · .09· · · · · · · · · · · · · · · · · · · · · · · ·-1.2 0.1151 0.1131 0.1112 0.1094 0.1075 · · · 0.0985-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 · · · 0.1170· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
P(Z ≤ 1.61) = 0.9463;P(Z > −1.12) = 1− P(Z ≤ −1.12) = 1− 0.1314 = 0.8686;P(−1.12 < Z ≤ 1.61) = P(Z ≤ 1.61)− P(Z ≤ −1.12) =0.9463− 0.1314 = 0.8149.
![Page 5: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/5.jpg)
Normal Distribution
Z ∼ N(0, 1), calculate (a)P(Z ≤ 1.61); (b)P(Z > −1.12); and(c)P(−1.12 < Z ≤ 1.61).
z .00 .01 .02 .03 .04 · · · .09· · · · · · · · · · · · · · · · · · · · · · · ·-1.2 0.1151 0.1131 0.1112 0.1094 0.1075 · · · 0.0985-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 · · · 0.1170· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
P(Z ≤ 1.61) = 0.9463;P(Z > −1.12) = 1− P(Z ≤ −1.12) = 1− 0.1314 = 0.8686;P(−1.12 < Z ≤ 1.61) = P(Z ≤ 1.61)− P(Z ≤ −1.12) =0.9463− 0.1314 = 0.8149.
![Page 6: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/6.jpg)
Normal Distribution
Z ∼ N(0, 1), calculate (a)P(Z ≤ 1.61); (b)P(Z > −1.12); and(c)P(−1.12 < Z ≤ 1.61).
z .00 .01 .02 .03 .04 · · · .09· · · · · · · · · · · · · · · · · · · · · · · ·-1.2 0.1151 0.1131 0.1112 0.1094 0.1075 · · · 0.0985-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 · · · 0.1170· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
P(Z ≤ 1.61) = 0.9463;
P(Z > −1.12) = 1− P(Z ≤ −1.12) = 1− 0.1314 = 0.8686;P(−1.12 < Z ≤ 1.61) = P(Z ≤ 1.61)− P(Z ≤ −1.12) =0.9463− 0.1314 = 0.8149.
![Page 7: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/7.jpg)
Normal Distribution
Z ∼ N(0, 1), calculate (a)P(Z ≤ 1.61); (b)P(Z > −1.12); and(c)P(−1.12 < Z ≤ 1.61).
z .00 .01 .02 .03 .04 · · · .09· · · · · · · · · · · · · · · · · · · · · · · ·-1.2 0.1151 0.1131 0.1112 0.1094 0.1075 · · · 0.0985-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 · · · 0.1170· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
P(Z ≤ 1.61) = 0.9463;P(Z > −1.12) = 1− P(Z ≤ −1.12) = 1− 0.1314 = 0.8686;
P(−1.12 < Z ≤ 1.61) = P(Z ≤ 1.61)− P(Z ≤ −1.12) =0.9463− 0.1314 = 0.8149.
![Page 8: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/8.jpg)
Normal Distribution
Z ∼ N(0, 1), calculate (a)P(Z ≤ 1.61); (b)P(Z > −1.12); and(c)P(−1.12 < Z ≤ 1.61).
z .00 .01 .02 .03 .04 · · · .09· · · · · · · · · · · · · · · · · · · · · · · ·-1.2 0.1151 0.1131 0.1112 0.1094 0.1075 · · · 0.0985-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 · · · 0.1170· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
P(Z ≤ 1.61) = 0.9463;P(Z > −1.12) = 1− P(Z ≤ −1.12) = 1− 0.1314 = 0.8686;P(−1.12 < Z ≤ 1.61) = P(Z ≤ 1.61)− P(Z ≤ −1.12) =0.9463− 0.1314 = 0.8149.
