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STATISTICAL TOOLS FOR EVALUATION

THE NORMAL CURVE

AND PROBABILITY

THE NORMAL CURVE

• SMOOTH, BILATERALLY SYMMETRICAL CURVE CENTERED AROUND A POINT THAT IS SIMULTANEOUSLY THE MODE, MEDIAN, AND MEAN

• HENCE, THE CENTER POINT IS BOTH THE MOST FREQUENT SCORE AND THE SCORE BELOW AND ABOVE WHICH HALF THE SCORES FALL

THE NORMAL CURVE• THE NORMAL CURVE HAS

A MEAN = 0 AND A STANDARD DEVIATION = 1

• THE NORMAL CURVE IS THE GRAPH OF AN INFINITE NUMBER OF Z-SCORES

• TO USE THE NORMAL CURVE TO MAKE PROBABILITY STATEMENTS, THINK OF THE AREA UNDER THE CURVE AS 100 EQUAL PORTIONS

• 50 LIE ON EACH SIDE OF THE MEAN

THE NORMAL CURVE AND PROBABILITY

THE NORMAL CURVE AND PROBABILITY

• WHAT IS THE PROBABILITY OF A Z SCORE EQUAL TO OR GREATER THAN 0?

• P (Z > 0 ) = 50 / 100 = 1/2 OR 50%

THE NORMAL CURVE AND PROBABILITY

• WHAT PERCENTAGE OF THE AREA UNDER THE NORMAL CURVE LIES BETWEEN O (Z = 0) AND 1.36 (Z = 1.36)?

THE NORMAL CURVE AND PROBABILITY

THE NORMAL CURVE AND PROBABILITY

• WHAT PERCENTAGE OF THE AREA UNDER THE NORMAL CURVE LIES BETWEEN O (Z = 0) AND 1.36 (Z = 1.36)?

• USING TABLE 2.9, Z OF 0 = 0 AND Z OF 1.36 = 41.31

• P (Z BETWEEN 0 AND 1.36) = 41.31 / 100

• P = .4131 OR 41%

THE NORMAL CURVE AND PROBABILITY

• WHAT IS THE PROBABILITY THAT Z IS EQUAL TO OR GREATER THAN 1.03?

THE NORMAL CURVE AND PROBABILITY

THE NORMAL CURVE AND PROBABILITY

• WHAT IS THE PROBABILITY THAT Z IS EQUAL TO OR GREATER THAN 1.03?

• P (Z ≥ 0) = 50 / 100 0R .50

• P (Z BETWEEN 0 AND 1.03) = 34.85 / 100 = .3485

• P (Z > 1.03) = .50 - .3485 = .1515 = .15 OR 15%

THE NORMAL CURVE AND PROBABILITY

• A TEACHER ALWAYS ADMINISTERS 100-POINT TESTS AND ALWAYS GIVES A’S TO SCORES OF 93 AND ABOVE. ON THE LAST EXAM THE MEAN WAS 72 AND THE STANDARD DEVIATION WAS 9. ASSUMING THAT THE TEST SCORES WERE NORMALLY DISTRIBUTED, WHAT WAS THE PROBABILITY OF RECEIVING AN A ON THAT TEST?

• FIRST NEED TO CALCULATE THE Z-SCORE

• Z = (SCORE - MEAN) / STANDARD DEVIATION

• Z = (93 - 72) / 9 = 21 / 9 = 2.33

THE NORMAL CURVE AND PROBABILITY

THE NORMAL CURVE AND PROBABILITY

• A TEACHER ALWAYS ADMINISTERS 100-POINT TESTS AND ALWAYS GIVES A’S TO SCORES OF 93 AND ABOVE. ON THE LAST EXAM THE MEAN WAS 72 AND THE STANDARD DEVIATION WAS 9. ASSUMING THAT THE TEST SCORES WERE NORMALLY DISTRIBUTED, WHAT WAS THE PROBABILITY OF RECEIVING AN A ON THAT TEST?

• P (Z ≥ 0) = 50 / 100 0R .50

• P (Z BETWEEN 0 AND 2.33) = 49.01 / 100 = .4901

• P (Z > 2.33) = .50 - .4901 = .0099 = .01 OR 1%

THE NORMAL CURVE AND PROBABILITY

• TO DEVELOP SOME PERFORMANCE STANDARDS, A TEACHER DECIDES TO USE THE NORMAL CURVE TO DETERMINE THE TEST SCORE ABOVE WHICH 7% OF THE SCORES SHOULD FALL

THE NORMAL CURVE AND PROBABILITY

• TO DEVELOP SOME PERFORMANCE STANDARDS, A TEACHER DECIDES TO USE THE NORMAL CURVE TO DETERMINE THE TEST SCORE ABOVE WHICH 7% OF THE SCORES SHOULD FALL

• FIRST DETERMINE THE Z-SCORE ABOVE WHICH 7% OF THE AREA UNDER THE NORMAL CURVE FALLS

• P (Z > ?) = 7 / 100• 50% OF TEST SCORES LIE ABOVE THE MEAN WHERE

Z = 0• IF WE ARE LOOKING FOR THE TOP 7%, 43% OF THE

TEST SCORES LIE BETWEEN THE MEAN AND THE SCORE FOR WHICH THE TEACHER IS LOOKING

• USING TABLE 2.9, WHAT Z-SCORE CORRESPONDS CLOSEST TO 43%?

• Z-SCORE OF 1.48 CORRESPONDING TO .4306 LIES THE CLOSET TO .43

THE NORMAL CURVE AND PROBABILITY

THE NORMAL CURVE AND PROBABILITY

• THEN DETERMINE THE TEST SCORE (X) ABOVE WHICH 7% OF THE SCORES SHOULD FALL USING THE FOLLOWING FORMULA TO CALCULATE X:

X = MEAN + Z(STANDARD DEVIATION)

ASSUMING THE MEAN IS 31.25, STANDARD DEVIATION IS 5, AND THE CALCULATED Z-SCORE = 1.48:

X = 31.25 + 1.48(5) = 31.25 + 7.4 = 38.65

QUESTIONS OR COMMENTS??

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