Section 7.1 Properties of the Normal Distribution In Chapter 6, talked about probability distributions. Coin flip problem: Difference of two spinners: The random variable x can only take on certain discrete values. In Chapter 7, will talk about The random variable x can take on ANY value. probability distributions. Interpreting Area as a Probability Example: Find the probability that a randomly thrown dart will land in the shaded region of the dartboard. Assume that the dart WILL land on the board, and that it is equally likely for the dart to land anywhere on the board. P(shaded) = KEY IDEA: Use to represent. Page 1 of 22
Probability Density Functions By definition, for any probability histogram or density curve, the total area under the curve is, because this represents of the probability. Discrete: Continuous: P(x = 1) = P(x > 0) = For the discrete probability histogram: Each column has unit width = and height = Area of each column = Sum of all = sum of = A density curve (graph of a probability density function) is basically the equivalent of this for a variable. It is a that represents the distribution of a continuous variable. Properties of Density Curves: 1. A density curve is always on or above the. 2. The total area under a density curve equals. Page 2 of 22
Continuous Uniform Distribution A continuous uniform distribution means that the random variable is to take on any value in a given interval. Example: A company manufactures 40 lb. bags of fertilier, and when the bags come off the manufacturing line their weight has a uniform distribution between 39.25 lbs and 40.85 lbs. Draw the distribution. 1. What is P(x), the height of the rectangle? P(x) Weight, lbs 2. What is the probability of randomly selecting one bag and having its weight be between 39.25 40 lbs? Find the area: P(x) Weight, lbs 3. What is the probability of randomly selecting one bag and having its weight be exactly 40 lbs? Find the area: Page 3 of 22
Key Difference Between Discrete and Continuous Distributions From the histogram of the 4 coin flips: P(1 H) = From the uniform distribution density curve: P(40 lbs) = Remember, for a continuous distribution, It is completely different! Discrete distributions: calculating the probability associated with particular value. Continuous distributions: calculating the probability associated with a of values. Cannot calculate the probability associated with a value, it is not defined. Page 4 of 22
Introduction to the Continuous Normal Distribution: Transition from discrete distribution to a smooth continuous distribution: Example: Pulse rates of students Pulse rates (bpm) Because the dot plots exhibit a bell-shaped distribution, we can model the data with a specific probability density function, called the probability density function. Page 5 of 22
Normal Probability Density Function A normal distribution is a mathematical model defined by the following equation: 1 y = e 2 (x μ σ )2 σ 2π Two constants in the equation: The entire of the distribution is defined by two parameters: Defines: Defines: Page 6 of 22
Properties of the Normal Density Curve 1. The normal curve is symmetric about. 2. The highest point occurs at. Also,. 3. It has inflection points at. 4. The area under the normal curve is. 5. The area under the curve to each side of the mean =. 6. The tails on either end of the curve are asymptotic 7. The curve follows the Empirical Rule: Page 7 of 22
: Special case of normal probability distribution Mean of data, = Standard deviation of data, = Total area under curve = Symmetric, bell-shaped curve 1 Curve described by: y = e 2 (x μ σ )2 σ 2π Which simplifies to: -scores: Number of standard deviations that a given value, x, is above or below. = Page 8 of 22
Section 7.2 Applications of the Normal Distribution Using -scores and Table V to find probabilities: Given a -score: Table V gives the probability associated with getting a -score that. Can t find the probability of getting that score. How to use Table V: Find -score using column and top row. Identify corresponding value in body of table. Value in body = area to left of that -score = Probability(less than ). MUST round -scores off correctly to two decimal places! Here is a small excerpt of the table: Example: = 2.74 Area to left = Page 9 of 22
Find the probability of selecting a -score equal to 1.42-4 -3-2 -1 0 1 2 3 4 Find the probability of selecting a -score less than = 1.42-4 -3-2 -1 0 1 2 3 4 Find the probability of selecting a -score less than = 0.60-4 -3-2 Standard -1 Normal 0 Distribution 1 2 3 4 Find the probability of selecting a -score greater than = 0.60-4 -3-2 -1 0 1 2 3 4 Find the probability of selecting a -score between = 1.42 and = 0.60-4 -3-2 -1 0 1 2 3 4 Page 10 of 22
