Finding probabilities involving Z scores
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1 Section 6 3B: Finding probabilities involving Z scores The probability that a number in the z distribution is less than a given z value is the area under the z curve that is LEFT of the z value The area under the z curve that is to the left of a given Z value represents the probability of selecting one number from the z distribution and having that number be less than the Z value. Find P( z < 2.1 ) If we want to know the probability of selecting one number from the z distribution and having that number be less than the negative z value of 2.1 we need to find the area to the left of z = 2.1 The yellow area to the left of z = 2.1 represents P( z < 2.1) z = Find P( z < 2.7 ) If we want to know the probability of selecting one number from the z distribution and having that number be less than the positive Z value of 2.7 we need to find the area to the left of z = 2.7 The yellow area to the left of z = 2.7 represents P( z < 2.7) 0 z= 2.7 We use two different tables to help find the area to the left of a given z value or z score. The Negative Z Scores Table The Positive Z Scores Table is used to find the area that is is used to to find the area that is to the left of a negative z value to the left of a positive z value z = z= 2.7 Section 6 3B Lecture Page 1 of Eitel
2 The Negative Z Scores Table The Negative Z Scores Table is used to find the area that is to the LEFT of a negative z value. z = The vertical line in the middle of the graph divides the total area of 1 in half. The area to the left of this line is All the shaded areas to the left of a negative Z value will be less than.5000 The 2 decimal place Z score is the negative 1 decimal place z value from the left column (in red) with an additional decimal place from the row on top (in red) The 4 decimal place number in yellow at the intersection of the left column z value (in red) and top row z value (in red) is the area to the LEFT of that given negative z score The Z Table only gives areas to the left of a Z value. Negative Z Scores Standard Normal (Z) Distribution: Area to the LEFT of Z This is only a portion of the entire Negative z Score Table Section 6 3B Lecture Page 2 of Eitel
3 Example 1A Finding the area to the left of a negative Z score Find P( z < 1.86 ) The area under the z curve that is to the left of a given z value represents the probability of selecting one number from the z distribution and having that number be less than the z value. number be less than the z value of 1.86 we need to find the area to the LEFT of z = 1.86 The number at the intersection of the 1.8 row and the 0.06 column is Negative Z Scores This means that the area to the left of z = 1.86 is the area to the left of z = 1.86 is.0314 left tail area =.0314 z = negative value If the area to the left of z = 1.86 is.0314 then P( z < 1.86 ) = Section 6 3B Lecture Page 3 of Eitel
4 Example 1B Finding the area to the Right of a negative Z score Find P( z > 1.86 ) The area under the z curve that is to the right of a given z value represents the probability of selecting one number from the z distribution and having that number be more than the z value. number be more than the z value of 1.86 we need to find the area to the RIGHT of z = 1.86 The number at the intersection of the 1.8 row and the 0.06 Negative Z Scores column is This means that the area to the LEFT of z = 1.86 is The vertical line drawn at the value of z divides the area under the z curve into two areas. The yellow area is to the left of the z value and the white area is to the right of the z value. the area to the left of z = 1.86 is.0314 The total of the yellow and white areas is 1 the area to the right of z = 1.86 is =.9686 left tail area =.0314 right tail area =.9686 z = so If the area to the Left of Z = 1.86 is.0314 then the area to the Right of Z = 1.86 is =.9686 P( z > 1.86 ) = =.9686 Section 6 3B Lecture Page 4 of Eitel
5 Example 2A Find the area to the left of a negative Z score Find P( z < 1.02 ) The area under the z curve that is to the left of a given z value represents the probability of selecting one number from the z distribution and having that number be less than the z value. number be less than the z value of 1.02 we need to find the area to the LEFT of z = 1.02 The number at the intersection of the 1.0 row and the 0.02 column is Negative Z Scores This means that the area to the left of z = 1.02 is the area to the left of z = 1.02 is.1539 left tail area =.1539 z = 1.02 negative value 0 If the area to the left of z = 1.86 is.1539 then P( z < 1.02 ) = Section 6 3B Lecture Page 5 of Eitel
6 Example 2B Finding the area to the Right of a negative Z score Find P( z > 1.02 ) The area under the z curve that is to the right of a given z value represents the probability of selecting one number from the z distribution and having that number be more than the z value. number be more than the z value of 1.02 we need to find the area to the RIGHT of z = 1.02 the area to the left of z = 1.02 is.1539 the area to the right of z = 1.02 is =.8461 left tail area =.1539 right tail area =.8461 z = If the area to the Left of Z = 1.02 is then the area to the Right of Z = 1.02 is P( z > 1.02 ) = =.8461 Section 6 3B Lecture Page 6 of Eitel
7 Example 3 Find the area to the left of a negative Z score A Special Case for z = P( z < ) number be less than the z value of we need to find the area to the left of z = Two negative Z scores with 3 decimal places are used in this course. These two 3 decimal place values and the areas to the left of them are listed separately at the bottom of the table. The cells at the bottom of the negative z scores table show that the area to the left of Z = is Negative Z Scores Z scores of 3.5 or less use.0001 AREA Z Score AREA Z Score This means that the area to the LEFT of z = 1.86 is P( z < ) = Finding the area to the Right of a negative Z score P( z > ) number be greater than the z value of we need to find the area to the right of z = If the area to the Left of Z = 1.02 is then the area to the Right of Z = is =.9950 P( z > ) = =.9950 Section 6 3B Lecture Page 7 of Eitel
