# For a continued random variable the probability of observing one particular value is zero.

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1 Chapter 7The Normal Probability Distribution Chapter 7.1 Uniform and Normal Distribution Objective A: Uniform Distribution A1. Introduction Recall: Discrete random variable probability distribution Special case: Binomial distribution Finding the probability of obtaining a success in n independent trials of a binomial experiment is calculated by plugging the value of a into the binomial formula as shown below: a P( x a) C p (1 p) Continuous Random variable For a continued random variable the probability of observing one particular value is zero. i.e. P( x a) 0 n a n a Continuous Probability Distribution We can only compute probability over an interval of values. Since P( x a) 0 and P( x b) 0 fora continuous variable, P( a x b) P( a x b) To find probabilities for continuous random variables, we use probability density functions. 1

2 Two common types of continuous random variable probability distribution: 1) Uniform distribution. 2) Normal distribution. A2. Uniform distribution 1 b a a b Note: The area under a probability density function is 1. Area of rectangle Height Width 1 Height ( b a) 1 Height for a uniform distribution ( b a) Example 1: A continuous random variable x is uniformly distributed with10 x 50. (a) Draw a graph of the uniform density function. Height = 1 (b a) Area of rectangle = Height x Width 1 = Height x(b - a) = 1 = 1 (50 10) 40 (b) What is P(20 x 30)? Area of rectangle = Height x Width = 1 * (30-20) 40 = 1 * = 1 =

3 (c) What is Px ( 15)? Area of rectangle = Height x Width P (x< 15) = P (x 15) = 1 = P (10 x 15) = 1 40 * 5 40 * (15 10) 1 40 = 1 40 = Objective B: Normal distribution Bell-shaped Curve 3

4 Example 1: Graph of a normal curve is given. Use the graph to identify the value of and X 730 Example 2: The lives of refrigerator are normally distributed with mean 14 years and standard deviation 2.5 years. (a) Draw a normal curve and the parameters labeled X (b) Shade the region that represents the proportion of refrigerator that lasts for more than 17 years. 4

5 (c) Suppose the area under the normal curve to the right x 17 is Provide twointerpretations of this result. Notation: P (x 17) = The area under the normal curve for any interval of values of the random variable x represents either: The proportions of the population with the characteristic described by the interval of values % of all refrigerators are kept for at least 17 years. the probability that a randomly selected individual from the population will have the characteristic described by the interval of values. The probability that a randomly selected refrigerator will be kept for at least 17 years is 11.51%. Chapter 7.2 Applications of the Normal Distribution Objective A: Area under the Standard Normal Distribution The standard normal distribution Bell shaped curve =0 and = Z The random variable for the standard normal distribution is Z. Negative Z Positive Z Use the Z table (Table V) to find the area under the standard normal distribution. Each value in the body of the table is a cumulative area from the left up to a specific Z -score. Probability is the area under the curve over an interval. The total area under the normal curve is 1. Z 0 Z 5

6 Under the standard normal distribution, (a) What is the area to the right 0? 0.5 (b) What is the area to the left 0?0.5 Example 1: Draw the standard normal curve with the appropriate shaded area and use StatCrunch to determine the shaded area. Open StatCrunch select Stat Calculators Normal select Standard select Input desired value for X compute record results (a) Find the shaded area that lies to the left of Z P( Z 1.38) (b) Find the shaded area that lies to the right of Similar steps as in part (a) except you want to select and input value, compute and record results Z P (Z> 0.56) =

7 (c) Find the shaded area that lies in between 1.85 and Open StatCrunch select Stat Calculators Normal select Between Input desired values for X range compute record results P (1.85 Z 2.47) = Objective B: Finding the Z-score for a given probability Z Area 0.5 Area 0. 5 Area 0. 5 Example 1: Draw the standard normal curve and the z -score such that the area to the left of the z -score is Use StatCrunch to find the z -score. Open StatCrunch Select Stat Calculator Normal Standard Input the value for P (x ) = Compute and record the results P (Z <-1.73) = Z? 0 7