![Page 9: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/9.jpg)
Normal Distribution
Many tables for the normal distribution contain only thenonnegative part.
z .00 .01 .02 .03 .04 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
What is P(Z < −1.63)?By symmetry of the pdf of Z , we know that P(Z < −1.63) =P(Z > 1.63) = 1− P(Z ≤ 1.63) = 1− 0.9484 = 0.0516
![Page 10: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/10.jpg)
Normal Distribution
Many tables for the normal distribution contain only thenonnegative part.
z .00 .01 .02 .03 .04 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
What is P(Z < −1.63)?By symmetry of the pdf of Z , we know that P(Z < −1.63) =P(Z > 1.63) = 1− P(Z ≤ 1.63) = 1− 0.9484 = 0.0516
![Page 11: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/11.jpg)
Normal Distribution
Many tables for the normal distribution contain only thenonnegative part.
z .00 .01 .02 .03 .04 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
What is P(Z < −1.63)?
By symmetry of the pdf of Z , we know that P(Z < −1.63) =P(Z > 1.63) = 1− P(Z ≤ 1.63) = 1− 0.9484 = 0.0516
![Page 12: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/12.jpg)
Normal Distribution
Many tables for the normal distribution contain only thenonnegative part.
z .00 .01 .02 .03 .04 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
What is P(Z < −1.63)?By symmetry of the pdf of Z , we know that P(Z < −1.63) =P(Z > 1.63) = 1− P(Z ≤ 1.63) = 1− 0.9484 = 0.0516
![Page 13: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/13.jpg)
Normal Distribution
Many tables for the normal distribution contain only thenonnegative part.
z .00 .01 .02 .03 .04 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · ·
What is P(Z < −1.63)?By symmetry of the pdf of Z , we know that P(Z < −1.63) =P(Z > 1.63) = 1− P(Z ≤ 1.63) = 1− 0.9484 = 0.0516
![Page 14: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/14.jpg)
Normal Distribution
Recall: The (100p)th percentile of the distribution of a continuousrv X , η(p), is defined by
p = F (η(p)) =
∫ η(p)
−∞f (y)dy
Similarly, the (100p)th percentile of the standard normal rv Z isdefined by
p = F (η(p)) =
∫ η(p)
−∞
1√2π
e−y2/2dy
We need to use the table for normal distribution to find (100p)thpercentile.
![Page 15: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/15.jpg)
Normal Distribution
Recall: The (100p)th percentile of the distribution of a continuousrv X , η(p), is defined by
p = F (η(p)) =
∫ η(p)
−∞f (y)dy
Similarly, the (100p)th percentile of the standard normal rv Z isdefined by
p = F (η(p)) =
∫ η(p)
−∞
1√2π
e−y2/2dy
We need to use the table for normal distribution to find (100p)thpercentile.
![Page 16: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/16.jpg)
Normal Distribution
Recall: The (100p)th percentile of the distribution of a continuousrv X , η(p), is defined by
p = F (η(p)) =
∫ η(p)
−∞f (y)dy
Similarly, the (100p)th percentile of the standard normal rv Z isdefined by
p = F (η(p)) =
∫ η(p)
−∞
1√2π
e−y2/2dy
We need to use the table for normal distribution to find (100p)thpercentile.
![Page 17: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/17.jpg)
Normal Distribution
Recall: The (100p)th percentile of the distribution of a continuousrv X , η(p), is defined by
p = F (η(p)) =
∫ η(p)
−∞f (y)dy
Similarly, the (100p)th percentile of the standard normal rv Z isdefined by
p = F (η(p)) =
∫ η(p)
−∞
1√2π
e−y2/2dy
We need to use the table for normal distribution to find (100p)thpercentile.
![Page 18: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/18.jpg)
Normal Distribution
e.g. Find the 95th percentile for the standard normal rv Z
z .00 .01 .02 .03 .04 0.5 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · · · · ·η(95) = 1.645, a linear interpolation of 1.64 and 1.65.
