5장 세상의 많은 현상들은 정규분포를 따른다. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

Presentation on theme: "5장 세상의 많은 현상들은 정규분포를 따른다. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분"— Presentation transcript:

1 5장 세상의 많은 현상들은 정규분포를 따른다. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분
5장 세상의 많은 현상들은 정규분포를 따른다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

2 학습할 내용 정규분포와 표준화란? 표준정규분포와 확률계산 표를 이용하여 표준정규분포 확률의 찾아라
표준정규분포로부터 정규분포의 확률을 계산 엑셀을 이용한 정규분포와 표준정규분포의 확률계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

3 정규분포는 아래 그림과 같이 종(bell)모양을 하고 있다.
정규분포(normal distribution)는 연속확률분포 중에서 가장 많이 활용되는 분포이며, 자연과학 현상이나 사회과학 현상에서 접하는 여러 데이터의 분포가 정규분포와 유사한 분포를 나타낸다. 정규분포는 아래 그림과 같이 종(bell)모양을 하고 있다. 총콜레스트롤 분포 (출처 : 국민건강보험공단, 2011년) © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

4 정규분포 연속확률변수 X의 확률밀도함수 f(x)가 다음과 같이 정해지는 확률분포를 정규분포라 하고, 정규분포는 평균 μ와 표준편차 σ에 의해 결정된다. 즉, 분포의 모양은 평균과 표준편차에 따라 달라진다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

5 정규분포의 특징 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

6 정규분포의 비교 평균 μ=5이고, 표준편차 σ=2인 정규분포의 확률밀도함수 f1(x)와 평균 μ=7이고, 표준편차 σ=1인 정규분포의 확률밀도함수 f2(x) 의 비교 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

7 5.1 정규분포의 표준화 정규분포에서 확률을 계산하기 위해서는 적분을 해야 하는데 정규분포의 적분은 계산이 어려운데다 평균과 표준편차에 따라 분포의 위치와 모양이 달라지는 분포이며, 평균과 표준편차가 다른 정규분포를 쉽게 비교하기위해 표준화를 사용한다. 평균과 표준편차가 다른 모든 정규분포의 확률변수X에 대해 표준화를 하면 하나의 동일한 분포가 되는데 이 분포는 평균 μ=0, 표준편차 σ=1 인 정규분포가 되며, 이를 표준정규분포(standard normal distribution)라 한다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

8 정규분포의 표준화와 표준정규분포 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

9 [참고] 표준화와 빅맥지수 맥도널드사의 햄버거 제품인 빅맥가격에 기초하여 120개 나라의 물가수준과 통화가치를 비교하는 지수로서 영국의 경제전문지 이코노미스트(The Economist)가 분기마다 한 번씩 발표. 맥도널드사의 햄버거는 세계적으로 품질·크기·재료가 표준화되어 있어 어느 곳에서나 값이 거의 일정한 빅맥가격을 기준으로 비교할 경우 각국의 통화가치가 어느 정도인지 알 수 있다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

10 Z 점수 평균 μ와 표준편차 σ가 다른 정규분포의 확률변수 X에 대해 표준화를 하면 이 분포는 평균 μ=0, 표준편차 σ=1인 표준정규분포 N(0,1)가 된다. 이때 표준화 식 을 통해 얻어진 값을 Z점수(Z score)라 한다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

11 [예제 5-1] Z 점수의 계산 풀이 평균 μ=150 이고, 표준편차 σ=10 인 정규분포 N(150, 102)에 대해
X=170에 대한 Z 점수를 계산하시오. 표준화 식 을 이용하면 이므로 점수는 2. 풀이 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

12 Z 점수의 의미 Z 점수는 표준화 식에서 표준편차(σ)로 나눈 값이므로 점수의 의미는 확률변수 가 평균을 중심으로 몇 배의 표준편차(σ) 만큼 떨어져 있는가를 나타낸다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

