# Normal Probability Distributions

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Normal Probability Distributions
Chapter 1 Normal Probability Distributions Larson/Farber 4th ed

Chapter Outline 1.1 Introduction to Normal Distributions and the Standard Normal Distribution 1.2 Normal Distributions: Finding Probabilities 1.3 Normal Distributions: Finding Values 1.4 Sampling Distributions and the Central Limit Theorem 1.5 Normal Approximations to Binomial Distributions Larson/Farber 4th ed

Introduction to Normal Distributions
Section 1.1 Introduction to Normal Distributions Larson/Farber 4th ed

Section 5.1 Objectives Interpret graphs of normal probability distributions Find areas under the standard normal curve Larson/Farber 4th ed

Properties of a Normal Distribution
Continuous random variable Has an infinite number of possible values that can be represented by an interval on the number line. Continuous probability distribution The probability distribution of a continuous random variable. Hours spent studying in a day 6 3 9 15 12 18 24 21 The time spent studying can be any number between 0 and 24. Larson/Farber 4th ed

Properties of Normal Distributions
A continuous probability distribution for a random variable, x. The most important continuous probability distribution in statistics. The graph of a normal distribution is called the normal curve. x Larson/Farber 4th ed

Properties of Normal Distributions
The mean, median, and mode are equal. The normal curve is bell-shaped and symmetric about the mean. The total area under the curve is equal to one. The normal curve approaches, but never touches the x-axis as it extends farther and farther away from the mean. x Total area = 1 μ Larson/Farber 4th ed

Properties of Normal Distributions
Between μ – σ and μ + σ (in the center of the curve), the graph curves downward. The graph curves upward to the left of μ – σ and to the right of μ + σ. The points at which the curve changes from curving upward to curving downward are called the inflection points. μ  3σ μ + σ μ  2σ μ  σ μ μ + 2σ μ + 3σ x Inflection points Larson/Farber 4th ed

Means and Standard Deviations
A normal distribution can have any mean and any positive standard deviation. The mean gives the location of the line of symmetry. The standard deviation describes the spread of the data. μ = 3.5 σ = 1.5 σ = 0.7 μ = 1.5 Larson/Farber 4th ed

Same Standard Deviations, Different Means
the curve on the right has a larger mean than the curve on the left the amount of the shift is equal to the difference in the means

Same Means, Different Standard Deviations
the lower curve has a larger standard deviation the spread of the curve increases with the standard deviation

Example: Understanding Mean and Standard Deviation
Which curve has the greater mean? Solution: Curve A has the greater mean (The line of symmetry of curve A occurs at x = 15. The line of symmetry of curve B occurs at x = 12.) Larson/Farber 4th ed

Example: Understanding Mean and Standard Deviation
Which curve has the greater standard deviation? Solution: Curve B has the greater standard deviation (Curve B is more spread out than curve A.) Larson/Farber 4th ed

Example: Interpreting Graphs
The heights of fully grown white oak trees are normally distributed. The curve represents the distribution. What is the mean height of a fully grown white oak tree? Estimate the standard deviation. Solution: σ = 3.5 (The inflection points are one standard deviation away from the mean) μ = 90 (A normal curve is symmetric about the mean) Larson/Farber 4th ed

The Standard Normal Distribution
A normal distribution with a mean of 0 and a standard deviation of 1. 3 1 2 1 2 3 z Area = 1 Any x-value can be transformed into a z-score by using the formula Larson/Farber 4th ed

The Standard Normal Distribution
If each data value of a normally distributed random variable x is transformed into a z-score, the result will be the standard normal distribution. m=0 s=1 z Standard Normal Distribution Normal Distribution x m s Use the Standard Normal Table to find the cumulative area under the standard normal curve. Larson/Farber 4th ed

Properties of the Standard Normal Distribution
The cumulative area is close to 0 for z-scores close to z = 3.49. The cumulative area increases as the z-scores increase. z 3 1 2 1 2 3 z = 3.49 Area is close to 0 Larson/Farber 4th ed

Properties of the Standard Normal Distribution
The cumulative area for z = 0 is The cumulative area is close to 1 for z-scores close to z = 3.49. z 3 1 2 1 2 3 Area is z = 0 z = 3.49 Area is close to 1 Larson/Farber 4th ed