Remark: If p does not appear in the table, we can either use thenumber closest to it, or use the linear interpolation of the closesttwo.
![Page 19: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/19.jpg)
Normal Distribution
e.g. Find the 95th percentile for the standard normal rv Z
z .00 .01 .02 .03 .04 0.5 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · · · · ·η(95) = 1.645, a linear interpolation of 1.64 and 1.65.
Remark: If p does not appear in the table, we can either use thenumber closest to it, or use the linear interpolation of the closesttwo.
![Page 20: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/20.jpg)
Normal Distribution
e.g. Find the 95th percentile for the standard normal rv Z
z .00 .01 .02 .03 .04 0.5 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · · · · ·
η(95) = 1.645, a linear interpolation of 1.64 and 1.65.
Remark: If p does not appear in the table, we can either use thenumber closest to it, or use the linear interpolation of the closesttwo.
![Page 21: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/21.jpg)
Normal Distribution
e.g. Find the 95th percentile for the standard normal rv Z
z .00 .01 .02 .03 .04 0.5 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · · · · ·η(95) = 1.645, a linear interpolation of 1.64 and 1.65.
Remark: If p does not appear in the table, we can either use thenumber closest to it, or use the linear interpolation of the closesttwo.
![Page 22: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/22.jpg)
Normal Distribution
e.g. Find the 95th percentile for the standard normal rv Z
z .00 .01 .02 .03 .04 0.5 · · · .09
· · · · · · · · · · · · · · · · · · · · · · · · · · ·1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 · · · 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 · · · 0.9633· · · · · · · · · · · · · · · · · · · · · · · · · · ·η(95) = 1.645, a linear interpolation of 1.64 and 1.65.
Remark: If p does not appear in the table, we can either use thenumber closest to it, or use the linear interpolation of the closesttwo.
![Page 23: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/23.jpg)
Normal Distribution
In statistical inference, the percentiles corresponding to right smalltails are heavily used.
Notationzα will denote the value on the z axis for which α of the areaunder the z curve lies to the right of zα.
zα
![Page 24: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/24.jpg)
Normal Distribution
In statistical inference, the percentiles corresponding to right smalltails are heavily used.
Notationzα will denote the value on the z axis for which α of the areaunder the z curve lies to the right of zα.
zα
![Page 25: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/25.jpg)
Normal Distribution
In statistical inference, the percentiles corresponding to right smalltails are heavily used.
Notationzα will denote the value on the z axis for which α of the areaunder the z curve lies to the right of zα.
zα
![Page 26: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/26.jpg)
Normal Distribution
Remark:1. zα is the 100(1− α)th percentile of the standard normaldistribution.2. By symmetry the area under the standard normal curve to theleft of −zα is also α.3. The zαs are usually referred to as z critical values.
Percentile 90 95 97.5 · · · 99.95α (tail area) 0.1 0.05 0.025 · · · 0.0005
zα 1.28 1.645 1.96 · · · 3.27
![Page 27: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/27.jpg)
Normal Distribution
Remark:1. zα is the 100(1− α)th percentile of the standard normaldistribution.
2. By symmetry the area under the standard normal curve to theleft of −zα is also α.3. The zαs are usually referred to as z critical values.
Percentile 90 95 97.5 · · · 99.95α (tail area) 0.1 0.05 0.025 · · · 0.0005
zα 1.28 1.645 1.96 · · · 3.27
![Page 28: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/28.jpg)
Normal Distribution
Remark:1. zα is the 100(1− α)th percentile of the standard normaldistribution.2. By symmetry the area under the standard normal curve to theleft of −zα is also α.
3. The zαs are usually referred to as z critical values.
Percentile 90 95 97.5 · · · 99.95α (tail area) 0.1 0.05 0.025 · · · 0.0005
zα 1.28 1.645 1.96 · · · 3.27
![Page 29: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/29.jpg)
Normal Distribution
Remark:1. zα is the 100(1− α)th percentile of the standard normaldistribution.2. By symmetry the area under the standard normal curve to theleft of −zα is also α.3. The zαs are usually referred to as z critical values.
