[Chapter 6] Probability

   Probability

Probability is used to predict the type of samples that are likely to be obtained from a population.

Inferential statistics rely on this connection when they use sample data as the basis of making conclusion about population.

확률은 모집단에서 얻을 수 있는 표본 유형을 예측하는 데 사용됩니다.

추리 통계는 표본 데이터를 모집단에 대한 결론을 내리는 기초로 사용할 때 이러한 연결에 의존합니다.

• Probability for any specific outcome: a fraction of a proportion of all the possible outcome.

• 

   Probability and Normal Distribution

• Normal distribution: theoretical distribution defined by a mathematical equation and would never be exactly duplicated by empirical data
• 정규 분포: 수학적 방정식에 의해 정의된 이론적 분포이며 경험적 데이터에 의해 정확히 복제되지 않습니다.
• //실제로는 존재하지 않는다는 뜻
  
• the sampling distribution tends to be normal if the sample sizes are large
• 표본 크기가 크면 표본 분포가 정규화되는 경향이 있습니다. // cf. 큰 수의 법칙

 

• The total area under the normal curve is 100%
• 정규 곡선 아래의 면적= 100%
• Convert raw scores into z values
• Center: mean (µ), and one standard deviation away from the center covers 34.13% (34*2=68%) and two standard deviation covers 95% of all outcomes

The Normal Distributions (Z scores)

   The normal curve

• Statistical Model: all possible outcomes in a given experimental situation
  통계 모델: 주어진 실험 상황에서 가능한 모든 결과
  
• For any frequency distribution the percentage of scores in a particular region corresponds to the probability of selecting a score from that region
  어떠한 도수분포표에서든, 특정 면적의 점수 백분율은 해당 지역에서 점수를 선택할 확률에 해당합니다.

   Characteristics of the Normal Curve

• Bell shape: high in the middle and low at both ends or tails
• Middle of the curve: close to the mean of the distribution, have a greater probability of occurring then values further away
• Unimodal and symmetric
• Table of the Standard Normal Distribution
  • a table of areas for normal distribution
  • provides percent area between mean and a Z score
  • ex) when X=700 mean=500, SD=100,
    p(X<700) = p(0<z<+2.00) = 2.28%

   Probabilities and Proportions for Scores from a Normal Distribution

• Due to the shape and properties of the normal curve, we can figure out percentiles from Z-Scores—remember those SAT scores?

1. Given a raw score, find the corresponding frequency or percent frequency
   1. Draw a rough diagram and indicate the area
   2. Convert the raw score to a z score: z = (X – μ) / σ
   3. Enter the normal curve table(p. 647 Appendix B at the end of the textbook, or from the snowboard statistical tables file, Table A) and read out the percent area from the mean to this z value

Example 1

• What percent of high school seniors in the population can be expected to have SAT scores between 500 and 625?
• Mean Score = 500
• Z scores for SAT = (X – μ) / σ

Answer: 39.44%

 

How to solve the problem?

1. Draw a diagram and shade the area

2. Convert the raw score into a Z score

3. Find the percent area in page 483 of the text book

 

Example 2

What percent of the population can be expected to have scores between 370 and 500?

40.32%

 

Example 3

What percent of the population can be expected to have scores between 367 and 540?

56.36%

 

Example 4

What percent of the population can be expected to have scores between 633 and 700?

6.9%

 

Example 4

What percent of population can be expected to have scores above 725?

1.22%