Summary of using -scores to find Probabilities: Probability = Table V only for STANDARD normal distribution 3 types of problems Table gives area to the of a given -score ( below the value) Area above a given -score = Area between two -scores = Notes: Lowest -score reported in Table V is = -3.49 Report any area to the LEFT of that as : Report any area to the RIGHT of that as : Highest -score reported in Table V is = 3.49 Report any area to the RIGHT of that as : Report any area to the LEFT of that as : Page 11 of 22
Reverse the process: For a certain probability find the corresponding Example: Find the -score that has an area of 0.90 to its left under the standard normal curve (same Standard as 90 Normal th percentile, Distribution or P90). Use Table V, BUT: Instead of looking up Look in the body of the table to find an. = Notes: Important: when looking up the area, choose the closest value and use its -score. Exception: if the desired value is EXACTLY in the middle of the two table entries, then take the of the two -scores in the table. Page 12 of 22
α Notation There are a couple of important -score values that will be used in analyses. α means: the score that has an area of α to its. Example: Find 0.05 Area to right = Area to left = Page 13 of 22
1. Determine the area under the standard normal curve that lies to the left of: (a) 0.87 (b) 2.43 (c) 5.07-4 -3-2 -1 0 1 2 3 4 2. Determine the area under the standard normal curve that lies to the right of: (a) 0.56 (b) 2.02-4 -3-2 -1 0 1 2 3 4 3. Determine the area under the standard normal curve that lies between: (a) 0.88 and 2.24 (b) 1.96 and 1.96-4 -3-2 -1 0 1 2 3 4-4 -3-2 -1 0 1 2 3 4 Page 14 of 22
4. Obtain the -score that has area 0.75 to its left. 5. Determine 0.015 Page 15 of 22
Working with Non-Standard Normally Distributed Variables Take any data that is normally distributed: Convert it to scores, and Continue to use Table V for probabilities. Do this by converting data values to standardied : Example: Women s heights are normally distributed, with: = 63.8 in = 2.6 in Question: What is the probability Standard Normal of randomly Distribution selecting a woman and having her be less than 5 feet (60 inches) tall? Convert the x-axis scale from inches to -scores by calculating a standardied -score: = Area to left of = Probability of randomly selecting a woman and having her be less than 5 ft tall is: Page 16 of 22
Example: The US Army requires women s heights to be between 58 in and 80 in. Find the percentage of women Standard meeting Normal Distribution that height requirement. Women s heights are normally distributed, with = 63.8 in and = 2.6 in. 1. Draw a picture. 2. Identify what we re trying to find 3. Convert to -scores 58 = 80 = 4. Find the area below each -score 5. Find area we want: 6. Answer the problem: Page 17 of 22
Reverse the process: For a certain probability find the corresponding Steps: 1. Given a probability, find from Table V. 2. Use -score to calculate. From formula for : = x rewrite as: Example: What height separates the upper 20% of women from the lower 80%? In other words, find the value of the 80 th percentile. Women s heights are normally distributed, with = 63.8 in and = 2.6 in. 1. Draw a picture 2. Find -score that corresponds to the given probability/area: 3. Use formula to find x: x = Page 18 of 22
Work in groups of 3 students. First, answer the following question: Normal Distribution SLEEP!! In the past 24 hours, to the nearest quarter-hour (0.25 h), how many hours did you sleep?. hours (Note: all three of you need to have a different answer). Data collected on Stat students in previous quarters suggests that X, the amount of sleep in hours in a 24-hour period, is normally distributed with a mean, of 6.75 hours and a standard deviation,, of 1.25 hours. Assume this describes the sleep habits of this quarter's Stat students as well. Consider the person in your group who slept the most. He/she slept. hours. Find the percentage of Stat students who slept longer than this person. Consider the person in your group who slept the least. He/she slept. hours. Find the percentage of Stat students who slept less than this person. Consider the person in your group who did not sleep the least or the most. He/she slept. hours. Find the percentage of Stat students who slept between the amount this person slept and the amount of the person who slept the most. Page 19 of 22
1. The Graduate Record Examination (GRE) is a standardied test that students usually take before entering graduate school. According to the document Interpreting Your GRE Scores, a publication of the Educational Testing Service, the scores on the verbal portion of the GRE are approximately normally distributed with a mean of 462 points and a standard deviation of 119 points. (a) Determine the percentage of students who score below 400 points. (b) Determine the percentage of students who score between 400 and 500 points. (c) What score would a student have to get to be in the top 5% of all test takers? In other words, find the value of the 95 th percentile. (d) Would it be unusual for someone to score 250 points on the GRE? Explain. Page 20 of 22
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