Percentile 90 95 97.5 · · · 99.95α (tail area) 0.1 0.05 0.025 · · · 0.0005
zα 1.28 1.645 1.96 · · · 3.27
![Page 30: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/30.jpg)
Normal Distribution
Proposition
If X has a normal distribution with mean µ and stadard deviationσ, then
Z =X − µσ
has a standard normal distribution. Thus
P(a ≤ X ≤ b) = P(a− µσ≤ Z ≤ b − µ
σ)
= Φ(b − µσ
)− Φ(a− µσ
)
P(X ≤ a) = Φ(a− µσ
) P(X ≥ b) = 1− Φ(b − µσ
)
![Page 31: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/31.jpg)
Normal Distribution
Proposition
If X has a normal distribution with mean µ and stadard deviationσ, then
Z =X − µσ
has a standard normal distribution. Thus
P(a ≤ X ≤ b) = P(a− µσ≤ Z ≤ b − µ
σ)
= Φ(b − µσ
)− Φ(a− µσ
)
P(X ≤ a) = Φ(a− µσ
) P(X ≥ b) = 1− Φ(b − µσ
)
![Page 32: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/32.jpg)
Normal Distribution
Example (Problem 38):There are two machines available for cutting corks intended for usein wine bottles. The first produces corks with diameters that arenormally distributed with mean 3cm and standard deviation 0.1cm.The second produces corks with diameters that have a normaldistribution with mean 3.04cm and standard deviation 0.02cm.Acceptable corks have diameters between 2.9cm and 3.1cm.Which machine is more likely to produce an acceptable cork?
P(2.9 ≤ X1 ≤ 3.1) = P(2.9− 3
0.1≤ Z ≤ 3.1− 3
0.1)
= P(−1 ≤ Z ≤ 1) = 0.8413− 0.1587 = 0.6826
P(2.9 ≤ X2 ≤ 3.1) = P(2.9− 3.04
0.02≤ Z ≤ 3.1− 3.04
0.02)
= P(−7 ≤ Z ≤ 3) = 0.9987− 0 = 0.9987
![Page 33: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/33.jpg)
Normal DistributionExample (Problem 38):There are two machines available for cutting corks intended for usein wine bottles. The first produces corks with diameters that arenormally distributed with mean 3cm and standard deviation 0.1cm.The second produces corks with diameters that have a normaldistribution with mean 3.04cm and standard deviation 0.02cm.Acceptable corks have diameters between 2.9cm and 3.1cm.Which machine is more likely to produce an acceptable cork?
P(2.9 ≤ X1 ≤ 3.1) = P(2.9− 3
0.1≤ Z ≤ 3.1− 3
0.1)
= P(−1 ≤ Z ≤ 1) = 0.8413− 0.1587 = 0.6826
P(2.9 ≤ X2 ≤ 3.1) = P(2.9− 3.04
0.02≤ Z ≤ 3.1− 3.04
0.02)
= P(−7 ≤ Z ≤ 3) = 0.9987− 0 = 0.9987
![Page 34: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/34.jpg)
Normal DistributionExample (Problem 38):There are two machines available for cutting corks intended for usein wine bottles. The first produces corks with diameters that arenormally distributed with mean 3cm and standard deviation 0.1cm.The second produces corks with diameters that have a normaldistribution with mean 3.04cm and standard deviation 0.02cm.Acceptable corks have diameters between 2.9cm and 3.1cm.Which machine is more likely to produce an acceptable cork?
