Graphing a Binomial Probability Distribution Histogram

Transcription

1 Chapter 6 8A: Using a Normal Distribution to Approximate a Binomial Probability Distribution Graphing a Binomial Probability Distribution Histogram Lower and Upper Class Boundaries are used to graph the rectangles that represent the discrete x values on an x axis that is continuous The Binomial distribution shown below has 5 discrete values for the discrete variable x. The variable x in the table is represented by discrete whole numbers that are 1 unit apart. In chapter two we used lower and upper class boundaries on the x axis to represent each rectangle on the x axis. The discrete x values of 1, 2, 3, 4 and 5 are not shown on the continuous x axis.! The rectangle that represents 1 in the table is labeled on the graph staring at.5 and ending at 1.5. The rectangle that represents 2 in the table is labeled on the graph staring at 1.5 and ending at 2.5. The rectangle that represents 3 in the table is labeled on the graph staring at 2.5 and ending at 3.5. The rectangle that represents 4 in the table is labeled on the graph staring at 3.5 and ending at 4.5. The rectangle that represents 5 in the table is labeled on the graph staring at 4.5 and ending at A Lecture! Page 1 of 8! 2018 Eitel

2 A Binomial Probability Distribution with 5 values for x In chapter 5 we found the value of P(x) 3 in the Binomial Probability Distribution above by adding the values P(3) + P(4) + P(5). That process worked if the number of rectangles was very small. What if the Binomial table had x values from 0 to 100 and we needed to find P(x) 30 Finding P(x) 30 would require the calculation of the sum of the all the values for P(x) from P(30) to P(100) or P(30) + P(31) + P(32) P(100). It would not be practical to find all the values for P(x) and then find their total. If a Binomial Probability Distribution has a large number of values it becomes desirable to look for a way to approximate the answer without computing each P(x) and adding all the values for P(x). 6 8A Lecture! Page 2 of 8! 2018 Eitel

3 Can we use the area under a Normal Curve of continuous x values to approximate the sum of the values of P(x)P(30) + P(31) + P(32) P(100) in the table? P(x) 30! P(x) 30 = P(30) + P(31) + P(32) P(99) + P(100)! the area under the curve to the right for a discrete number of P(x) values! of 30 for a continuous normal curve! = Can we use a Normal Distribution of continuous x values to approximate a Binomial Probability Distribution? Yes we can, under some conditions and with an adjustment to the value of x. 6 8A Lecture! Page 3 of 8! 2018 Eitel

4 Continuity Corrections There is one step that increases that accuracy of the approximation. It involves the way in which we use the lower and upper class boundaries to label the rectangles on the x axis. To find the value of P(x) 3 in a binomial distribution we add the values for P(3) + P(4) + P(5). It would seem that we would then find P(x) 3 in a normal distribution by finding the area under the curve to the right of 3 but that is not true. We must adjust the value of 3 to reflect the class boundaries we use when we graph the rectangles on the x axis. We call this the Correction for Continuity. It can be a challenge to understand this correction so read the next pages carefully. If P(x a) change to P(x a.5) Binomial Probability P(x 3)! Normal probability P(x > 2.5)!! For the Binomial Probability P(x 3) means to start at the rectangle for x = 3 and add all the rectangles to the right. The rectangle for x = 3 starts at 2.5 and we want to add it and the ones to the right of it. For the normal curve we need to start at 2.5 on x axis and go to the right x > A Lecture! Page 4 of 8! 2018 Eitel

5 If P(x > a) change to P(x > a +.5) Binomial Probability P(x > 3)! Normal probability P(x > 3.5)!!!! For the Binomial Probability P(x > 3) means to start at the rectangle for x = 4 and add all the rectangles to the right. The rectangle for x = 4 starts at 3.5 and we want to add it and the ones to the right of it. For the normal curve we need to start at 3.5 on x axis and go to the right x > 3.5 If P(x a) change to P(x a +.5) Binomial Probability P(x 3)! Normal probability P(x < 3.5)!!! For the Binomial Probability P(x 3) means to start at the rectangle for x = 3 and add all the rectangles to the left. The rectangle for x = 3 starts at 3.5 and we want to add it and the ones to the left of it. For the normal curve we need to start at 3.5 on x axis and go to the left x < A Lecture! Page 5 of 8! 2018 Eitel

6 If P(x < a) change to P(x < a.5) Binomial Probability P(x < 3)! Normal probability P(x < 2.5)!!! For the Binomial Probability P(x < 3) means to start at the rectangle for x = 2 and add all the rectangles to the left. The rectangle for x = 2 starts at 2.5 and we want to add it and the ones to the left of it. For the normal curve we need to start at 2.5 on x axis and go to the left x < A Lecture! Page 6 of 8! 2018 Eitel

7 Continuity Corrections Example 1 P( x > 40 )! Example 2 P( x > 40 ) P(x a) is adjusted to P(x a.5)! P(x > a) is adjusted to P(x a +.5) P(x 40) is adjusted to P(x > 39.5)! P(x > 40) is adjusted to P(x > 40.5)! A) at least 40! A) greater than 40 B) no less than 40! B) more than 40 C) greater than or equal to 40! Example 3 P( x < 40 )! Example 4 P( x < 40 ) If P(x a) is adjusted to P(x a +.5)! P(x < a) is adjusted to P(x < a.5) P(x 40) is adjusted to P( x < 40.5)! P(x < 40) is adjusted to P(x < 39.5)!! A) at most 40! A) fewer than 40 B) no more than 40! B) less than 40! C) less than or equal to 40! 6 8A Lecture! Page 7 of 8! 2018 Eitel

8 Using a Normal Distribution to Approximate a Binomial Distribution! 1. Check to be sure the binomial distribution is approximately normal. If n p 5 and n q 5 Then the Binomial Distribution can be considered approximatley normal 2. Find the mean and standard deviation for the normal distribution. Use the population mean µ = n p and Standard Deviation σ = n p q 3. Find the continuity correction for x Continuity Corrections If P(x a) is adjusted to P(x a.5)! If P(x a) is adjusted to P(x a +.5) If P(x > a) is adjusted to P(x > a +.5)! If P(x < a) is adjusted to P(x a.5)!! Write the probability question in algebraic terms using the corrected value for a. 4. Show the set up to convert the adjusted x value to z and then shade the area on the z graph that corresponds to that specified probability. 5. State the answer to the probability question. Round P(x) to 2 decimal places 6 8A Lecture! Page 8 of 8! 2018 Eitel