Larson/Farber 4th ed

Example: Using The Standard Normal Table
Find the cumulative area that corresponds to a z-score of 1.15. Solution: Find 1.1 in the left hand column. Move across the row to the column under 0.05 The area to the left of z = 1.15 is Larson/Farber 4th ed

Finding Areas Under the Standard Normal Curve
Sketch the standard normal curve and shade the appropriate area under the curve. Find the area by following the directions for each case shown. To find the area to the left of z, find the area that corresponds to z in the Standard Normal Table. The area to the left of z = 1.23 is Use the table to find the area for the z-score Larson/Farber 4th ed

Finding Areas Under the Standard Normal Curve
To find the area to the right of z, use the Standard Normal Table to find the area that corresponds to z. Then subtract the area from 1. The area to the left of z = 1.23 is Subtract to find the area to the right of z = 1.23:  = Use the table to find the area for the z-score. Larson/Farber 4th ed

Finding Areas Under the Standard Normal Curve
To find the area between two z-scores, find the area corresponding to each z-score in the Standard Normal Table. Then subtract the smaller area from the larger area. The area to the left of z = is Subtract to find the area of the region between the two z-scores:  = The area to the left of z = 0.75 is Use the table to find the area for the z-scores. Larson/Farber 4th ed

Example: Finding Area Under the Standard Normal Curve
Find the area under the standard normal curve to the left of z = Solution: 0.99 z 0.1611 From the Standard Normal Table, the area is equal to Larson/Farber 4th ed

Example: Finding Area Under the Standard Normal Curve
Find the area under the standard normal curve to the right of z = 1.06. Solution: 1.06 z 1  = 0.8554 From the Standard Normal Table, the area is equal to Larson/Farber 4th ed

Example: Finding Area Under the Standard Normal Curve
Find the area under the standard normal curve between z = 1.5 and z = 1.25. Solution: 0.8944 = 1.25 z 1.50 0.8944 0.0668 From the Standard Normal Table, the area is equal to Larson/Farber 4th ed

Section 1.1 Summary Interpreted graphs of normal probability distributions Found areas under the standard normal curve Larson/Farber 4th ed

Normal Distributions: Finding Probabilities
Section 1.2 Normal Distributions: Finding Probabilities Larson/Farber 4th ed

Section 1.2 Objectives Find probabilities for normally distributed variables Larson/Farber 4th ed

Probability and Normal Distributions
If a random variable x is normally distributed, you can find the probability that x will fall in a given interval by calculating the area under the normal curve for that interval. μ = 500 σ = 100 600 μ =500 x P(x < 600) = Area Larson/Farber 4th ed

Probability and Normal Distributions
Standard Normal Distribution 600 μ =500 P(x < 600) μ = σ = 100 x 1 μ = 0 μ = 0 σ = 1 z P(z < 1) Same Area P(x < 500) = P(z < 1) Larson/Farber 4th ed

Example: Finding Probabilities for Normal Distributions
A survey indicates that people use their computers an average of 2.4 years before upgrading to a new machine. The standard deviation is 0.5 year. A computer owner is selected at random. Find the probability that he or she will use it for fewer than 2 years before upgrading. Assume that the variable x is normally distributed. Larson/Farber 4th ed

Solution: Finding Probabilities for Normal Distributions
Standard Normal Distribution -0.80 μ = 0 σ = 1 z P(z < -0.80) 2 2.4 P(x < 2) μ = σ = 0.5 x 0.2119 P(x < 2) = P(z < -0.80) = Larson/Farber 4th ed

Example: Finding Probabilities for Normal Distributions
A survey indicates that for each trip to the supermarket, a shopper spends an average of 45 minutes with a standard deviation of 12 minutes in the store. The length of time spent in the store is normally distributed and is represented by the variable x. A shopper enters the store. Find the probability that the shopper will be in the store for between 24 and 54 minutes. Larson/Farber 4th ed

Solution: Finding Probabilities for Normal Distributions
-1.75 z Standard Normal Distribution μ = 0 σ = 1 P(-1.75 < z < 0.75) 0.75 24 45 P(24 < x < 54) x 0.7734 0.0401 54 P(24 < x < 54) = P(-1.75 < z < 0.75) = – = Larson/Farber 4th ed

