Calculating Normal Distribution Probabilities

Transcription

1 Calculating Normal Distribution Probabilities Lecture 13 February 12, 2018 Four Stages of Statistics Data Collection Displaying and Summarizing Data Probability Basics of Probability Probability Distributions Binomial Distribution Normal Distribution Sampling Distributions Inference Review: Empirical Rule Empirical Rule:approximation of percentage of observations that lie within a certain number of standard deviations from the mean in data that is normal (symmetric and unimodal) Observations from a normal dataset have: 68% within 1 standard deviation of the mean 95% within 2 standard deviations of the mean 99.7% within 3 standard deviations of the mean Example #1: Empirical Rule Scenario:Female heights normally distributed with mean 65in. and std. dev. 4in. selected female is between 61 and 69 inches? 61 and 69 are from the mean

2 Example #2: Empirical Rule Scenario:Female heights normally distributed with mean 65in. and std. dev. 4in. selected female is between 59 and 71 inches? _ Best approximation: Normal Distribution Normal Distribution:continuous probability distribution that describes data whose histogram is unimodal and symmetric Defined by two parameters Mean: Standard deviation: Typically denoted by either or Entire area under curve equals 1 Result: Areas can be interpreted as probabilities Allows us to calculate probability that an observation falls in a certain range from the mean Normal Distribution Examples Standard Normal Distribution Standard Normal Distribution:special case of the normal distribution with mean 0and standard deviation 1 Always denoted by Values referred to as Z-scores or Z-statistics Probabilities found using standard normal table Important Note: Z-scores tell you how many standard deviations an observation is from the mean.

3 Standardization A normal random variable with mean and standard deviation can be standardizedinto a standard normal random variable using: Observation Mean Standard Deviation Important Note: Probabilities for any normal distribution can be found by standardizing because standardization tells you how many standard deviations an observation is from its population mean. Example #3: Standardizing Scenario:Compare final exam scores of three students in different classes: Student A: Scored 80 in class with 71and 6 Student B: Scored 83 in class with 74and 9 Student C: Scored 87 in class with 85and 4 Task:Order the students from best to worst relative to the rest of their class. Student A: Student B: Student C: Calculating Normal Distribution Probabilities Normal distributions can have any mean and any nonnegative standard deviation. Result: Infinitely many normal distributions No easy-to-use function to calculate probabilities Instead convert all normal random variables into standard normal random variables Use a table to find probabilities

4 Standard Normal Table To find a probability: Divide the Z-score into two parts: Ones and tenths Hundredths Look up ones and tenths place in first column Look up hundredths place in first row. Intersect row and column to get area (probability) to the leftof the Z-score. Example #4: Lower Tail Scenario:Student A scored 80 on the final exam in a class with 71and 6. Question:What proportion of students scored worse on the exam? Example #4: Lower Tail Scenario:Student A scored 80 on the final exam in a class with 71and 6. Question:What proportion of students scored worse on the exam? Example #5: Upper Tail Scenario:Yearly car insurance for 25 year old males is normal with $2350and $425. Question:What is the probability a 25 year old male spends more than $2800 on insurance?

5 Example #5: Upper Tail Scenario:Yearly car insurance for 25 year old males is normal with $2350and $425. Question:What is the probability a 25 year old male spends more than $2800 on insurance? Example #7: Middle of Distribution Scenario:Weights of newborn babies normally distributed with 7.7lbs. and 1.2lbs. selected newborn weights between 6 and 10 lbs? Example #6: Sampling Two Observations Scenario:Suppose two 25 year old males are sampled independent of one another. Question:What is the probability both spend more than $2800 on insurance? Each has probability Use _ Example #7: Middle of Distribution Scenario:Weights of newborn babies normally distributed with 7.7lbs. and 1.2lbs. selected newborn weights between 6 and 10 lbs? _ _

6 Example #8: Both Tails Scenario:High temperature on June 7 in Pgh. is normally distributed with 77and 5.2. Question:What is the probability the high temperature is at least 6 degrees from average? Example #9: Shifting Mean Scenario:High temperature on Dec. 7 in Pgh. is normally distributed with 42and 5.2. Question:What is the probability the high temperature is at least 6 degrees from average? Still want values Mean standard deviation Example #8: Both Tails Scenario:High temperature on June 7 in Pgh. is normally distributed with 77and 5.2. Question:What is the probability the high temperature is at least 6 degrees from average? Summary Normal Distribution:continuous probability distribution that described data whose histogram is unimodal and symmetric Standard Normal Distribution:special case of normal distribution with 0and 1 Probabilities calculated using standard normal table Standardization:transforms normal random variable to standard normal using Four Types of Problems: area in lower tail, area in upper tail, area between two values, area in both tails