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The Normal Distribution Normal and Skewed Distributions Normal Distribution • Definition: A continuous, symmetric, bell-shaped distribution of a variable. • The shape and position of the curve depend on 2 variables. ( The mean and the standard deviation) • The larger the deviation the more dispersed, or spread out, the distribution is. • The area under the curve is more important than the frequencies themselves. When pictured the y-axis is usually omitted. • Z-value: The number of standard deviations that a particular X value is away from the mean. Other Properties of Normal Distribution • 1. The normal distribution curve is unimodal. • 2. Theoretically, no matter how far it extends, it never touches the x-axis. It just gets increasingly closer. • 3. The total area, under the curve, is equal to 1.00 or 100%. • 4. The area under the curve follows the Empirical Rule. One deviation about 68% of the curves area. Two deviations about 95% of curves area. Finally 3 deviations about 99.7% of curves area. Standard Normal Distribution • Differs slightly from normal distribution in the fact that its’ mean value is 0 and the standard deviation value is 1. Finding Areas Under the Curve • • • • • • • • There are 7 basic types of problems. 1. Between 0 and any z-value. 2. In any tail of the curve. 3. Between two z-values on the same side of the mean. 4. Between z-values on opposite sides of the mean. 5. Less than any z-value to the right of the mean. 6. Greater than any z-value to the left of the mean 7. In any two tails of the curve. How to Solve for the Area • Step 1: Draw the picture of the curve. • Step 2: Shade the desired area. • Step 3: Depending on the area shaded, you may either do nothing, add, or even subtract appropriate z-values from Table E. Examples • • • • • #1: Find the area between z = 0 and z = - 2.4 #2: Find the area to the left of z = 1.25 #3: Find the area between z = .85 and z = 1.7 #4: Find the area between z = - .54 and z = 2 #5: Find the area to the left of z = - 2.07 and to the right of z = 3.04 Probability Questions • Area under the curve corresponds to a probability. • Problems involving probabilities are solved in the same manner as previous area problems. • Examples: • P( 0 < n < 2.34 ) • P ( n > - 1.78 ) • P ( n < .85 ) Find the Z • May need to find the specific z-value for a given area under the normal distribution. • Work backwards: • Step 1: Find the area in Table E. • Step 2: Read the correct z-value in the left column and in the top row and add two values together. • Example 1: Find the z-value that corresponds to the area 0.4931. • Example 2: Find the z-value that corresponds to the area 0.2794. Other Applications • The standard normal distribution curve can be used to solve a wide variety of problems. Including probabilities and finding specific data values for given percentages. Only requirement is that the variable be normally or approximately distributed. • To solve problems we need to transform the original variable into a standard normal distribution variable and then use Table E to solve the problems. • To determine z-value we use the same formula from Chapter 3. z = ( value – mean ) ÷ sd. Examples • #1 The avg. hourly wage of Detroit car workers is $13.50. If standard deviation is $1.80 find the probabilities for a randomly selected worker. Assume variable is normally distributed. • P(The worker earns more than $15.60) • P(The worker earns less than $9.00) • #2 The avg. age of CEOs is 62 years. If standard deviation is 3 years find the probability for randomly selected CEOs will fall in the following ranges. Assume variable is normally distributed. • Between 64 and 70 years old. • Between 57 and 62 years old. Steps to finding specific data values • Work backwards. • Step 1: Find the area in Table E. • Step 2: Read the correct z-value in the left column and in the top row and add two values together. • Step 3: Use the formula X = z ● 𝜎 + 𝜇 • 𝜎 = 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 • 𝜇 = 𝑀𝑒𝑎𝑛 Examples • The scores on a test has a mean of 90 and standard deviation of 7. If a teacher wishes to select the top 30% of students who took the test find the cutoff score. Assume variable is normally distributed. • A pet shop owner decides to sell snakes that appeal to the middle 50% of customers. The mean for the snakes is $37.90 and the standard deviation is $5.60. Find the minimum and maximum prices of the snakes the owner should sell. Assume variable is normally distributed. Distribution of Sample Means • Along with knowing how data values vary about a mean for a population, statisticians are also interested in knowing about the distribution of the means of samples taken. • Sampling distribution of sample means: A distribution obtained by using the means computed from random samples of a specific size taken from a population. • If randomly selected, the sample means, for most part are somewhat different from the population mean. These differences are caused by sampling error. • Sampling Error: The difference between the sample