Normal Distribution Normal Distribution Normal distribution is a
Normal Distribution
Normal Distribution • Normal distribution is a distribution with a bell-shaped appearance. In a normal distribution, the mean = median = mode
Distribution of Correct Answers of 19 Students who Participated in a Math Contest Number of Correct Answers Frequency 1 1 2 2 3 4 4 5 5 4 6 2 7 1 N = 19
Distribution of Correct Answers of 19 Students who Participated in a Math Contest Number of Correct Answers Frequency 1 0 2 0 3 1 4 2 5 4 6 9 7 3 N = 19
Distribution of Correct Answers of 19 Students who Participated in a Math Contest Number of Correct Answers Frequency 1 3 2 9 3 4 4 2 5 1 6 0 7 0 N = 19
Skewness • Skewness refers to the degree of symmetry or asymmetry of a distribution.
Skewness • A distribution is skewed to the left if the mean is less than its median. The bulk of the distribution is on the right. This is otherwise known as negatively skewed • A distribution is skewed to the right if the mean is greater than its median. The bulk of the distribution is on the left. This is otherwise known as positively skewed
Exercise Determine whether the distribution is normal, skewed to the left or skewed to the right. 1. 19, 17, 14, 11, 10, 8, 7, 7 2. 5, 3, 4, 2, 1
Properties of Normal Curve • Mean = Median = Mode • It is symmetrical about the mean. • The tails or ends are asymptotic relative to the horizontal axis. • The total area under the normal curve is equal to 1 or 100%. • The normal curve area may be subdivided into standard deviations, at least 3 units to the left and 3 units to the right of the vertical line.
Standard Normal Distribution • Standard Normal distribution is a distribution with a mean of zero and a standard deviation of one. • (Graph of standard normal distribution)
Z-score • Z-score measures how many standard deviations a particular value is above or below the mean. • (formula) • (Illustrate a distribution with a population mean of 80 and a standard deviation of 5)
In which test did Clifford perform better? Population Mean Standard Deviation Score Algebra 80 English 75 5 5 85 82
Solve the following problems: 1. Who performed better? a. Lucy whose z-score is 2 or Lily whose z -score is 1. 75 b. Angelo whose z-score is -1. 5 or Antonio whose z-score is -2.
Solve the following problems: 2. In an examination , the mean grade is 81 and the standard deviation is 6. Find the z-score of the grades of the following students: a. Peter, 75 b. Paul, 95 c. Mark, 87 d. John, 93
Solve the following problems: 3. Charles sells stamps. He earns an average of P 55 a day with a standard deviation of P 11. 25. How much did Charles earn on a particular day if his zscore is -0. 8?
Areas under the Normal Curve Ø Find the area to the right of Z = 0 Ø Find P(Z ≥ 0) can be read as “ Find the probability that z will take the values greater than or equal to zero”
Areas under the Normal Curve Ø Find the area to the left of Z = 0 Ø Find P(Z ≤ 0) can be read as “ Find the probability that z will take the values less than or equal to zero”
Areas under the Normal Curve Ø Find the area from z = -2. 41 to z =1. 98 Ø Find P(-2. 41 ≤ Z ≤ 1. 98) can be read as “ Find the probability that z will take the values greater than or equal to -2. 41 but less than or equal to 1. 98” Ø (give the table of areas of normal curve)
Examples: 1. Find the area from z = 0 to z = -2. 3 2. Find P(-2 ≤ Z ≤ 2. 5) 3. Find P(z ≥ -2. 43) 4. Find the area to the left of z = - 1. 53 5. Find P(0. 5 < z < 2. 5)
Problem-solving: The IQ of 300 students in a certain school is approximately normally distributed with a mean of 100 and standard deviation of 15. a. What is the probability that a randomly selected student will have an IQ of 115 and above? b. How many students have an IQ between 85 and 120?
Problem-solving: In an examination in Statistics, the mean grade is 72 and the standard deviation is 6. Find the probability that a particular student will have a score: a. of higher than or equal to 75 b. from 65 to 80 c. lower than or equal to 60
Problem-solving: • Suppose a borderline hypertensive is defined as a person whose DBP is between 90 and 95 mm Hg inclusive, and the subjects are 35 -44 -year-old males whose BP is normally distributed with mean 80 and variance 144. What is the probability that a randomly selected person from this population will be a borderline hypertensive?
Problem-solving: • Suppose that total carbohydrate intake in 12 -14 year old males is normally distributed with mean 124 g/1000 cal and SD 20 g/1000 cal. • a) What percent of boys in this age range have carbohydrate intake above 140 g/1000 cal? • b) What percent of boys in this age range have carbohydrate intake below 90 g/1000 cal?
Problem-solving: • Assume that among diabetics the fasting blood level of glucose is approximately normally distribute with a mean of 105 mg per 100 ml and SD of 9 mg per 100 ml. • a) What proportions of diabetics have levels between 90 and 125 mg per 100 ml? • b) What proportions of diabetics have levels below 87. 4 mg per 100 ml? • c) What level cuts of the lower 10% of diabetics? • d) What are the two levels which encompass 95% of diabetics?
Problem-solving: • If adult male cholesterol is normally distributed with a mean of 200 and standard deviation 25, what is the probability of selecting male whose cholesterol is: • Less than 165 • Greater than 165 • Between 165 and 220 • Greater than 220
THE 68– 95– 99. 7 RULE • In the Normal distribution with mean μ and standard deviation σ: • Approximately 68% of the observations fall within σ of the mean μ. • Approximately 95% of the observations fall within 2σ of μ. • Approximately 99. 7% of the observations fall within 3σ of μ.
THE 68– 95– 99. 7 RULE • The distribution of heights of young • women aged 18 to 24 is approximately Normal with mean μ = 64. 5 inches • and standard deviation σ = 2. 5 inches.
THE 68– 95– 99. 7 RULE • The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. Use the 68– 95– 99. 7 rule to answer the following questions. • (a) Between what values do the lengths of the • middle 95% of all pregnancies fall? • (b) How short are the shortest 2. 5% of all • pregnancies? How long do the longest 2. 5% • last?
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