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Modular 12 Ch 7.2 Part II to 7.3 Ch 7.2 Part II Applications of the Normal Distribution Objective B : Finding the Z-score for a given probability Objective C : Probability under a Normal Distribution Objective D : Finding the Value of a Normal Random Variable Ch 7.3 Assessing Normality Ch 7.4 The Normal Approximation to the Binomial Probability Distribution (Skip) Objective A : Continuity Correction Objective B : A Normal Approximation to the Binomial Ch 7.2 Applications of the Normal Distribution Objective B : Finding the Z-score for a given probability Area 0 . 5 Area 0 . 5 Area 0 . 5 Example 1 : Find the Z-score such that the area under the standard normal curve to its left is 0.0418. 1 . 73 ) 0.0418 P ( Z _____ From Table V 0 . 03 0 . 0418 1 .7 Z ? 0 0 . 0418 Example 2 : Find the Z-score such that the area under the standard normal curve to its right is 0.18. From Table V 0 . 92 P ( Z ______ ) 0.18 0 . 02 1 0 . 18 0 . 82 0 . 18 0 .9 0 . 8186 0 . 8212 (Closer to 0.82) Z ? Example 3 : Find the Z-score such that separates the middle 70%. 1 . 04 Z _____ 1 . 04 ) 0.70 P ( _____ 70 % Z ? From Table V 0 . 04 Z ? 15 % 15 % or 0 . 15 or 0 . 15 Z 1 . 04 1 .0 Z 1 . 04 ( Due to symmetry ) 0 . 1515 0 .1 4 9 2 (Closer to 0.15) Ch 7.2 Part II Applications of the Normal Distribution Objective B : Finding the Z-score for a given probability Objective C : Probability under a Normal Distribution Objective D : Finding the Value of a Normal Random Variable Ch 7.3 Assessing Normality Ch 7.4 The Normal Approximation to the Binomial Probability Distribution Objective A : Continuity Correction Objective B : A Normal Approximation to the Binomial Ch 7.2 Applications of the Normal Distribution Objective C : Probability under a Normal Distribution Step 1 : Draw a normal curve and shade the desired area. Step 2 : Convert the values of X to Z – scores using Z X . Step 3 : Use Table V to find the area to the left of each Z – score found in Step 2. Step 4 : Adjust the area from Step 3 to answer the question if necessary. Example 1 : Assume that the random variable X is normally distributed with mean 50 and a standard deviation 7 . (Note: This is not a standard normal curve because 0 and 1 .) (a) P ( X 58 ) Z X 58 50 50 7 8 1 . 14 58 X 0 .8 7 2 9 7 P ( Z 1 . 14 ) From Table V 0 .8 7 2 9 0 1 . 14 Z (b) P ( 45 X 63 ) X 63 X 45 Z X X Z 45 50 0 . 2389 63 50 7 5 7 Z 0 . 71 0 . 9686 7 13 7 Z 1 . 86 P ( 0 . 71 Z 1 . 86 ) From Table V 0 . 71 0 . 2389 1 . 86 0 . 9686 0 . 9686 0 . 2389 0 . 7297 ( the whole blue area ) 0 . 71 0 1 . 86 Z Example 2 : GE manufactures a decorative Crystal Clear 60-watt light bulb that it advertises will last 1,500 hours. Suppose that the lifetimes of the light bulbs are approximately normal distributed, with a mean of 1,550 hours and a standard deviation of 57 hours, what proportion of the light bulbs will last more than 1650 hours? P ( X 1650 ) X 1650 , 1550 , 57 Z 0 . 0401 X 0 1650 1550 57 0 . 9599 100 57 Z 1 . 75 P ( Z 1 . 75 ) From Table V 1 . 75 0 . 9599 1 0 . 9599 0 . 0401 1 . 75 Z Ch 7.2 Part II Applications of the Normal Distribution Objective B : Finding the Z-score for a given probability Objective C : Probability under a Normal Distribution Objective D : Finding the Value of a Normal Random Variable Objective E : Applications Ch 7.3 Assessing Normality Ch 7.4 The Normal Approximation to the Binomial Probability Distribution Objective A : Continuity Correction Objective B : A Normal Approximation to the Binomial Ch 7.2 Applications of the Normal Distribution Objective D : Finding the Value of a Normal Random Variable Step 1 : Draw a normal curve and shade the desired area. Step 2 : Find the corresponding area to the left of the cutoff score if necessary. Step 3 : Use Table V to find the Z – score that corresponds to the area to the left of the cutoff score. Step 4 : Obtain x from Z by the formula Z X or x z . Ch 7.2 Applications of the Normal Distribution Objective D : Find the Value of a Normal Distribution Example 1 : The reading speed of 6th grade students is approximately normal (bell-shaped) with a mean speed of 125 words per minute and a standard deviation of 24 words per minute. (a) What is the reading speed of a 6th grader whose reading speed is at the 90% percentile? 125 , 24 , Z 1 . 28 90 % or 0 . 9 Solve for X Z 1 . 28 From Table V 0 . 08 Z 1 . 28 X X 125 24 1 . 28 ( 24 ) X 125 1 .2 0 . 8997 (Closer to 0.9) 0 . 9015 X 1 . 28 ( 24 ) 125 X 155 . 72 words per minute (b) Determine the reading rates of the middle 95% percentile. 125 , 24 , solve for X 95 % or 0 . 95 Z 1.96 Z ? Z ? 2 .5 % 2 .5 % or 0 . 025 or 0 . 025 Z 1.96 Z 1.96 due to sym m etry Z 1.96 X X 125 24 1.96 (24) X 125 From Table V 0 .0 6 X 1.96 (24) 125 X 7 7 .9 6 1 .9 0 . 025 words per minute Z 1 .9 6 Z 1.96 X X 125 24 1.96 (24) X 125 X 1.96 (24) 125 X 1 7 2 .0 4 words per minute Ch 7.2 Part II Applications of the Normal Distribution Objective B : Finding the Z-score for a given probability Objective C : Probability under a Normal Distribution Objective D : Finding the Value of a Normal Random Variable Ch 7.3 Assessing Normality Ch 7.4 The Normal Approximation to the Binomial Probability Distribution Objective A : Continuity Correction Objective B : A Normal Approximation to the Binomial Ch 7.3 Assessing Normality Ch 7.3 Normality Plot Recall: A set of raw data is given, how would we know the data has a normal distribution? Use histogram or stem leaf plot. Histogram is designed for a large set of data. For a very small set of data it is not feasible to use histogram to determine whether the data has a bell-shaped curve or not. We will use the normal probability plot to determine whether the data were obtained from a normal distribution or not. If the data were obtained from a normal distribution, the data distribution shape is guaranteed to be approximately bell-shaped for n is less than 30. Perfect normal curve. The curve is aligned with the dots. Almost a normal curve. The dots are within the boundaries. Not a normal curve. Data is outside the boundaries. Example 1: Determine whether the normal probability plot indicates that the sample data could have come from a population that is normally distributed. (a) Not a normal curve. The sample data do not come from a normally distributed population. (b) A normal curve. The sample data comes from a normally distribute population.