measure and the corresponding population measure due to the fact sample is not a perfect representation of the population Properties of the Distribution of Sample Means • If all possible samples of a specific size are selected from a population, the distribution of the sample means has 2 important properties. • #1: The mean of the sample means will be the same as the population mean. • #2: The standard deviation of the sample means will be smaller than the standard deviation of the population. It will be equal to the population standard deviation divided by the square root of the sample size. • Standard deviation of the sample means is called the standard error of the mean. Central Limit Theorem • A third property of the sampling distribution of sample means pertains to the shape of the distribution. • CLT definition: As sample size increases, the shape of the distribution of the sample means taken from a population with mean 𝜇 and standard deviation 𝜎 approach a normal distribution. • Can be used to answer questions about sample means in the same way that normal distributions answer questions about individual values. • Major difference: z = ( Value - 𝜇 ) ÷ ( 𝜎 ÷ √ n ) Examples • The avg. age of teachers is 46 years, with a standard deviation of 6 years. Find the probability, that the mean of the sample of 7 randomly selected teachers is greater than 50 years old. Assume the variable is normally distributed. • The mean weight of 13 year old males is 108 pounds and the standard deviation is 11. If a sample of 35 males is selected find the probability that the mean of the sample will be less than 104 pounds. Assume the variable is normally distributed. Examples • A survey found that Americans generate an average of 17.2 pounds of glass garbage each year. The standard deviation is 2.5 pounds. Find the probability that the mean of a sample of 55 families will be between 17 and 18 pounds. • Average rainfall for Des Moines is 30.83 inches with a standard deviation of 5 inches. If a random sample of 10 years is selected find the probability that the mean will be between 32 and 33 inches. • Mean score on a dexterity test for 12-year olds is 30. Standard deviation is 5. If test administered to 22 students find probability mean of samples will be between 27-31. The Correction Factor • The formula for standard error of the mean, 𝜎 ÷ √n, is accurate for samples drawn with/without replacement from very large populations. • The correction factor is needed if relatively large samples are taken from a small population. • The sample mean will more accurately estimate the population mean and there will be less error in the estimation. • Generally used when the sample is greater than 5% of the population. The Correction Factor Formulas • Formula for correction factor: 𝑁 −𝑛 • 𝑁 −1 • Formula for computing the z-value: Value - 𝜇 𝑁 −𝑛 (𝜎 ÷ √n ) ● 𝑁 −1 Examples • #1: A study of 100 renters showed that the average rent of their apartments was $1,200 and the standard deviation was $250. If 28 renters are selected, find the probability the average rent of their apartments was greater than $1,300. • #2: The average price of a new suit at Macy’s is $275 and a standard deviation of $10. If 17 suits are sold from a lot of 70, find the probability that the average price will be less than $270? The Normal Approximation to the Binomial Distribution • The normal distribution is used to solve problems that involve the binomial distribution due to the fact that when n is rather large the calculations get to difficult to do by hand. • When p is close to 0.5 and as n increases the binomial distribution becomes similar to the normal distribution. • When p is close to 0 or 1 and n is smaller the normal approximation becomes inaccurate. • Due to the fact above the normal approximation should only be used when n ● p and n ● q are both greater than or equal to 5. Formulas for Mean and Standard Deviation of Binomial Distribution • Mean: 𝜇 = 𝑛 ● 𝑝 • Standard Deviation: 𝑛 ● 𝑝 ● 𝑞 Steps for Normal Approximation to the Binomial Distribution • Step 1: Check to see whether normal approximation can be used. • Step 2: Find the mean and standard deviation. • Step 3: Write the problem in probability notation. Remember to use the continuity correction factor, and show corresponding area under the curve. • Step 5: Find the corresponding z-values. • Step 6: Find solution. • Correction factor: Employed when a continuous distribution is used to approximate a discrete distribution. Works like lower and upper boundaries. Examples • Of all 5 to 6 year-old children, in NJ, 87% are enrolled in school. If a sample of 300 such children are randomly selected, find the probability that at least 247 will be enrolled in school. • A survey found that 17% of teenage drivers text while driving. If 400 drivers are selected at random, find the probability that exactly 65 say they text while driving. Examples • Use the normal approximation to the binomial to find the probabilities for specific value/s of X. • #1: n = 25, p = .65, X = 12 • #2: n = 80, p = .4, X = 34 • #3: n = 30, p = .5 X ≤ 8