13 [예제 5-2] 어느 과목을 더 잘 본 것인가? 어느 학생이 수학은 80점, 영어는 85점을 받았다고 할 때 어느 과목을 더 잘했다고 볼 수 있는가? 만약 학교전체의 수학점수 평균과 영어점수 평균이 같다고 한다면 점수만으로 비교하여 영어 과목을 더 잘 보았다고 할 수 있다. 그러나 수학점수와 영어점수의 평균과 표준편차가 다르다면 어떻게 비교해야 할까? 수학점수와 영어점수는 정규분포를 따른다고 가정. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

14 [예제 5-2] 어느 과목을 더 잘 본 것인가? 표준화 분포상에서 비교
어느 학생이 수학은 80점, 영어는 85점을 받았다고 할 때 어느 과목을 더 잘했다고 볼 수 있는가? 수학점수의 평균과 표준편차는 60과 10이고, 영어점수의 평균과 표준편차는 80과 5라 할 때 표준화 분포상에서 비교 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

15 [참고] T 점수(T score)란? T=10Z+50
원 점수에 대한 표준점수로 Z점수와 T점수를 많이 사용한다. Z점수는 앞서와 같이 음수 값을 갖고 소수점을 포함하므로 이러한 단점을 없애기 위해 Z점수에 대해 다음과 같은 변환을 한 표준점수이다. T=10Z+50 Z점수는 N(0, 1)의 정규분포를 따르므로 T=10Z+50에 의해 T점수는 평균이 50, 표준편차가 10인 정규분포를 따르게 된다. Z점수가 -3≤ Z ≤3 에 해당한다면 T점수는 20≤ T ≤80의 범위를 갖게 되므로 일반적인 100점 만점의 단위로 사용할 수 있으며, 여러 다른 집단 간의 점수를 비교하거나 통합하는 경우에 사용된다. [예제 5-2]에 대해 Z 점수 1(영어)과 2(수학)를 T 점수로 변환하면 각각 60과 70이 된다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

16 5.2 표준정규분포에서 확률계산 표준정규분포는 평균(Z=0)을 중심으로 좌우 대칭이므로 확률을 계산할 때 [표 5-1]을 기초로 확률을 계산. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

17 5.2 표준정규분포에서 확률계산 표준정규분포는 평균(Z=0)을 중심으로 좌우 대칭이므로 확률을 계산할 때 [표 5-1]을 기초로 확률을 계산. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

18 [예제 5-3] 표준정규분포에서 P(-2≤Z≤1)의 확률계산방법
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

19 [예제 5-4] 표준정규분포에서 P(Z≤-1)의 확률계산방법
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

20 [예제 5-5] 표준정규분포에서 P(-2≤Z≤-2)의 확률계산방법
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

21 5.3 표를 이용하여 표준정규분포 확률을 찾는 방법 부록의 표준정규분포표에는 Z가 0 이상의 값을 갖는 경우의 확률(면적)만
5.3 표를 이용하여 표준정규분포 확률을 찾는 방법 부록의 표준정규분포표에는 Z가 0 이상의 값을 갖는 경우의 확률(면적)만 표시되어 있다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

22 [예제 5-6] P(0≤Z≤1.28)의 확률계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

23 [예제 5-7] P(-1.5≤Z≤0.5)의 확률계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

24 [예제 5-7] P(-1.5≤Z≤0.5)의 확률계산 계산방법
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

25 [예제 5-7] P(-1.5≤Z≤0.5)의 확률계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

26 [예제 5-7] P(-1.5≤Z≤0.5)의 확률계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

27 [예제 5-7] P(-1.5≤Z≤0.5)의 확률계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

28 [예제 5-8] P(Z≤Za)=0.9를 만족하는 Za는?
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

29 [예제 5-8] P(Z≤Za)=0.9를 만족하는 Za는?
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

30 [예제 5-8] P(Z≤Za)=0.9를 만족하는 Za는?
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

31 [예제 5-9] P(Z≥Za)=0.1을 만족하는 Za는?
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