P(2.9 ≤ X1 ≤ 3.1) = P(2.9− 3
0.1≤ Z ≤ 3.1− 3
0.1)
= P(−1 ≤ Z ≤ 1) = 0.8413− 0.1587 = 0.6826
P(2.9 ≤ X2 ≤ 3.1) = P(2.9− 3.04
0.02≤ Z ≤ 3.1− 3.04
0.02)
= P(−7 ≤ Z ≤ 3) = 0.9987− 0 = 0.9987
![Page 35: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/35.jpg)
Normal Distribution
Example (Problem 44):If bolt thread length is normally distributed, what is the probabilitythat the thread length of a randomly selected bolt is (a)within 1.5SDs of its mean value? (b)between 1 and 2 SDs from its meanvalue?
P(µ− 1.5σ ≤ X1 ≤ µ+ 1.5σ) = P(µ− 1.5σ − µ
σ≤ Z ≤ µ+ 1.5σ − µ
σ)
= P(−1.5 ≤ Z ≤ 1.5)
= 0.9332− 0.0668 = 0.8664
2 · P(µ+ σ ≤ X1 ≤ µ+ 2σ) = 2P(µ+ σ − µ
σ≤ Z ≤ µ+ 2σ − µ
σ)
= 2P(1 ≤ Z ≤ 2)
= 2(0.9772− 0.8413) = 0.0.2718
![Page 36: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/36.jpg)
Normal DistributionExample (Problem 44):If bolt thread length is normally distributed, what is the probabilitythat the thread length of a randomly selected bolt is (a)within 1.5SDs of its mean value? (b)between 1 and 2 SDs from its meanvalue?
P(µ− 1.5σ ≤ X1 ≤ µ+ 1.5σ) = P(µ− 1.5σ − µ
σ≤ Z ≤ µ+ 1.5σ − µ
σ)
= P(−1.5 ≤ Z ≤ 1.5)
= 0.9332− 0.0668 = 0.8664
2 · P(µ+ σ ≤ X1 ≤ µ+ 2σ) = 2P(µ+ σ − µ
σ≤ Z ≤ µ+ 2σ − µ
σ)
= 2P(1 ≤ Z ≤ 2)
= 2(0.9772− 0.8413) = 0.0.2718
![Page 37: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/37.jpg)
Normal DistributionExample (Problem 44):If bolt thread length is normally distributed, what is the probabilitythat the thread length of a randomly selected bolt is (a)within 1.5SDs of its mean value? (b)between 1 and 2 SDs from its meanvalue?
P(µ− 1.5σ ≤ X1 ≤ µ+ 1.5σ) = P(µ− 1.5σ − µ
σ≤ Z ≤ µ+ 1.5σ − µ
σ)
= P(−1.5 ≤ Z ≤ 1.5)
= 0.9332− 0.0668 = 0.8664
2 · P(µ+ σ ≤ X1 ≤ µ+ 2σ) = 2P(µ+ σ − µ
σ≤ Z ≤ µ+ 2σ − µ
σ)
= 2P(1 ≤ Z ≤ 2)
= 2(0.9772− 0.8413) = 0.0.2718
![Page 38: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/38.jpg)
Normal DistributionExample (Problem 44):If bolt thread length is normally distributed, what is the probabilitythat the thread length of a randomly selected bolt is (a)within 1.5SDs of its mean value? (b)between 1 and 2 SDs from its meanvalue?
P(µ− 1.5σ ≤ X1 ≤ µ+ 1.5σ) = P(µ− 1.5σ − µ
σ≤ Z ≤ µ+ 1.5σ − µ
σ)
= P(−1.5 ≤ Z ≤ 1.5)
= 0.9332− 0.0668 = 0.8664
2 · P(µ+ σ ≤ X1 ≤ µ+ 2σ) = 2P(µ+ σ − µ
σ≤ Z ≤ µ+ 2σ − µ
σ)
= 2P(1 ≤ Z ≤ 2)
= 2(0.9772− 0.8413) = 0.0.2718
![Page 39: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/39.jpg)
Normal Distribution
Proposition
{(100p)th percentile for N(µ, σ2)} =µ + {(100p)th percentile for N(0, 1)} · σ
Example (Problem 39)The width of a line etched on an integrated circuit chip is normallydistributed with mean 3.000 µm and standard deviation 0.140.What width value separates the widest 10% of all such lines fromthe other 90%?