8 Example 4 Find the area to the left of a negative Z score A Special Case for z = P( z < ) number be less than the z value of we need to find the area to the left of z = The cells at the bottom of the negative z scores table show that the area to the left of Z = is Negative Z Scores Z scores of 3.5 or less use.0001 AREA Z Score AREA Z Score This means that the area to the left of z = is P( z < ) = Finding the area to the Right of a negative Z score P( z > ) number be greater than the z value of we need to find the area to the right of z = If the area to the Left of Z = is then the area to the Right of Z = is =.9500 P( z > ) =.9500 Section 6 3B Lecture Page 8 of Eitel
9 The Positive Z Scores Table The Positive Z Scores Table is used to find the area that is to the left of a positive z value. z = 2.1 The vertical line in the middle of the graph divides the total area of 1 in half. The area to the left of this line is All the shaded areas to the left of a positive z value will be more than.5000 The 2 decimal place z score is the positive 1 decimal place z value from the left column (in red) with an additional decimal place from the row on top (in red) The 4 decimal place number in yellow at the intersection of the left column z value (in red) and top row z value (in red) stands for the area to the LEFT of that given z score The Z Table only gives areas to the left of a Z value. Positive Z Scores Standard Normal (Z) Distribution: Area to the LEFT of Z This is only a portion of the entire Positive z Score table Section 6 3B Lecture Page 9 of Eitel
10 Example 5 Finding the area to the left of a positive Z score P( z < 1.41 ) number be less than the z value of 1.41 we need to find the area to the LEFT of z = 1.41 The number at the intersection of the 1.4 row and the 0.01 Positive Z Scores column is This means that the area to the left of z = 1.41 is the area to the left of z = 1.41 is.9207 left tail area =.1539left tail area = z =1.41 positive value P( z < 1.41 ) = Finding the area to the Right of a positive Z score P( z > 1.41 ) number be more than the z value of 1.41 we need to find the area to the RIGHT of z = 1.41 The total of the yellow and white areas is 1 so the right area (in white) = 1 the left area If the area to the Left of Z = 1.41 is then the area to the Right of Z = 1.41 is =.0793 P( z > 1.41 ) = =.0793 Example 6 Section 6 3B Lecture Page 10 of Eitel
11 Finding the area to the left of a positive Z score P( z < 2.33 ) number be less than the z value of 2.33 we need to find the area to the LEFT of z = 2.33 The number at the intersection of the 2.3 row and the 0.03 column is Positive Z Scores This means that the area to the left of z = 2.33 is the area to the left of z = 2.33 is.9901 left tail area =.1539left tail area = z = 2.33 positive value P( z < 2.33 )= Finding the area to the Right of a positive Z score P( z > 2.33 ) number be more than the z value of 2.33 we need to find the area to the RIGHT of z = 2.33 If the area to the Left of Z = 2.33 is.9901 then the area to the Right of Z = 2.33 is =.0099 P( z > 2.33 ) = =.0099 Section 6 3B Lecture Page 11 of Eitel
12 Example 7 Finding the area to the left of a positive Z score A Special Case for z = P( z < ) number be less than the z value of we need to find the area to the LEFT of z = The cells at the bottom of the positive z scores table show that the area to the left of Z = is Positive Z Scores Z scores of 3.5 or more use.9999 AREA Z Score AREA Z Score This means that the area to the left of z = is left tail area =.9500 right tail area = z= P( z < ) =.9500 Finding the area to the Right of a positive Z score P( z > ) If the area to the Left of Z = is.9500 then the area to the Right of Z = is =.0500 P( z > ) = =.0500 Section 6 3B Lecture Page 12 of Eitel
13 Finding the area in yellow between two Z scores P( z 1 < Z < z 2 ) z 1 z 2 z 1 subtract the blue area to the left of z 1 from the yellow area to the left of z 2 z 2 z 1 z 2 subtract the blue area to the left of z 1 from the yellow area to the left of z 2 z 1 z 2 The area in yellow between Two Z scores z 1 and z 2 P( z 1 < Z < z 2 ) = the Area to the left of z 2 the area to the left of z 1 Section 6 3B Lecture Page 13 of Eitel
14 Example 8 Find the area between z = 2.48 and z = 2.87 P( 2.48 < z < 2.87 ) number be BETWEEN the z value of 2.48 the z value of 2.87 we need to find the area BETWEEN z = 2.87 and z = 2.87 Positive Z Scores z = 2.48 z = Positive Z Scores If the area to the left of Z = 2.87 is and the area to the left of Z = 2.48 is then the area BETWEEN Z = 2.87 and Z = 2.48 is =.9913 P( 2.48 < z < 2.87 ) =.9913 Section 6 3B Lecture Page 14 of Eitel
15 Example 9 Find the area between Z = 1.75 and Z = 1.88 P( 1.75 < z < 1.88 ) number be BETWEEN the z value of 1.75 the z value of 1.88 we need to find the area BETWEEN z = 1.75 and z = 1.88 Positive Z Scores z = 1.75 z = Negative Z Scores If the area to the left of Z = 2.87 is and the area to the left of Z = 2.48 is then the area BETWEEN Z = 1.75 and Z = 1.88 is =.9298 P( 1.75 < z < 1.88 ) =.9298 Section 6 3B Lecture Page 15 of Eitel
16 Example 10 Find the area between Z = 2.51 and Z = P( 2.51 < z < number be BETWEEN the z value of 2.51 the z value of we need to find the area BETWEEN z = 2.51 and z = z = 2.51 z = Positive Z Scores the area to the left of Z = is.9950 Z scores of 3.5 or more use.9999 AREA Z Score AREA Z Score Negative Z Scores the area to the left of Z = 2.51 is If the area to the left of Z = is and the area to the left of Z = 2.51 is then the area BETWEEN Z = and Z = 2.51 is =.9890 P( 2.51 < z < ) =.9890 Section 6 3B Lecture Page 16 of Eitel
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