32 [실습문제] 표준정규분포에 대해서 다음의 확률을 계산하시오. P(Z≤-2)의 확률은?
P(Z≤-2)=P(Z≥2)이고, P(Z≥2)=0.5-P(0≤Z≤2) = =0.0228 P(-1.5≤Z≤1.5)=P(-1.5≤Z≤0)+P(0≤Z≤1.5)이고, P(-1.5≤Z≤0)=P(0≤Z≤1.5)가 되므로 P(-1.5≤Z≤1.5)=2×P(0≤Z≤1.5) =2×0.4332=0.8664 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

33 [실습문제] 표준정규분포에 대해서 다음의 확률을 계산하시오. 3) P(Z≤0.5)의 확률은?
P(Z≤0.5)=P(Z≤0)+P(0≤Z≤0.5) P(Z≤0.5)= P(Z≤0) + P(0≤Z≤0.5) = = © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

34 [실습문제] 표준정규분포에 대해서 다음의 확률을 계산하시오. 4) P(0≤Z≤Za)=0.45를 만족하는 Za는?
1.64 또는 1.65가 된다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

35 [실습문제] 표준정규분포에 대해서 다음의 확률을 계산하시오. 5) P(Z≤Za)=0.05를 만족하는 Za는?
P(Za≤Z≤0)=P(0≤Z≤Zb)=0.45가 되고 P(0≤Z≤Zb)=0.45를 만족하는 Zb를 [실습 1]의 (4)에와 같이 1.64 또는 1.65가 된다. 그러나 문제에서 요구하는 Za는 0보다 작은 음수이므로 -1.64 또는 -1.65가 된다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

36 5.4 표준정규분포를 이용한 정규분포의 확률계산 정규분포에 대해 표준화를 하면 평균이 0, 표준편차가 1인 정규분포가 되고, 이를 표준정규분포라 한다. 현실적인 문제에서 정규분포에 대한 확률계산 문제는 다음과 같은 단계로 진행하면서 확률을 계산. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

37 [예제 5-10] 임의로 선택한 학생의 월 생활비가 400,000원 이상일 확률 계산
[예제 5-10] 임의로 선택한 학생의 월 생활비가 400,000원 이상일 확률 계산 경영학과 학생들의 월 생활비는 평균이 350,000원, 표준편차가 50,000원인 정규분포를 따른다고 한다. 이 가정 하에 임의로 선택한 학생의 월 생활비가 400,000원 이상일 확률을 계산하시오. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

38 [단계 1] 문제에서 계산하려는 확률(면적)을 정규분포 상에 표시
[단계 1] 문제에서 계산하려는 확률(면적)을 정규분포 상에 표시 경영학과 학생들의 월 생활비는 평균이 350,000원, 표준편차가 50,000원인 정규분포를 따른다고 한다. 이 가정 하에 임의로 선택한 학생의 월 생활비가 400,000원 이상일 확률을 계산하시오. 다음 그림과 같이 평균 인 정규분포를 그리고, 계산하고자 하는 확률은 P(X≥400000)이므로 구분하여 나타낸다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

39 [단계 2] 표준화를 통해 계산하려는 확률(면적)을 표준정규분포 상에 표시
[단계 2] 표준화를 통해 계산하려는 확률(면적)을 표준정규분포 상에 표시 평균인 과 계산하려는 기준 값 에 대해 표준화를 하면 다음과 같이 0과 1로 변환된다. 이 값을 표준정규분포 상에서 그림으로 나타내고 정규분포와 비교하면 다음과 같다. 가정에서 평균은 35000, 표준편차는 50000이므로 표준화 Z는 다음과 같이 계산. 확률 계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

40 [예제 5-11] 임의로 선택한 학생의 월 생활비가 250,000원 이상 400,000원 이하가 될 확률 계산
[예제 5-10]의 가정(평균이 350,000원, 표준편차가 50,000원인 정규분포)하 에 임의로 선택한 학생의 월 생활비가 250,000원~400,000원 사이에 있을 확률을 계산하시오. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