ηN(3,0.1402)(90) = 3.0+0.140·ηN(0,1)(90) = 3.0+0.140·1.28 = 3.1792
![Page 40: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/40.jpg)
Normal Distribution
Proposition
{(100p)th percentile for N(µ, σ2)} =µ + {(100p)th percentile for N(0, 1)} · σ
Example (Problem 39)The width of a line etched on an integrated circuit chip is normallydistributed with mean 3.000 µm and standard deviation 0.140.What width value separates the widest 10% of all such lines fromthe other 90%?
ηN(3,0.1402)(90) = 3.0+0.140·ηN(0,1)(90) = 3.0+0.140·1.28 = 3.1792
![Page 41: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/41.jpg)
Normal Distribution
Proposition
{(100p)th percentile for N(µ, σ2)} =µ + {(100p)th percentile for N(0, 1)} · σ
Example (Problem 39)The width of a line etched on an integrated circuit chip is normallydistributed with mean 3.000 µm and standard deviation 0.140.What width value separates the widest 10% of all such lines fromthe other 90%?
ηN(3,0.1402)(90) = 3.0+0.140·ηN(0,1)(90) = 3.0+0.140·1.28 = 3.1792
![Page 42: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/42.jpg)
Normal Distribution
Proposition
{(100p)th percentile for N(µ, σ2)} =µ + {(100p)th percentile for N(0, 1)} · σ
Example (Problem 39)The width of a line etched on an integrated circuit chip is normallydistributed with mean 3.000 µm and standard deviation 0.140.What width value separates the widest 10% of all such lines fromthe other 90%?
ηN(3,0.1402)(90) = 3.0+0.140·ηN(0,1)(90) = 3.0+0.140·1.28 = 3.1792
![Page 43: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/43.jpg)
Normal Distribution
Proposition
Let X be a binomial rv based on n trials with success probability p.Then if the binomial probability histogram is not too skewed, X hasapproximately a normal distribution with µ = np and σ =
√npq,
where q = 1− p. In particular, for x = a posible value of X ,
P(X ≤ x) = B(x ; n, p) ≈(
area under the normal curve
to the left of x+0.5
)= Φ(
x+0.5− np√
npq)
In practice, the approximation is adequate provided that bothnp ≥ 10 and nq ≥ 10, since there is then enough symmetry in theunderlying binomial distribution.
![Page 44: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/44.jpg)
Normal Distribution
Proposition
Let X be a binomial rv based on n trials with success probability p.Then if the binomial probability histogram is not too skewed, X hasapproximately a normal distribution with µ = np and σ =
√npq,
where q = 1− p. In particular, for x = a posible value of X ,
P(X ≤ x) = B(x ; n, p) ≈(
area under the normal curve
to the left of x+0.5
)= Φ(
x+0.5− np√
npq)
In practice, the approximation is adequate provided that bothnp ≥ 10 and nq ≥ 10, since there is then enough symmetry in theunderlying binomial distribution.