8 Example 2: Draw the standard normal curve and the Z-score such that the area to the right of the Z-score is 0.18.Use StatCrunch to find the Z-score. Similar to example 1, input P (x ) = 0.18, compute P (Z > ) = Z? Example 3: Draw the standard normal curve and two Z-scores such that the middle area of the standard normal curve is Use StatCrunch to find the two Z-scores. If the middle area is 0.70, the total tailed areas is 0.30 (1-0.70) and the left tailed area is 0.15 (0.30/2). We will use StatCrunch to find the z score for the lower bound then use the symmetric concept to find the z score for the upper bound. Open StatCrunch Select Stat Calculator Normal Standard Input the value for P (x ) = 0.15 Compute and record the results % 0.15 P(-1.04< Z <1.04 ) = 0.70 Z 1.04 Z 1.04 (Due to symmetry) Objective C: Probability under a Normal Distribution Step 1: Draw a normal curve and shade the desired area. X Step 2: Convert the values X to Z -scores using Z. Step 3: Use StatCrunch to find the desired area. 8

9 Example 1: Assume that the random variable X is normally distributed with mean 50 and a standard deviation 7. (Note: this is not the standard normal curve because 0 and 1.) (a) PX ( 58) Z X X P( Z 1.14) Z (b) P(45 X 63) X 45 Z X Z 0.71 X 63 Z X Z 1.86 P (-0.71 Z 1.86) =

10 Example 2: Redo Example 1 Use StatCrunch and random variable X directly without converting to Z first. (a) PX ( 58) Open StatCrunch Select Stat Calculator Normal Standard Input the values for Mean, Std. Dev. and P (x 58) = Compute P (x 58) = (b) P(45 X 63) Open StatCrunch Select Stat Calculator Normal Between Input the values for Mean, Std. Dev. and P (45 x 63) = Compute P (45 x 63) =

11 Example 3: 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, use StatCrunch to find what proportion of the light bulbs will last more than 1650 hours? Open StatCrunch Select Stat Calculator Normal Standard Input the values for Mean, Std. Dev. and P (x 1650) = Compute P( X 1650) Objective D: Finding the Value of a Normal Random Variable Step 1: Draw a normal curve and shade the desired area. Step 2: Use StatCrunch to find the appropriate cutoff Z -score. X Step 3: Obtain X from Z by the formula Z or X Z. 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 graderwhose reading speed is at the 90 percentile? Open StatCrunch Select Stat Calculator Normal Standard Input the value for Mean = 0, Std. Dev. = 1, and P (x ) = 0.90 Compute and record the results 11

12 X Z X (24) X (b) Determine the reading rates of the middle 95 percentile. 95% in the middle means each tail is 5% divided by 2 = 2.5% = Open StatCrunch Select Stat Calculator Normal Standard Input the value for Mean = 0, Std. Dev. = 1, and P (x ) = Compute and record the results X Z X 125 ( ) (24) X Open StatCrunch Select Stat Calculator Normal Standard Input the value for Mean = 0, Std. Dev. = 1, and P (x ) = Compute and record the results X Z X 125 ( ) (24) X The middle 95% reading speed are between words per minute to words per minute. 12

13 Example 2: Redo Example 1 Use StatCrunch to find X directly without converting from Z to X. Open StatCrunch Select Stat Calculator Normal Standard Input the values for Mean = 125, Std. Dev. = 24, and P (x ) = 0.90 Compute (a) What is the reading speed of a 6th grader whose reading speed is at the 90 percentile? X (b) Determine the reading rates of the middle 95% percentile. If the middle area is 0.95, the total tailed areas is 0.05 and the left tailed area is (0.05/2). We will use StatCrunch to find the X score for the lower bound then change the inequality sign to find the X score for the upper bound. Open StatCrunch Select Stat Calculator Normal Standard Input the value for Mean = 125, Std. Dev. = 24, and P (x ) = > Compute X = words per minute 13

14 Open StatCrunch Select Stat Calculator Normal Standard Input the value for Mean = 125, Std. Dev. = 24, and P (x ) = > Compute X = words per minute The middle 95% reading speed are between words per minute to words per minute. Chapter 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 hasa 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

15 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. 15

16 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. There is no guarantee that this sample data set is normally distributed. (b) A normal curve. The sample data come from a normally distribute population. There is a guarantee that this sample data set is approximately normally distributed. 16

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