41 [단계 2] 표준화를 통해 계산하려는 확률(면적)을 표준정규분포 상에 표시
[예제 5-10]의 가정(평균이 350,000원, 표준편차가 50,000원인 정규분포)하 에 임의로 선택한 학생의 월 생활비가 250,000원~400,000원 사이에 있을 확률을 계산하시오. 다음 그림과 같이 평균이 인 정규분포를 그리고, 계산하고자 하는 확률은 P(250000≤X≤400000)이므로 구분하여 나타낸다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

42 [단계 2] 표준화를 통해 계산하려는 확률(면적)을 표준정규분포 상에 표시
확률 계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

43 [실습 문제 6] 정규분포와 표준정규분포 상에 확률을 표시
[예제 5-10]의 가정 (평균이 350,000원, 표준편차가 50,000원인 정규분포) 하에 임의로 선택한 학생의 월 생활비가 250,000원보다 적을 확률을 정규분포와 표준정규분포 상에 나타내고 계산하시오. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

44 [단계 1] 문제에서 계산하려는 확률(면적)을 정규분포 상에 표시
[예제 5-10]의 가정 (평균이 350,000원, 표준편차가 50,000원인 정규분포) 하에 임의로 선택한 학생의 월 생활비가 250,000원보다 적을 확률을 정규분포와 표준정규분포 상에 나타내고 계산하시오. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

45 [단계 2] 표준화를 통해 계산하려는 확률(면적)을 표준정규분포 상에 표시
확률 계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

46 [예제 5-12] 확률이 주어진 상태에서 확률변수 X의 값을 찾는 방법
경영통계학을 수강한 100명의 성적은 평균이 75이고, 표준편차가 10인 정규분포를 따른다고 할 때 어느 한 학생이 상위 10%안에 들려면 최소 몇 점을 받아야 하는가? © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

47 [단계 1] 문제에서 계산하려는 확률(면적)을 정규분포 상에 표시
상위 10% 안에 든다는 것은 아래의 정규분포 그림에서 오른쪽에 채워진 면적인 P(X≥xa)=0.1을 만족하는 것이며, 여기서 xa의 값을 찾는 것이다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

48 [단계 2] 표준 정규분포 상에 표시 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

49 확률 계산 표준정규분포표에서 P(0≤Z≤Za)=0.4을 만족하는 Za를 찾으면 가장 근사한 값은 P(0≤Z≤1.28)= 이므로 Za=1.28 Za는 xa를 표준화하여 얻어진 값이므로 표준화 식에 대입하면 의 등식이 성립하므로 X를 계산하면 88 따라서 어느 한 학생이 상위 10%안에 들려면 최소 88점을 받아야 하며 이는 을 의미. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

50 Z 점수의 범위와 확률 6.1 절에서 점수는 표준화 식에서 표준편차(σ)로 나눈 값이므로 점수의 의미는 확률변수 X가 평균을 중심으로 몇 배의 표준편차 만큼 떨어져 있는가를 나타낸다고 하였다. 점수의 범위에 따른 확률(면적)은 다음과 같다. 그림 표시 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

51 [실습 문제 7] 최소 매출실적 계산 어느 기업에서 여름기간의 매출실적이 상위 20%에 해당하는 직원에게 특별 휴가를 준다고 할 때 특별 휴가를 받기위한 최소 매출실적은 얼마인가? 직원들의 매출실적 평균은 1500만원이고 표준편차는 500만원인 정규분포를 따른다고 가정한다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

52 [단계 1] 상위 20%에 해당하는 매출 실적 X와 상위 면적 20%를 만족하는 Z값은?
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

53 [단계 2] 표준정규분포 상에서 za를 찾는다 표준정규분포에서 P(0≤Z≤Za)=0.3을 만족하는 Za를 찾는다.
따라서 다음의 등식이 성립한다. 위 식에 각각 대입하면 계산하면 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

54 5.5 엑셀을 이용한 정규분포와 표준정규분포의 확률계산
5.5 엑셀을 이용한 정규분포와 표준정규분포의 확률계산 엑셀에서 정규분포의 확률밀도함수나 확률은 함수 NORMDIST를 이용하여 계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