Example: Finding Probabilities for Normal Distributions
Find the probability that the shopper will be in the store more than 39 minutes. (Recall μ = 45 minutes and σ = 12 minutes) Larson/Farber 4th ed

Solution: Finding Probabilities for Normal Distributions
Standard Normal Distribution μ = 0 σ = 1 P(z > -0.50) z -0.50 39 45 P(x > 39) x 0.3085 P(x > 39) = P(z > -0.50) = 1– = Larson/Farber 4th ed

Example: Finding Probabilities for Normal Distributions
If 200 shoppers enter the store, how many shoppers would you expect to be in the store more than 39 minutes? Solution: Recall P(x > 39) = 200(0.6915) =138.3 (or about 138) shoppers Larson/Farber 4th ed

Example: Using Technology to find Normal Probabilities
Assume that cholesterol levels of men in the United States are normally distributed, with a mean of 215 milligrams per deciliter and a standard deviation of 25 milligrams per deciliter. You randomly select a man from the United States. What is the probability that his cholesterol level is less than 175? Use a technology tool to find the probability. Larson/Farber 4th ed

Section 1.2 Summary Found probabilities for normally distributed variables Larson/Farber 4th ed

Normal Distributions: Finding Values
Section 1.3 Normal Distributions: Finding Values Larson/Farber 4th ed

Section 1.3 Objectives Find a z-score given the area under the normal curve Transform a z-score to an x-value Find a specific data value of a normal distribution given the probability Larson/Farber 4th ed

Finding values Given a Probability
In section 1.2 we were given a normally distributed random variable x and we were asked to find a probability. In this section, we will be given a probability and we will be asked to find the value of the random variable x. 5.2 x z probability 5.3 Larson/Farber 4th ed

Solution: Finding a z-Score Given an Area
Locate in the body of the Standard Normal Table. The z-score is 1.24. The values at the beginning of the corresponding row and at the top of the column give the z-score. Larson/Farber 4th ed

Example: Finding a z-Score Given a Percentile
Find the z-score that corresponds to P5. Solution: The z-score that corresponds to P5 is the same z-score that corresponds to an area of 0.05. z 0.05 The areas closest to 0.05 in the table are (z = -1.65) and (z = -1.64). Because 0.05 is halfway between the two areas in the table, use the z-score that is halfway between and The z-score is Larson/Farber 4th ed

Transforming a z-Score to an x-Score
To transform a standard z-score to a data value x in a given population, use the formula x = μ + zσ Larson/Farber 4th ed

Example: Finding an x-Value
The speeds of vehicles along a stretch of highway are normally distributed, with a mean of 67 miles per hour and a standard deviation of 4 miles per hour. Find the speeds x corresponding to z-sores of 1.96, -2.33, and 0. Solution: Use the formula x = μ + zσ z = 1.96: x = (4) = miles per hour z = -2.33: x = 67 + (-2.33)(4) = miles per hour z = 0: x = (4) = 67 miles per hour Notice mph is above the mean, mph is below the mean, and 67 mph is equal to the mean. Larson/Farber 4th ed

Example: Finding a Specific Data Value
Scores for a civil service exam are normally distributed, with a mean of 75 and a standard deviation of 6.5. To be eligible for civil service employment, you must score in the top 5%. What is the lowest score you can earn and still be eligible for employment? Solution: ? z 5% 75 x An exam score in the top 5% is any score above the 95th percentile. Find the z-score that corresponds to a cumulative area of 0.95. 1 – 0.05 = 0.95 Larson/Farber 4th ed

Solution: Finding a Specific Data Value
From the Standard Normal Table, the areas closest to 0.95 are (z = 1.64) and (z = 1.65). Because 0.95 is halfway between the two areas in the table, use the z-score that is halfway between 1.64 and That is, z = 5% z 1.645 75 x ? Larson/Farber 4th ed

Solution: Finding a Specific Data Value
Using the equation x = μ + zσ x = (6.5) ≈ 85.69 5% z 1.645 75 x 85.69 The lowest score you can earn and still be eligible for employment is 86. Larson/Farber 4th ed

Section 1.3 Summary Found a z-score given the area under the normal curve Transformed a z-score to an x-value Found a specific data value of a normal distribution given the probability Larson/Farber 4th ed