![Page 45: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/45.jpg)
Normal Distribution
A graphical explanation for
P(X ≤ x) = B(x ; n, p) ≈(
area under the normal curve
to the left of x+0.5
)= Φ(
x+0.5− np√
npq)
![Page 46: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/46.jpg)
Normal Distribution
A graphical explanation for
P(X ≤ x) = B(x ; n, p) ≈(
area under the normal curve
to the left of x+0.5
)= Φ(
x+0.5− np√
npq)
![Page 47: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/47.jpg)
Normal Distribution
A graphical explanation for
P(X ≤ x) = B(x ; n, p) ≈(
area under the normal curve
to the left of x+0.5
)= Φ(
x+0.5− np√
npq)
![Page 48: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/48.jpg)
Normal Distribution
Example (Problem 54)Suppose that 10% of all steel shafts produced by a certain processare nonconforming but can be reworked (rather than having to bescrapped). Consider a random sample of 200 shafts, and let Xdenote the number among these that are nonconforming and canbe reworked. What is the (approximate) probability that X isbetween 15 and 25 (inclusive)?In this problem n = 200, p = 0.1 and q = 1− p = 0.9. Thusnp = 20 > 10 and nq = 180 > 10
P(15 ≤ X ≤ 25) = Bin(25; 200, 0.1)− Bin(14; 200, 0.1)
≈ Φ(25 + 0.5− 20√
200 · 0.1 · 0.9)− Φ(
15 + 0.5− 20√200 · 0.1 · 0.9
)
= Φ(0.3056)− Φ(−0.2500)
= 0.6217− 0.4013
= 0.2204
![Page 49: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/49.jpg)
Normal DistributionExample (Problem 54)Suppose that 10% of all steel shafts produced by a certain processare nonconforming but can be reworked (rather than having to bescrapped). Consider a random sample of 200 shafts, and let Xdenote the number among these that are nonconforming and canbe reworked. What is the (approximate) probability that X isbetween 15 and 25 (inclusive)?
In this problem n = 200, p = 0.1 and q = 1− p = 0.9. Thusnp = 20 > 10 and nq = 180 > 10
P(15 ≤ X ≤ 25) = Bin(25; 200, 0.1)− Bin(14; 200, 0.1)
≈ Φ(25 + 0.5− 20√
200 · 0.1 · 0.9)− Φ(
15 + 0.5− 20√200 · 0.1 · 0.9
)
= Φ(0.3056)− Φ(−0.2500)
= 0.6217− 0.4013
= 0.2204
![Page 50: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/50.jpg)
Normal DistributionExample (Problem 54)Suppose that 10% of all steel shafts produced by a certain processare nonconforming but can be reworked (rather than having to bescrapped). Consider a random sample of 200 shafts, and let Xdenote the number among these that are nonconforming and canbe reworked. What is the (approximate) probability that X isbetween 15 and 25 (inclusive)?In this problem n = 200, p = 0.1 and q = 1− p = 0.9. Thusnp = 20 > 10 and nq = 180 > 10
P(15 ≤ X ≤ 25) = Bin(25; 200, 0.1)− Bin(14; 200, 0.1)
≈ Φ(25 + 0.5− 20√
200 · 0.1 · 0.9)− Φ(
15 + 0.5− 20√200 · 0.1 · 0.9
)
= Φ(0.3056)− Φ(−0.2500)
= 0.6217− 0.4013
= 0.2204
![Page 51: Normal Distribution - Home - Math - The University of Utahlzhang/teaching/3070spring2009... · 2009-06-11 · Normal Distribution Table A.3 Standard Normal Curve Areas z .00 .01 .02](https://reader033.vdocument.in/reader033/viewer/2022060500/5f1a11f740976e123e5d946f/html5/thumbnails/51.jpg)
Normal DistributionExample (Problem 54)Suppose that 10% of all steel shafts produced by a certain processare nonconforming but can be reworked (rather than having to bescrapped). Consider a random sample of 200 shafts, and let Xdenote the number among these that are nonconforming and canbe reworked. What is the (approximate) probability that X isbetween 15 and 25 (inclusive)?In this problem n = 200, p = 0.1 and q = 1− p = 0.9. Thusnp = 20 > 10 and nq = 180 > 10
P(15 ≤ X ≤ 25) = Bin(25; 200, 0.1)− Bin(14; 200, 0.1)
≈ Φ(25 + 0.5− 20√
200 · 0.1 · 0.9)− Φ(
15 + 0.5− 20√200 · 0.1 · 0.9
)
= Φ(0.3056)− Φ(−0.2500)
= 0.6217− 0.4013
= 0.2204