55 정규분포의 확률밀도함수 계산 평균이 0이고, 표준편차가 1인 정규분포에 대해 X=-1인 경우의 확률밀도함수 f(-1)의 값은?
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

56 정규분포의 누적확률 계산 평균이 5이고, 표준편차가 2인 정규분포에 대해
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

57 표준정규분포의 누적확률 계산 표준정규분포의 누적확률은 함수 NORMSDIST를 이용하여 계산
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

58 표준정규분포의 누적확률 계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

59 표준정규분포의 누적확률 계산 표준정규분포에 대해 의 계산
표준정규분포에 대해 의 계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

60 [실습 문제] 엑셀 함수를 이용하여 다음의 확률을 계산하시오.
[실습 문제] 엑셀 함수를 이용하여 다음의 확률을 계산하시오. 8) 평균 μ=172와 표준편차 σ=15인 정규분포에 대해 P(X>180)의 확률은? 9) 평균 =70과 표준편차 =10인 정규분포에 대해 P(X<55)의 확률은? © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

61 [실습 문제] 엑셀 함수를 이용하여 다음의 확률을 계산하시오.
[실습 문제] 엑셀 함수를 이용하여 다음의 확률을 계산하시오. 10) P(Z≤-1.5)의 확률은? 11) P(2≤Z≤2.5)의 확률은? 12) P(Z≤-0.5)의 확률은? 엑셀 파일 열기 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

62 [단원정리] 1/8 정규분포 연속확률분포 중에서 가장 많이 활용되는 분포이며, 종(bell)모양을 하고 있다. 정규분포는 평균 μ와 표준편차 σ에 의해 결정되며 분포의 모양은 평균 와 표준편차 에 따라 달라진다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

63 [단원정리] 2/8 정규분포의 특징 평균( )을 중심으로 좌우 대칭이며 종 모양(bell shape)의 형태이다.
정규분포의 모양은 평균 와 표준편차 에 의해 결정된다. 평균=중앙값=최빈값 확률밀도함수 f(x)의 곡선 아래 부분과 x축 사이의 면적은 항상 1이다. 확률밀도함수 f(x)는 x축에 무한대로 접근하므로 -∞ < x < ∞ © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

64 [단원정리] 3/8 정규분포의 표준화와 표준정규분포
정규분포를 따르는 확률변수 X에 대해 표준화 을 하면 확률변수 는 표준정규분포 N(0, 1)을 따른다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

65 [단원정리] 4/8 표준정규분포에서의 확률(면적)계산 part I
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

66 [단원정리] 5/8 표준정규분포에서의 확률(면적)계산 part II
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

67 [단원정리] 6/8 표준정규분포표를 이용한 확률(면적)계산
예를 들어 P(0≤Z≤1.5)의 확률계산에 있어서 가 1.5가 되는 위치는 표준정규분포표에서 먼저 표의 좌측 첫 번째 열에서 1.5를 찾고 오른쪽으로 수평선을 표시하고, 소수 이하 둘째 자리의 값이 0.00이므로 첫 번째 행에서 Z가 0.00이 되는 지점에서 수직선을 표시하여, 두 직선이 교차하는 지점이 되며 이때 찾으려는 확률(면적) P(0≤Z≤1.5)는 가 된다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

68 [단원정리] 7/8 현실적인 문제에서 정규분포에 대한 확률계산
정규분포에 대한 확률계산 문제는 우선 해당 확률변수 X가 정규분포를 따른다는 가정이 필요하며 다음과 같은 단계로 확률을 계산한다. © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분

69 [단원정리] 8/8 엑셀을 이용한 확률 계산 정규분포의 확률밀도함수나 확률은 함수 NORMDIST를 이용
표준정규분포의 누적확률은 함수 NORMSDIST를 이용하여 계산 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION. 2019년 7월 31일 오후 6시 46분2019년 7월 31일 오후 6시 46분